Journal of Food Engineering 92 (2009) 261–268

Contents lists available at ScienceDirect

Journal of Food Engineering

journal homepage: www.elsevier .com/locate / j foodeng

Oil migration in chocolate: A case of non-Fickian diffusion

J. Román Galdámez a, Kristin Szlachetka c, J. Larry Duda d,�, Gregory R. Ziegler b,*

a Sabic Innovative Plastics Mt. Vernon, Inc., Mt. Vernon, IN 47602, United Statesb Department of Food Science, Pennsylvania State University, 341 Food Science Building, University Park, PA 16802, United Statesc The Solae Company, St. Louis, MO 63188, United Statesd Department of Chemical Engineering, United States

a r t i c l e i n f o

Article history:Received 28 April 2006Received in revised form 9 October 2008Accepted 2 November 2008Available online 28 November 2008

Keywords:ChocolateOil migrationPhase behaviorTortuosityModel

0260-8774/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.jfoodeng.2008.11.003

* Corresponding author. Tel.: +1 814 863 2960; faxE-mail address: grz1@psu.edu (G.R. Ziegler).

� Deceased.

a b s t r a c t

Oil migration through filled chocolate products during storage periods is inevitable and is responsible forthe loss of the original quality of the product. A model that predicts the extent of oil migration can helpfood engineers optimize the post-production period, thereby delivering an acceptable quality product toconsumers.

In this work, a predictive model is proposed based on an explicit formulation of the diffusion problemin terms of the molecular diffusivity and the internal microstructure of chocolate. This work overcomesthe limitations of the previous models by including a moving boundary, which allows for swelling of thechocolate slab and replacing the effective diffusion coefficient with a more physically meaningful one,which includes a tortuosity term that varies with oil concentration. The model has been validated againstexperimental mass uptake curves.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The chocolate confectionery market is dominated by compositeproducts such as pralines, filled chocolates or snack bars, where achocolate or confectionery coating layer is in direct contact witha fat-based cream, biscuit, nut or nut paste. The marketable life-time of these products is often limited by fat or oil migration, e.g.the fillings may contain large amounts of highly mobile oils, suchas peanut or hazelnut oil that migrates into the chocolate coatingcausing quality loss (see Fig. 1). Quality defects arising from oilmigration include softening of the coating, hardening of the filling,deterioration in sensory quality and a greater tendency to fatbloom. For this reason, oil migration has been extensively studied(Aguilera et al., 2004; Khan and Rousseau, 2006; Ziegleder et al.,1996a,b; Choi et al., 2007).

1.1. Mechanism of oil migration

Dark chocolate is a mixture of cocoa butter, chocolate liquor,and sugar. Milk chocolate contains milk solids as well. Commercialdark chocolate typically contains between 25% and 40% cocoa but-ter with the rest being variable amounts of sugar and solid cocoapowder. Chocolate is a dispersion of solid cocoa particles, sugarcrystals, and milk powder (in the case of milk chocolate) in a con-

ll rights reserved.

: +1 814 863 6132.

tinuous phase of solid and liquid fat, whose proportions depend ontemperature (Ziegler et al., 2004).

Cocoa butter comprises a mixture of saturated and unsaturatedfats; oleic acid (O), stearic acid (S), and palmitic acid (P) account formore than 95% of the fatty acids in cocoa butter. Cocoa butter con-tains high levels of the triacylglycerols (TAGs) POS, POP, and SOSthat are crystalline at normal storage temperatures; some of theunsaturated TAGs in cocoa butter have low melting points makingit partly liquid at room temperature.

The triacylglycerols in nut-based fillings, which often containhazelnut or almond oils, such as triolein (OOO), LOO, LLO, POO,and SOO, where L stands for lauric acid, are predominately liquidat room temperature. Originally, the driving force for diffusionwas assumed to be a difference in liquid fat content, but recentlydiffusion has been attributed to a gradient in triacylglycerol con-centration within some domains of the product (Ghosh et al.,2002). In a series of experiments where a bar of chocolate wasbrought into contact with a nougat filling, Ziegleder et al.(1996a,b) showed that after sufficient time the main triacylglyce-rols of hazelnut oil had almost evenly distributed between choco-late and filling. On the other hand, little of the triacylglycerols ofthe cocoa butter had migrated into the filling.

Molecular or Fickian diffusion is widely used by food engineersas a general model for mass transfer. In the literature, there areseveral attempts to model fat migration using simplified solutionsto Fick’s Second Law of diffusion (Ziegleder et al., 1996a; Choi et al.,2007). As the first approximation, Ziegleder et al. (1996a)employed the short-time solution with a constant diffusion

Nomenclature

CFPðtÞ hazelnut concentration in filter paper [kg/m3]Deff effective diffusion coefficient [m2/s]D0 diffusion coefficient in the liquid phase of the cocoa but-

ter [m2/s]jvHZ diffusivity flux of hazelnut oil with respect to the vol-

ume average velocity [kg/m2 s]K partition distribution constantL(t) slab thickness expressed as a function of time [m]�L dimensionless slab thicknesst Time [s]�t dimensionless timerSF!LF rate of crystal dissolution [kg/m3 s]V̂ i specific volume of species i [m3/g]wFAT

HZ weight fraction of hazelnut oil in the cocoa butter phasex position [m]

Greek lettersa aspect ratio

k time lag [hours]n immobilization variableqT

HZ mass concentration of hazelnut oil in the total chocolate[kg/m3]

qTSF mass concentration of solid crystals in the total choco-

late [kg/m3]qLF

HZ mass concentration of hazelnut oil in the liquid cocoabutter [kg/m3]

qs0 mass concentration established instantaneously at thesurface at time zero [kg/m3]

�q dimensionless mass concentrations global tortuosity factor/Fat

SF volume fraction of solid fat in the cocoa butterphase

/TNF volume fraction of non-fat solid in chocolate

/TLF volume fraction of liquid fat in chocolate

mv volume average velocity

Fig. 1. Schematic representation of hazelnut oil migration into chocolate.

262 J.R. Galdámez et al. / Journal of Food Engineering 92 (2009) 261–268

coefficient. Effective diffusion coefficients were extracted from theexperimental data of the mass of the migrated oil plotted againstthe square root of time. Ziegleder et al. (1996b) observed a linearcorrelation between the logarithm of the effective diffusion coeffi-cients so obtained and the liquid fat content in the chocolate.

Models based on the so-called Fickian diffusion with a constantdiffusivity fail to accurately describe the migration of oil into choc-olate (Choi et al., 2007), and because of this some have suggestedthat a mechanism other than diffusion, e.g. capillary pressure, isresponsible for oil migration. However, it is equally likely that sim-plified models fail to accurately reproduce experimental data be-cause simplifying assumptions are not fulfilled. In addition toconstant diffusivity, these simplified solutions neglect swellingand interaction between the oil and the cocoa butter. Recent obser-vations of oil migration through chocolate using magnetic reso-nance imaging have shown that the dominant mechanism isdiffusion, and that if capillary pressure is involved it is a minor con-tributor (Deka et al., 2006).

Shetty (2004) proposed a new model that modified the effectivediffusivity depending on the fat phase behavior. The idea behindthis model is simple: the migrating oil disturbs the equilibrium be-tween the liquid and solid phase present in the fat phase of choc-

olate. Oil migration decreases the solid fat content and therebyincreases diffusion. Shetty’s (2004) model used the same expres-sion for the relationship between the effective diffusion coefficientand the liquid fat content obtained by Ziegleder et al. (1996b). Thedifference in the Shetty (2004) model was that the liquid fat con-tent was calculated independently through an experimental phasediagram.

On the basis that the exact mechanism of oil migration in choc-olate remains poorly understood, the present paper proposes anew model, based on molecular diffusion, to try to help elucidatethe mechanism underlying fat migration. The model will accountfor structural parameters such as fat crystal microstructure andtortuosity and allows for chocolate swelling in order to overcomesome of the drawbacks present in the previous models.

1.2. Phase behavior

Most chocolate confectionery products contain between 18%and 40% cocoa butter. Cocoa butter, as with most natural fats, con-tains at least two phases, i.e. liquid and solid, whose proportionsvary with temperature. Cocoa butter can crystallize in six differentpolymorphic forms, referred to as I–VI, with b forms V and VI being

J.R. Galdámez et al. / Journal of Food Engineering 92 (2009) 261–268 263

the most stable. In good quality chocolate products, the formationof form V crystals is enhanced during manufacturing by a processcalled tempering. During tempering numerous small fat crystalsare promoted.

The solid fat content (SFC) depends not only on temperature butalso on the presence of other fats. Fats like peanut or hazelnut oil,common components in filling products, have high levels of lowmelting point TAGs that can dissolve the crystals existing in cocoabutter. At each temperature and composition, there is a maximumamount of the solid phase that is soluble in the liquid phase. Thisequilibrium phase behavior is described by a phase diagram. Here,a simplification is necessary because of the complexity of the fatcomposition. Although numerous triacylglycerol species are pres-ent in cocoa butter and the migrating oil, the phase behavior is of-ten assumed to be a pseudo-binary system. Shetty (2004)constructed an experimental phase diagram that related theamount of solid phase in cocoa butter – hazelnut oil mixtures tocomposition and temperature. At a given temperature, the weightfraction of solid can be related to the weight fraction of hazelnut oilin cocoa butter using the following relationships (Shetty, 2004):

wFatSF ¼ 0:554� 1:190 � ðwFat

HZ Þ þ 0:638 � ðwFatHZ Þ

2 at 26 �C

wFatSF ¼ 0:653� 1:041 � ðwFat

HZ Þ þ 0:375 � ðwFatHZ Þ

2 at 23 �C

wFatSF ¼ 0:694� 1:067 � ðwFat

HZ Þ þ 0:342 � ðwFatHZ Þ

2 at 20 �C

ð1Þ

1.3. Model development

1.3.1. Model assumptionsIt was assumed that:

1. Chocolate comprised four species: liquid fat (LF), solid fat(SF), non-fat solids (NF), hazelnut oil (HZ).

2. The diffusion process was considered to be isothermal andone-directional.

3. Diffusion takes place only in the liquid phase of the cocoabutter and the driving force for diffusion is a gradient inhazelnut oil concentration in that phase.

4. Solids and fat crystals are impermeable to hazelnut oil andno adsorption occurs on their surfaces.

5. Local equilibrium between the solid and liquid phases, i.e.dissolution was fast in comparison with diffusion.

6. Vapor pressure of hazelnut oil is negligible and bloom, orcrystal formation on the chocolate’s surface was neglected(none was observed).

7. Counter-diffusion of cocoa butter into the filter paper wasnegligible.

8. Solid fat content is temperature dependent according to theexperimental phase diagram for the solid fat content in equi-librium with a certain amount of hazelnut oil (Shetty, 2004).

9. Specific volumes are constant and independent of concen-tration of HZ.

10. Species continuity equations can be formulated in a volume-averaged velocity coordinate frame.

11. Diffusion of fat crystals and other non-fat solids wasnegligible.

12. A moving boundary can be used to accommodate the changein volume of the sample (jump mass balances are used todescribe the movement of the boundary).

13. Convection due to the difference in density of the liquid andsolid fat is negligible.

14. A constant aspect ratio for crystals was appropriate to for-mulate the tortuosity factor.

The diffusion process within the chocolate slab was formulatedrelative to the volume average velocity (Cussler, 1997, pp. 52–54).

Implicitly, it is assumed that partial specific volumes of the compo-nents are independent of composition so there is no volumechange on mixing, and combined with assumptions 6 and 13, thisleads to a zero volume average velocity throughout the chocolateslab and elimination of the convective terms in the continuityequations. Given these simplifying assumptions, the continuityequations for the relevant components such us hazelnut oil and so-lid fat can be written as

oqTHZ

ot¼ ojv

HZ

oxð2Þ

oqTSF

ot¼ �rSF!LF ð3Þ

In these equations, qTi is the mass concentration of the species i in

the total volume, jvi is the diffusive flux of species i, and rSF!LF is

the rate of transformation of the solid fat to liquid fat due to thepresence of hazelnut oil. Since hazelnut oil diffuses only in the li-quid phase of chocolate, i.e. the liquid cocoa butter, it is necessaryto express the flux of this component in Eq. (2) as a function ofthe concentration gradient of hazelnut oil in the liquid fat. Previousmodels incorrectly set the driving force as the gradient of HZ in thetotal chocolate. The expressions for the HZ flux in the liquid and therelationship between different diffusive fluxes relative to the vol-ume average velocity are

jvHZ ¼ Deff

oqLFHZ

oxð4ÞX

i

jvi V̂ i ¼ 0 ð5Þ

where Deff is the effective diffusion coefficient based on informationregarding the internal microstructure of the chocolate.

Modeling of diffusion through heterogeneous materials is not asobvious as one could anticipate. The presence of particles, whetherthey are fat crystals or non-fat solids, forces the migrating speciesto follow a longer path. Factors such as the geometry of the dis-persed phase (particle shape, size and size distribution) contributeto a considerable change in the net effect of these particles (Cuss-ler, 1997). To describe the effective diffusion in these complexmaterials two models are extensively studied in the literature:the Maxwell model (Maxwell, 1873), developed for a dilute disper-sion of spheres, and the model to describe diffusion in the presenceof asymmetric flakes (DeRocher et al., 2004; Moggridge et al.,2002). Whereas the Maxwell model does not have any parameterregarding the size of the spheres, the flake model depends stronglyon the flake aspect ratio, a, i.e. the width to thickness ratio.

In this work, a new model accounting for tortuosity is proposed,where the tortuosity depends on the volume fraction of both thefat and non-fat solids present in the chocolate.

Deff ¼ D0s ¼ D0fT1ð/TNFÞfT2ð/Fat

SF Þ ð6Þ

Here, D0 is the diffusivity of HZ in the liquid fat and s is the effectivetortuosity. s is described as the product of the tortuosity that thediffusing species ‘‘sees” by the presence of the non-fat solids, fT1,

and the tortuosity that the diffusing species experiences withinthe fat phase due to fat crystals, fT2. Expressions for these two termsare

fT1 ¼1� /T

NF

1þ 0:5/TNF

for spheres ð7Þ

f�1T2 ¼ 1þ ða/Fat

SF Þ2

1� /FatSF

for flakes ð8Þ

We assumed that fat crystals are more likely to resemble flakesthan perfect spheres. Both leaf-like structures of classic form VIcrystals and needle-like crystals have been observed in chocolate

264 J.R. Galdámez et al. / Journal of Food Engineering 92 (2009) 261–268

products (Marangoni, 2005). Trial predictions using the Maxwellmodel (spheres), consistently reached equilibrium significantlyfaster than the measured data. For this reason, the flake modelwas used. The diffusion of HZ into chocolate changes the solid fatcontent, decreasing the amount of solid crystals and, consequently,increasing the effective diffusion coefficient.

For a coupled system like this, it is important to consider therelative rates of diffusion of the oil throughout the liquid phaseversus dissolution of the solid phase. If the dissolution of crystalsis relatively fast compared to diffusion, then the solid and liquidphase can reach equilibrium quickly at each position as the oil pen-etrates. In those cases diffusion is the rate-limiting mechanism.However, if diffusion is fast enough, hazelnut oil can diffuse intothe chocolate without a significant change in the solid fat contentdue to dissolution. In that case, the diffusion process will be slowedbecause the tortuous path will remain longer as the solid fat con-tent has a lower rate of dissolution. Unfortunately, there are notmany studies on dissolution of fat crystals during migration to dis-cern which mechanism is dominant.

Ziegler et al. (2004) and Shetty (2004) suggest that a change inthe rate-limiting mechanism, from dissolution to diffusion, occurssomewhere around 20 �C. Below this temperature, they observedthat chocolate was mostly solid and little dissolution occurs. Abig drop in solid fat content was observed at higher temperature,suggesting that the activation energy for dissolution is close tothe thermal energy available at those temperatures, and therefore,the dissolution process is relatively fast. As the first approximation,the model will assume that diffusion is the rate-limiting mecha-nism, and, therefore, Eq. (3) can be re-written as

qTSF ¼ qT

SFðwFATHZ Þ ð9Þ

where the weight fraction of fat crystals is related to temperatureand hazelnut weight fraction by means of the relationships shownin Eq. (1) (see Appendix).

1.3.1.1. Initial and boundary conditions. Initially, it was assumedthat chocolate contains no hazelnut oil

qTHZðt ¼ 0Þ ¼ 0 ð10Þ

For the boundary conditions, assumption 6 effectively meansthat there is no flux of HZ on the slab surface in contact with air so

dqLFHZ

dx

����x¼LðtÞ

¼ 0 ð11Þ

For the side in contact with the HZ oil, several alternativeboundary conditions were considered. The simplest one is to keepthe concentration of HZ constant at the chocolate surface:

qTHZðt; x ¼ 0Þ ¼ qT

HZðt ¼ 1Þ ð12Þ

This boundary condition assumes that the concentration at thechocolate surface immediately rises to the equilibrium value whenit is contacted with hazelnut oil in the filter paper. A more generalform for the boundary condition is setting the concentration at thesurface in equilibrium with the concentration of hazelnut oil left inthe filter paper:

qTHZðt; x ¼ 0Þ ¼ KCFPðtÞ ð13Þ

In this equation, K is the partition coefficient and CFP is thehazelnut oil concentration in the filter paper at any given time.Both equations neglect any resistance at the surface between thefilter paper and chocolate. If the surface resistance becomes impor-tant, Eqs. (12) and (13) can be combined with an exponential mod-el developed by Long and Richman (1960):

qTHZðtÞ � qs0

qTHZðt; x ¼ 0Þ � qs0

¼ 1� e�tk ð14Þ

where qTHZðt; x ¼ 0Þ is the equilibrium surface concentration given

by Eq. (12) or (13), qs0 is the concentration established instanta-neously at the surface at time zero, and k is the characteristic timefor the surface resistance.

The model includes a moving boundary that allows the choco-late slab to expand as a result of the diffusion of HZ oil. The move-ment of the boundary was obtained from a jump mass balance atthe chocolate interface (Duda and Vrentas, 1969):

dLðtÞdt¼ D0s

/LF

dqLFHZ

oxþ mv

����x¼LðtÞ

ð15Þ

These equations were transformed into dimensionless formmaking use of the following dimensionless variables for concentra-tion, temperature, time, and space.

�qLFHZ ¼

qLFHZ � qLF

HZ t!0

qLFHZ t!1 � qLF

HZ t!0

ð16Þ

�t ¼ tD0

L2t¼0

ð17Þ

�L ¼ LðtÞLt¼0

ð18Þ

In addition, we include a new variable, n, to immobilize theboundary as explained elsewhere (Duda and Vrentas, 1969)

n ¼ xLðtÞ ð19Þ

The final equations to solve in dimensionless form are

o½�qTHZ�

o�t� n

�Ld�Ld�t

o½�qTHZ�

o�t¼ 1

�L2

o

onD0s

o½�qLFHZ�

on

� �ð20Þ

�Ld�Ld�t¼ D0s

/LF

d�qLFHZ

on

� �����n¼1

ð21Þ

A finite difference scheme was used to solve this set of differen-tial equations. For the special case where the diffusion coefficientdepends on concentration, an implicit Crank–Nicolson method isused to convert the Eqs. (20) and (21) into a set of non-linear alge-braic equations. This method improves the accuracy and conver-gence of the solution procedure. The resulting equations weresolved using the DNEQNF subroutine included in the IMSL library(Visual Numerics, http://www.vni.com/).

2. Materials and methods

Lindt Bittersweet Thins (World Wide Chocolate, www.world-widechocolate.com) and hazelnut oil (pure unrefined oil manufac-tured from roasted hazelnuts) from Huilerie J. Leblanc (GourmetCountry, www.gourmetcountry.com) were used in the model sys-tem to represent chocolate products with nut-based filling. Choco-late samples weighed 2.84 ± 0.36 g and were 3.8 cm in length andwidth, and 1.4 mm in thickness, allowing diffusion to be consid-ered uni-directional since the thickness was much less than thelength and width. Two different amounts of hazelnut oil (0.57and 1.14 g) were used in the experiments – one equivalent to theamount of fat found in the chocolate sample (1.14 g) and one con-taining half the amount of fat found in the chocolate sample(0.57 g).

An empty Petri dish (disposable Fisherbrand 100 � 15 mm,Fisher Scientific, Fair Lawn, NJ) was weighed on a lab scale (MettlerToledo AB104-S Scale, Switzerland) and tared. The desired amountof hazelnut oil was pipetted into the tared Petri dish. Filter paperswere cut to 3.3125 in. circles to fit entirely in the bottom of the Pet-ri dishes. One piece of filter papers was then placed on top of thehazelnut oil inside the Petri dish. The Petri dish was rotated byhand in order to allow for an even distribution of hazelnut oil

0.00

0.20

0.40

0.60

0.80

1.00

0.0 5.0 10.0 15.0 20.0 25.0 30.0

time0.5 [hours0.5]

aver

age

cum

ulat

ive

wei

ght g

ain

[g]

α= 15

α= 17

Fig. 2. Comparison between experimental mass uptake curves at 20 �C (initial loadof hazelnut oil in filter paper: N 1.14 g, d 0.57 g) and model predictions (solid lines).Eqs. (12) and (14) were used for the boundary condition.

J.R. Galdámez et al. / Journal of Food Engineering 92 (2009) 261–268 265

across the filter paper. Filter papers were used to ensure the choc-olate sample would have a uniform supply of oil at the chocolate-oil interface, rather than having the chocolate ‘‘float” in the oil. Inthis model system, the filter paper represented the non-oil portionof a nut-based filling. One chocolate sample was weighed on thesame lab scale and placed in the Petri dish at the center of the filterpaper. The dishes were covered and placed in an incubator (Impe-rial III Model 310 Lab Line Instruments, Inc Melrose Park, IL) at twodifferent temperatures (20 and 23 �C). The samples were removedfrom the Petri dishes at regular time intervals, weighed and mea-sured to record any dimensional changes, then placed back in theincubator. In order to obtain the weight of the individual chocolatesample, a sheet of weighing paper was tared on the lab scale. Petridishes containing the samples were removed from the incubatorand the chocolate samples were weighed one at a time by remov-ing the dish’s lid and carefully lifting one corner of the chocolatefrom the filter paper using a spatula. Then, the sample was‘‘flipped” onto a piece of weighing paper so that the side of thechocolate that had been in contact with the hazelnut oil was facingupward. The weighing paper with the chocolate was weighed onthe lab scale and returned to the Petri dish with the side in contactwith the HZ returned to its original position. This continued untilthe weight gain of the sample was negligible or the integrity ofthe sample was compromised and it was impossible to removethe sample without physical damage. Each concentration-temper-ature combination was run in duplicate.

Nuclear magnetic resonance (NMR) (Bruker Optics MinispecAnalyzer, Model mq20) was used to obtain the diffusion coeffi-cients for cocoa butter (Cadbury Trebor-Allan, Inc., Toronto, Can-ada), hazelnut oil, and mixtures of hazelnut oil and cocoa butterusing the ‘‘diffusio” application (Minispec Software v2.57Rev.09/NT/XP, Bruker Optics, Inc.). Cocoa butter was held at 60 �C in anincubator (VWR Model 1325F, Sheldon Manufacturing, Inc.) fortwo weeks to ensure any solids would settle out. Six mixtures ofcocoa butter and hazelnut oil (0%, 20%, 40%, 60%, 80% and 100%)were created to total 1 cm of height in each NMR tube (diame-ter = 1 cm) for a total mass of 0.56 g/sample. The desired amountsof hazelnut oil and cocoa butter were added to the tube and mixedon the Maxi Mix II (Barnstead/Thermolyne Model M37615). Mea-surements were taken at six temperatures (35, 40, 45, 50, 55 and60 �C) using the NMR temperature control system (Bruker N2, Bru-ker Optics, Inc.). The diffusion coefficient was measured six timesfor each sample at each temperature using a gradient amplitudeof 90% (180 G/cm). D0 had a weak temperature dependence inthe experimental range.

0.0

0.2

0.4

0.6

0.8

0.0 5.0 10.0 15.0 20.0 25.0 30.0

time0.5 [hours0.5]

aver

age

cum

ulat

ive

wei

ght g

ain

[g]

α= 20

α= 22

Fig. 3. Comparison between experimental mass uptake curves at 20 �C (initial loadof hazelnut oil in filter paper: N 1.14 g, d 0.57 g) and model predictions (solid lines).Eqs. (13) and (14) were used for the boundary condition.

3. Results

Though the model can predict both the spatial and temporal HZconcentration, in the current analyses the output of this model isthe total mass uptake of HZ to compare with the experimental dataavailable. With minimal modifications, however, the program canproduce concentration profiles of HZ oil or calculate the averagesolid fat content in order to reproduce the experimental data ob-tained from other experimental techniques, such as magnetic res-onance imaging (MRI). First, the model was validated with theexperimental data taken in this work and the effect of the bound-ary conditions used was examined. Second, the model is used tounderstand the effect of other variables such as temperature.Lastly, the model was used to produce concentration profiles todemonstrate its potential for comparison with other experimentaltechniques not used in this work.

One way to delay migration and fat bloom in confectioneryproducts is to control the filling to chocolate ratio. High filling ratioproducts, those with a thin layer of chocolate surrounding the fill-

ings, develop fat bloom faster because filling oils have to migrate asmaller distance to reach the outside surface. In the same way, thelarger proportion of fat in fillings relative to chocolate increases theconcentration gradient of fat and speeds up the process. Experi-ments were designed to study this effect and that of temperature.These data are shown in Figs. 2–5, where the solid symbols repre-sent the average value of both replicates and the bars are the dif-ference between the average and the experimental values. It isshown that by decreasing the filling ratio, i.e. having less hazelnutoil relative to cocoa butter, leads to a slower diffusion process. Theparameters used in the simulations for Figs. 2–5 are summarized inTable 1. The model accurately predicted the experimental trends.

The experimental mass uptake curves (Fig. 2) show an initialtime lag period where no important weight gain is observed. Thesame type of ‘‘S”-shape curves has been observed in systems wherethe surface resistance becomes important (Nielsen and Hansenm,2005). k was estimated experimentally from the uptake curves asthe point where a back extrapolation line, from the middle pointsof the curve, crosses the x-axis and qs0 was zero for all the simula-tions. By combining Eqs. (12) and (14) for the boundary condition,the model can predict both uptakes curves with only one parame-ter, the aspect ratio (if the time lag is not considered an adjustableparameter). A constant aspect ratio of 15 can fit the experimental

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.0 5.0 10.0 15.0 20.0

time0.5 [hours0.5]

aver

age

cum

ulat

ive

wei

ght g

ain

[g]

α= 7

α= 9

Fig. 4. Comparison between experimental mass uptake curves at 23 �C (initial loadof hazelnut oil in filter paper: j 1.14 g, e 0.57 g) and model predictions (solidlines). Eqs. (12) and (14) were used for the boundary condition.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0 5 10 15 20 25

t0.5 [hours0.5]

aver

age

cum

ulat

ive

wei

ght g

ain

[g]

T=26°C

T=23°C

T=21°C

Fig. 5. Hazelnut mass uptake curves at different temperatures. All curves wereobtained using a constant concentration at the hazelnut-chocolate boundary (Eq.(12)) with an aspect ratio of a = 5.

Table 1Parameters used for the simulations carried out in this work.

Fig. 2 Fig. 3 Fig. 4T [�C] 20 20 23

Load ofHZ [g]

1.14 0.57 1.14 0.57 1.14 0.57

D0 [m/s2]

2 � 10�10 2 � 10�10 2 � 10�10 2 � 10�10 2 � 10�10 2 � 10�10

k [hour] 10 25 10 25 1.2 9w 0.2 0.12 – – 0.2 0.11K – – 0.7 0.7 – –a 15 17 20 22 7 9

266 J.R. Galdámez et al. / Journal of Food Engineering 92 (2009) 261–268

average values for the 1.14 g of hazelnut oil experiment, but for thelowest hazelnut amount a slightly better fit is achieved with 17.For comparison, mica, a well-known material for composite mem-branes, has an aspect ratio of 30 (DeRocher et al., 2004). Thesenumbers are only empirical and semi-quantitative because noinformation regarding the structure and alignment of the fat crys-tals was available, and are the model’s only fitted parameter. Nev-ertheless, the values obtained are reasonable ones and someinformation about the crystal shape may be inferred.

The partition distribution coefficient, K, was obtained experi-mentally from the weight gain at equilibrium of the two experi-ments shown in Fig. 2. Fig. 3 shows the results from the modelusing these values of K as the input in conjunction with Eqs. (13)and (14). In order to fit the data using this construction, higher val-ues for the aspect ratio, a, where needed (a = 20 for the 1.14 g of HZoil experiment and a = 22 for the 0.57 g). Higher values for a wereexpected since the concentration at the surface and, therefore, theconcentration gradient, was bigger than the case using Eq. (12)alone. With Eq. (12), the model uses the lowest possible concentra-tion at the surface during the whole process. In contrast, theboundary condition represented by Eq. (13) has a higher initialconcentration at the surface, which decreases as the concentrationof hazelnut in the filter paper decreases because it has migratedinto the chocolate. Only at the end will both cases show the sameconcentration at the boundary.

Increasing the temperature accelerated the diffusion process asis shown in Fig. 4. Experimental data reach the equilibrium pla-teau at shorter times than experiments at lower temperature,Fig. 2. The model, using Eqs. (12) and (14), was used to fit thedata. Lower values for the aspect ratio parameter were needed,decreasing from 15 to 7 for the experiment with 1.14 g or HZand from 17 to 9 for the other set of experiments. Temperaturehas a two-fold effect on diffusion. First, it decreases the amountof crystals and, second, the remaining crystals are smaller witha lower aspect ratio, i.e. they are less asymmetric. It is notablethat the fitted aspect ratio, a, decreases with an increase in thetemperature as anticipated.

Given that the model provided an accurate representation of theexperimental data with reasonable values of the only fitted param-eter, a, it was used to simulate migration under other conditions.Temperature is a critical factor during storage. A poor selectionof storage or distribution temperature may accelerate the diffusionprocess and shorten the life of the chocolate product. To study thiseffect, the model was run at three temperatures, 26, 23 and 21 �C.Eq. (12) was used to model the boundary condition without masstransfer resistance. Fig. 5 shows the mass uptake of hazelnut oilat these temperatures, with no time lag and the same final concen-tration in each run. The lower the temperature the longer it takesto reach the equilibrium concentration. This is the result of cou-pling the phase equilibrium behavior of the fat phase with the dif-fusion model. At lower temperatures, more crystals are present inthe fat phase and, therefore, increase the tortuosity of the diffusionpath. A significant drop in solid fat content occurs between 23 and26 �C (Shetty, 2004), which results in a much more rapid diffusionof oil into the chocolate at this temperature (Fig. 5).

The model was used to predict liquid phase profiles inside thechocolate wafers for qualitative comparison to published MRI datawhich can provide spatially resolved concentration profiles (Dekaet al., 2006; Choi et al., 2007). Figs. 6 and 7 show the model predic-tions. In both cases, the simulation was carried out at 21 �C with a5 mm thick slab of chocolate containing 40% of cocoa butter. Fig. 7was obtained using a fixed concentration at the surface, i.e. usingEq. (12) alone. In this case, there is an instantaneous change inthe concentration at the boundary and it is kept at this value dur-ing the whole process. If Eqs. (13) and (14) are used instead, a con-tinuous concentration built up at the surface was observed (Fig. 7).For both cases a sharp gradient is established similar to the onesfound in the literature. Experimentally, Deka et al. (2006) haveshown an increase in the concentration at the boundary betweenthe chocolate and the oil with time, reinforcing the idea of a signif-icant surface resistance during migration of oils when filter paperis used to hold the hazelnut oil. Therefore, surface resistanceshould be taken into consideration for analyzing any diffusionexperiment, unless this effect has been reduced by choosing anadequate experimental set up.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7position [cm]

Wei

ght f

ract

ion

of li

quid

pha

se time = 1 htime = 10 htime = 30 htime = 50 htime = 100 h

Fig. 6. Concentration profiles for the liquid phase content (HZ oil and liquid cocoabutter) at different times. The model uses a fix boundary condition (Eq. (12)) withan aspect ratio, a = 15.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

position [cm]

wei

ght f

ract

ion

of li

quid

pha

se time = 1h

time = 10h

time = 30h

time = 50h

time = 100h

Tim

e

Fig. 7. Concentration profiles for the liquid phase content (HZ oil and liquid cocoabutter) at different times. The model uses Eqs. (12) and (14) for describing surfaceresistant at the boundary and an aspect ratio, a = 15.

J.R. Galdámez et al. / Journal of Food Engineering 92 (2009) 261–268 267

4. Discussion and conclusion

Controlling migration of liquid oil into chocolate with filledproduct is critical for maintaining an acceptable quality. Modelsthat predict the migration of external fats can be extremely usefulfor food engineers for optimizing the manufacturing process andguaranteeing the best quality of the final product for consumers.Published models are highly empirical, and usually introduce somelump parameters encompassing all possible forms of mass transfer.A more general form of oil migration models will be of great use inoptimizing or developing new processes.

This work proposes a new model that represents an importantimprovement over existing ones. It is conceived with a more real-istic formulation of the problem, where in the driving force formigration is the concentration gradient of liquid oils in the liquidphase of the chocolate. Mass transfer is described by molecular dif-fusivity, and the presence of solid particles which affect the diffu-sion through a tortuosity term that varies with oil concentrationand temperature. With independent measurements of the equilib-rium between solid fat crystals in cocoa butter and hazelnut oil theproposed model is able to predict, within experimental error, themass pickup of a chocolate wafer exposed to a source of liquid oil.

As oil migration proceeded, manipulation of the chocolate wa-fers without causing physical damage became difficult. Therefore,

experimental measurement of swelling was not attempted. Anestimation of the final thickness of the sample was made from dig-ital images. The model overestimated swelling by 30–40%, whichcan be attributed to the fact that swelling occurred in all threedimensions in the experimental sample, i.e. the sample increasedin length and width as well as thickness. However, the model al-lowed swelling only in thickness.

The model is based on the assumption of local equilibrium be-tween solid fat and the migrating oil. This assumption is rathercontroversial because it implicitly assumes that diffusion is therate-limiting mechanism, although it may not be true, especiallyat low temperatures. Nevertheless, it is an important starting pointfor future models when the necessary information regarding therates of dissolution of crystals are available. This work addressesthe importance of selecting an appropriate boundary condition be-cause most of the experimental work shows a significant surfaceresistance and reinforces the idea that temperature not onlychanges the amount of fat crystals in the chocolate, but also theirsize and shape.

This study has demonstrated that a diffusion-based model canadequately capture the migration of liquid oil through chocolate.No other mass transfer mechanisms are required. Further improve-ments would include a model that also predicts changes within thefilling phase and might include interactions of the oil with the dis-persed particulate solids.

Appendix

The model developed in this manuscript solves the continuityequation coupled with a thermodynamic equilibrium between thedifferent fat phases as described by Shetty (2004). To incorporatethe equilibrium (Eq. (1)) into the constitutive equation (Eq. (4))the following relationship was used. The mass concentration canbe expressed in terms of the weight mass fraction based on the to-tal chocolate mass as follows:

qi ¼wTotal

iPj

wTotalj � V̂ j

ðA1Þ

Here, j is any of the four species in the model as stated in assump-tion 1, wTotal

j is the weight mass fraction with respect to the totalmass, and V̂ j is the specific volume of each species.

The phase equilibrium between the hazelnut oil and the choco-late fat only involves the fat fraction of the chocolate. For this rea-son, the equilibrium relationship shown in Eq. (1) is expressed interms of the weight fraction based on the total amount of fat. Toincorporate the equilibrium into Eq. (A1) the following expressionswere derived to relate the weight mass fraction based on the fatand on the total chocolate mass.

wFatHZ ¼

wTotalHZ

1þ ð1� FATÞðwTotalHZ � 1Þ

ðA2Þ

Here, FAT is the initial fraction of fat in the chocolate. Similarexpressions could be derived to relate the weight fraction of theothers species in terms of wTotal

HZ , so that Eq. (A1) has only onedependence. Finally, the volume fractions used in Eq. (6) can becalculated as follows:

/Totali ¼ wTotal

i � V̂ iPj

wTotalj � V̂ j

where j ¼ LF; SF; HZ and NF ðA3Þ

/Fati ¼

wFati � V̂ iP

jwFat

j � V̂ j

where j ¼ LF; SF and HZ ðA4Þ

268 J.R. Galdámez et al. / Journal of Food Engineering 92 (2009) 261–268

References

Aguilera, J.M., Michel, M., Mayor, G., 2004. Fat migration in chocolate: diffusion orcapillary flow in a particle solid? – A hypothesis paper. Journal of Food Science69, 167–174.

Choi, Y.J., McCarthy, K.L., McCarthy, M.J., Kim, M.H., 2007. Oil migration inchocolate. Applied Magnetic Resonance 32, 205–220.

Cussler, E.L., 1997. Diffusion: Mass Transfer in Fluid Systems, second ed. Cambridge,UK. pp. 174–175.

Deka, K., MacMillan, B., Ziegler, G.R., Marangoni, A.G., Newling, B., Balcom, B., 2006.Spatial mapping of solid and liquid lipid in confectionery products using a 1Dcentric SPRITE MRI technique. Food Research International 39, 365–371.

DeRocher, J.P., Gettelfinger, B.T., Wang, J., Nuxoll, E.E., Cussler, E.L., 2004. Barriermembranes with different sizes of aligned flakes. Journal of Membrane Science257, 21–30.

Duda, J.L., Vrentas, J.S., 1969. Perturbation solutions of diffusion-controlled movingboundary problems. Chemical Engineering Science 24, 461–470.

Ghosh, V., Ziegler, G.R., Anantheswaran, R.C., 2002. Fat, moisture, and ethanolmigration through chocolate and confectionery coatings. Critical Reviews inFood Science and Nutrition 42, 583–626.

Khan, R., Rousseau, D., 2006. Hazelnut oil migration in dark chocolate-kinetic,thermodynamic and structural considerations. European Journal of LipidScience and Technology 108 (5), 434–443.

Long, F.A., Richman, D., 1960. Concentration gradients for diffusion of vapors inglassy polymers and their relation to time dependent diffusion phenomena.Journal of the American Chemical Society 82, 513–519.

Marangoni, A.G., 2005. Fat Crystal Networks. Marcel Dekker, New York.Maxwell, J.C., 1873. A Treatise on Electricity and Magnetism, vol. 1. Clarendon Press,

Oxford. p. 365.Moggridge, G.D., Lape, N.K., Yang, C., Cussler, E.L., 2002. Barrier films using flakes

and reactive additives. Organic Coatings 1239, 1–10.Nielsen, T.B., Hansenm, C.M., 2005. Significance of surface resistance in

absorption by polymers. Industrial and Engineering Chemistry Research44, 3959–3965.

Shetty, A., 2004. Oil Migration in Fat-based Confectionery. M.S. Thesis. ThePennsylvania State University.

Ziegleder, G., Moser, C., Geiger-Greguska, J., 1996a. Kinetics of fat migration inchocolate products Part 1: Principles and analytical aspects. Fett/Lipid 98, 196–199.

Ziegleder, G., Moser, C., Geiger-Greguska, J., 1996b. Kinetics of fat migration inchocolate products Part 2: Influence of storage temperature, diffusioncoefficient, solid fat content. Fett/Lipid 98, 253–256.

Ziegler, G.R., Shetty, A., Anantheswaran, R.C., 2004. Nut oil migration throughchocolate. Manufacturing Confectioner 84 (9), 118–126.