modeling the rheological properties of cheddar cheese with different fat contents at various...

MODELING THE RHEOLOGICAL PROPERTIES OFCHEDDAR CHEESE WITH DIFFERENT FAT CONTENTS ATVARIOUS TEMPERATURESjtxs_283 331..348

XIN YANG1, NEAL ROBERT ROGERS, TRISTAN KENDRICKS BERRY and EDWARD ALLEN FOEGEDING

Department of Food, Bioprocessing and Nutrition Sciences, North Carolina State University, Box 7624, Raleigh, NC 27695-7624

KEYWORDSCheese, composite gel, fat globule,mathematical model, storage modulus,temperature

1Corresponding author. TEL: 908-575-6248;FAX: 919-513-8023; EMAIL:[email protected]

Accepted for Publication January 3, 2011

doi:10.1111/j.1745-4603.2011.00283.x

ABSTRACT

Cheddar cheese consists of a gel phase with imbedded fat particles and can be repre-sented as a particle-filled gel. The storage modulus (G′) of Cheddar cheese contain-ing different fat contents was fitted to 12 theoretical models for particle-filled gels.Models that included the G′ of fat particles and their interactions best describedcheese G′. The estimated G′ of fat particle (Gf′) was larger than that of gel matrix(Gm′) at 10, 15 and 20C, corresponding to a reinforcing effect of fat on cheese G′.However, Gf′ decreased at a faster rate than Gm′ with increasing temperature, result-ing in a weakening effect at 25C. Cheese rheological properties were dominated bythe solid fat phase at 10 and 15C and showed no significant change with aging. Incontrast, cheese G′ at 20 and 25C decreased after aging cheeses for 12 weeks, corre-sponding to decreases of Gm′ as a result of changes in the protein network.

PRACTICAL APPLICATIONS

Fat is critical to cheese texture. The rheological properties of cheeses depend onthe two phases: fat particles and protein gel matrix. However, limited works havebeen done to quantitatively evaluate the contributions from the two phases. In thisarticle, the rheological properties of Cheddar cheese containing a range of fatcontent were fitted to 12 mathematical models for particle-filled gels. The advan-tages and limitations of these theoretical models were compared. This studyextended the application of the mathematical models in Cheddar cheese, showinga quantitative evaluation on the role of fat in cheese rheology. This provides quan-titative information for developing fat substitutes to function as fat particles incheese texture.

INTRODUCTION

Cheese in various varieties is found almost everywhere milk isconsumed as a food. The appearance, flavor and texture ofcheese are the results of a series of chemical and physical pro-cesses that occur during primary manufacturing and aging.The physical properties of most cheeses allow them to bemodeled from the perspective of soft condensed matterphysics (Mezzenga et al. 2005). Cheddar, like most cheeses,starts as a weak milk gel, is cut into particles, and then througha series of de-watering and particle re-knitting steps, aprimary cheese structure is formed. This consists of close-packed gel particles (i.e., curds) with pockets of interparticle

fluid. As time progresses, interparticle fluid equilibrates andparticle–particle junctions disappear, leaving a fairly homo-geneous gel phase with imbedded fat particles (Fox et al.2000; Everett 2007). This allows for Cheddar cheese to be rep-resented as a particle-filled gel (also called gelled emulsion orcomposite gel) composed of two phases – a protein gel matrixand fat globules (Fig. 1) (Visser 1991; Fox et al. 2000; Rogerset al. 2010).

Theoretical models have been applied to predict the rheo-logical properties of model food composite gels (Richardsonet al. 1981; Brownsey et al. 1987; van Vliet 1988; Chen andDickinson 1998; Jampen et al. 2001; Kim et al. 2001; Koidiset al. 2002; Sok Line et al. 2005; Rosa et al. 2006; Manski et al.

Journal of Texture Studies ISSN 0022-4901

331Journal of Texture Studies 42 (2011) 331–348 © 2011 Wiley Periodicals, Inc.

2007). In these studies, the gel networks are usually made offood proteins, e.g., whey proteins, sodium caseinate, gelatinand soy proteins, or polysaccharides, e.g., gellan. Syntheticpolymer aqueous solution was also used as a suspendingmedium for a composite suspension (Pal 2002, 2008). Thefillers are particles with a range of rheological properties,including rigid fillers, e.g., glass beads, and deformable fillers,e.g., protein gel particles, oil droplets and fat globules. Theeffect of filler on the composite gel stiffness depends on theratio of the filler stiffness to the gel matrix stiffness and ismore significant when the gel matrix stiffness is lower (vanVliet 1988; Chen and Dickinson 1998; Rosa et al. 2006;Manski et al. 2007). This characteristic is illustrated in thetheoretical models, which will be presented in the theory part(Kerner 1956; van der Poel 1958; Palierne 1990). The modelestablished by Kerner (1956) for a composite gel, which canbe transformed to the same equation as van der Poel (1958),and the model developed by Palierne (1990) for a concen-trated suspension, are two well-known models being appliedin many studies. These models can be used to predict therheological properties of a composite gel when the fillervolume fraction is not high (van Vliet 1988; Ahmed and Jones1990). However, they failed to describe the experimentalresults when the volume fraction of the fillers reached arelatively high level, because of the aggregation of fillers(Brownsey et al. 1987; Chen and Dickinson 1998; Kim et al.2001). This was improved by introducing a self-crowdingfactor, which corresponds to the interactions among fillers, ora maximum packing volume fraction of fillers to the theoreti-cal models (Mooney 1951; Lewis and Nielsen 1970; Pal 2002,2008).

Although the theoretical models have been frequently usedin model food systems, they were seldom applied to predict

the rheological properties of a real food – such as cheese. Theinfluences of the compositional factors, including moisture,salt, fat, temperature, pH and aging, on the rheology ofGouda cheese were analyzed using a particle-filled gel modeland attributed to variations in the two phases – casein gelmatrix and fat particles (Luyten 1988; Visser 1991). The influ-ence of temperature depends on fat content and is mainlybecause of a change in the fat globule rigidity in the tempera-ture range from 14 to 26C (Luyten 1988; Visser 1991). Luytenand van Vliet (1990) used a theoretical model from van derPoel (1958) to calculate the ratio of the compression modulusat 20C to that at 26C for Gouda cheeses containing 10, 48 and60% (w/w) fat in dry matter and found that this ratioincreased with increasing fat content. The results from theo-retical calculation were similar to the values from experi-ments, suggesting that the van der Poel (1958) model can beapplied to predict cheese rheology (Luyten and van Vliet1990). Besides the model of van der Poel (1958), others havebeen proposed to describe filler effects in gels and compositematerials. Therefore, the goals of this study were to extend theapplication of the mathematical models to Cheddar cheese,and to evaluate a range of models.

Previously, we demonstrated that the sensory texture oflow-fat cheese is related to both the strength and breakdownpattern of the cheese structure (Rogers et al. 2009). It was alsoshown that cheese rheological properties are dominated bythe fat particles at 10 and 15C, and by the gel network at 20and 25C (Rogers et al. 2010). In this study, we used theoreticalmodels to predict cheeses’ rheological properties using thedata from Rogers et al. (2010), and determined which modelsbest fit Cheddar cheese containing 3 to 33% fat at 10 to 25C.The advantages and limitations of various theoretical modelsin predicting cheese rheology were evaluated.

Cheese Microscopic imagesSchematic illusions of the cheese structure

3% f

The fat phase The protein phase The fat particles The protein gel matrix

at

20% fat

33% fat

FIG. 1. THE SCHEMATIC ILLUSIONS OF THECHEDDAR CHEESE STRUCTUREThe confocal laser scanning microscopic images(100 mm ¥ 100 mm) of Cheddar cheese arefrom Rogers et al. (2010).

MODELING CHEDDAR CHEESE PROPERTIES X. YANG ET AL.

332 Journal of Texture Studies 42 (2011) 331–348 © 2011 Wiley Periodicals, Inc.

THEORY

The simplest approximation of the shear modulus of a two-phase composite gel is a volume-weighted average of themoduli of the individual phases (Allen et al. 2008):

G G Gm f f f= − +( )1 φ φ (1)

where G, Gm and Gf are the shear moduli of the composite gel(cheese), the gel matrix (casein gel matrix) and the filler (fatglobule), respectively; and ff is the volume fraction of thefiller. In this model, both phases – the gel matrix and the filler– are assumed uniformly distributed throughout the com-posite gel and contribute to the gel shear modulus in propor-tion to their volume fractions (Allen et al. 2008). Note thatinteractions between the gel matrix and fillers are not consid-ered in this model and, as moduli are assumed to be constant,it is an isothermal model.

An analogy was proposed between the enhancement in theshear modulus of a composite gel and the increases in the vis-cosity of a suspension because of the addition of fillers, i.e.,hc/hm = Gc/Gm, where hc and hm are viscosities of the suspen-sion and the continuous phase, and Gc and Gm are shearmoduli of the composite gel and the matrix (Smallwood1944; Hill and Power 1956; Kerner 1956; van der Poel 1958;Uemura and Takayanagi 1966; Nielsen 1967; Theocaris andSideridis 1984; Ahmed and Jones 1990). Therefore, the major-ity of the theories for the shear modulus of a particle-filled gelwere developed from the Einstein’s equation (Eq. 2) for theviscosity of a dilute suspension with uniform, monodisperseand rigid spherical inclusions (Einstein 1911):

η η φc m E fK= +( )1 (2)

where KE is the Einstein coefficient which is equal to 2.5 forspheres; ff is the volume fraction of particulate inclusions.The shear modulus of a composite gel can be expressed in asimilar way (Theocaris and Sideridis 1984; Ahmed and Jones1990):

G Gc m f= +( . )1 2 5φ (3)

Several assumptions were made for this analogous formula(Eq. 3): (1) fillers are completely adherent to the matrix, i.e.,active fillers; (2) fillers are rigid particles (Gf →•); (3) fillersare spherical particles of the same size; (4) fillers are homoge-neously dispersed; and (5) the volume fraction of fillers issmall enough that the interactions among fillers can beneglected (Theocaris and Sideridis 1984; van Vliet 1988).However, the interactions among fillers cannot be ignored ifthe filler concentration reaches a non-dilute level. Therefore,many authors modified the Einstein equation and extendedits application for a suspension of infinitely dilute concentra-tion to a nondilute concentration (Guth 1945; Mooney 1951;Brinkman 1952). Guth (1945) computed the suspension vis-

cosity as a power series of ff and introduced a particle inter-action term to the Einstein model:

G Gc m f f= + +( . . )1 2 5 14 1 2φ φ (4)

where the term of 14.1ff2 corresponds to the mutual interac-

tion of pairs of particles. Mooney (1951) included a self-crowing factor – an interaction constant – to the Einsteinequation, taking into account the effects of the filler agglom-erations:

G GS

c mf

f

=−

⎛⎝⎜

⎞⎠⎟

exp.2 5

1

φφ (5)

where S is a self-crowding factor, expressing the ratio of theapparent volume occupied by the filler over its own truevolume (Mooney 1951). Experimental values of S range fromabout 1.2 to over 2.0 (Nielsen 1967), while the theoreticalvalue of S is from 1 to 1.91, depending on the packing geom-etry of the fillers (Mooney 1951). For the infinite dilute fillerconcentration, S is equal to 1 (no filler aggregation); for theclosely packing of spheres, S = 1/0.74 = 1.35; and for the cubicpacking of spheres, S = 6/p = 1.91 (Mooney 1951).A differentform of the self-crowding factor is S = 1/fmax, where fmax is themaximum volume fraction of the fillers. So a different form ofthe Mooney’s equation is

G Gc mf

f

=−

⎛⎝⎜

⎞⎠⎟

exp.

max

2 5

1

φφ φ (6)

The value of fmax varies with the packing geometry of thefillers. Theoretically, it is either 0.64 for a random packing or0.74 for a hexagonal close packing of uniform spheres (Scottand Kilgour 1969). Eilers (1941) introduced fmax to the Ein-stein equation for the suspension viscosity using an empiricalapproach, which can be transformed for the composite gelshear modulus as

G

Gc

m

f

f

= +−

⎛⎝⎜

⎞⎠⎟

11 25

1

2.

max

φφ φ

(7)

Equations (3)–(7) are established for suspension viscosityand extended to the composite gel shear modulus, in which thefillers are assumed to be rigid spherical particles with an infi-nite shear modulus. However, in many cases, the fillers are vis-coelastic materials and have a finite shear modulus – Gf ,whichisnotconsidered intheviscositymodels (Eqs. 3–7).Therefore,Palierne (1990) introduced the Gf to the Einstein equation forthe shear modulus of an infinitely dilute suspension:

G G H HM

Mc m f= + = −( )

+( . ),1 2 5

2 1

2 3φ where (8)

In this model, the reinforcing effect of the filler particle isdescribed using the ratio of the moduli of filler and gel matrix

X. YANG ET AL. MODELING CHEDDAR CHEESE PROPERTIES


or a parameter of M = Gf /Gm. The Palierne’s model shows thesame formula as the Einstein’s model if M→• (Gf →•) andsuggests Gc = Gm at any ff if M = 1 (Gf = Gm). Palierne (1990)also extended this equation for a suspension with finite fillerconcentration (Eq. 9) using a self-consistent treatmentsimilar to the Lorentz sphere method in electricity, where theeffect of other inclusions to a sphere is taken into accountwithin a larger effective sphere radius:

G GH

HH

M

Mc m

f

f

=+

−

⎛

⎝⎜⎜

⎞

⎠⎟⎟ = −( )

+

132

1

2 1

2 3

φ

φwhere (9)

However, the Palierne’s model (Eq. 9) does not considerthe limit of the filler packing volume fraction and generallyunderpredicts the shear modulus of a composite material at ahigh concentration of fillers (Brownsey et al. 1987; Chen andDickinson 1998; Kim et al. 2001; Pal 2002). Therefore, Pal(2002) introduced fmax to the Palierne’s model and developedtwo new models – Pal’s model 1 (Eq. 10) and Pal’s model 2(Eq. 11) – for the concentrated suspensions.

G

G

G G M

Mc

m

c m f

f

= −−

⎛⎝⎜

⎞⎠⎟ −

⎛⎝⎜

⎞⎠⎟1

2 5

1

2 5.

max

exp. φ

φ φ(10)

G

G

G G M

Mc

m

c m f= −−

⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

−

11

2 5 2 5.

max

. maxφφ

φ

(11)

The two models differ in the way of modifying the diver-gence in Gc at ff = fmax. The Pal’s model 1 (Eq. 10) applied theMooney (1951) approach and the Pal’s model 2 (Eq. 11) usedthe Krieger and Dougherty (1959) method. The two modelsshow the same formulas as in Mooney (1951) and Krieger andDougherty (1959) if M→• (Gf →•).

All the previous models (Eqs. 1–11) assume the matrix tobe an incompressible material with a Poisson’s ratio of 0.5. Awell-known formula for prediction of the shear modulus of acomposite, which takes into account the Poisson’s ratio of thematrix – vm, was developed by van der Poel (1958) and simpli-fied by Smith (1974, 1975):

G

G

v M

v M v v Mc

m

m f

m m m f

=− −

− + − − − −+

15 1 1

8 10 7 5 8 10 11

( )( )

( ) ( )( )

φφ

(12)

The effect of vm on this model was discussed in Nielsen(1970) and Dickie (1976), in which a lower vm correspondedto a lower predicted value of the composite gel shearmodulus. In most studies, the vm is assumed to be 0.5 as the gelshould behave as an incompressible material in the linear vis-coelastic region of the rheological experiments (van Vliet1988; Rosa et al. 2006). For an incompressible matrix with aPoisson’s ratio of 0.5, the van der Poel’s model (Eq. 12) is

equivalent to the Palierne’s model (Eq. 9). Kerner (1956) andUemura and Takayanagi (1966) also developed the sameformula as Eq. 12. However, this model (Eq. 12) does not takeinto account the maximum allowable packing volume frac-tion of the filler and always predicted lower values than thereal data, especially when the filler volume fraction reached ahigh value (Brownsey et al. 1987; Chen and Dickinson 1998;Kim et al. 2001; Pal 2002). Therefore, Lewis and Nielsen(1970) modified Eq. 12 by introducing an additional func-tion – the effective volume fraction of fillers (yff) in a manneranalogous to the introduction of a self-crowding factor inMooney (1951):

G

G

v M

v M v v Mc

m

m f

m m m f

=− −

− + − − − −+

15 1 1

8 10 7 5 8 10 11

( )( )

( ) ( )( )

φψφ

(13)

The function yff represents the effective volume fractionof fillers, which takes into account the crowding effect offillers, and has two simplest forms (Lewis and Nielsen 1970):

ψφφ

φ φff

f

= −−

−⎛⎝⎜

⎞⎠⎟1

1exp

max

(14)

ψφ φφ

φ φf f f= + −⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

11

2

max

max

(15)

Slight differences exist between the two forms of yff

(Fig. 2). The two forms of yff correspond to modifiedKerner’s model 1 (Eqs. 13, 14) and modified Kerner’s model2 (Eqs. 13, 15), respectively. Pal (2008) also modified thePalierne’s model (Eq. 9) in a similar way as Lewis andNielsen (1970) did on the Kerner’s model by substitutingthe actual volume fraction of fillers (ff) with the effectivevolume fraction of fillers (yff):

FIG. 2. THE TWO SIMPLEST FORMS OF THE EFFECTIVE VOLUMEFRACTION yff ASSUMING fmax = 0.64The figure is redrawn from Lewis and Nielsen (1970).



G GH

HH

M

Mc m

f

f

=+

−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ = −( )

+

132

1

2 1

2 3

ψφ

ψφwhere (16)

where the two forms of yff are defined by Eqs. 14, 15,corresponding to the modified Palierne’s model 1 (Eqs. 16,14) and the modified Palierne’s model 2 (Eqs. 16, 15). Thedevelopment of these theoretical models is summarized inFig. 3.

MATERIALS AND METHODS

Rheological Analysis

The complex moduli (G*), storage moduli (G′) and lossmoduli (G′′) of cheeses are from Rogers et al. (2010). Stresssweeps of cheese at 10 Hz and 25C showed that the limit(strain and stress) of the linear viscoelasticity, where consecu-tive measurement (taken every 26.3 s) of G* started todecrease, decreased with fat content (Rogers et al. 2010). Fre-quency sweeps were conducted from 0.1 to 10 Hz at 150 Pa ona Stress Tech controlled stress rheometer (ATS Rheosystems,Bordentown, NJ) using 20 mm parallel plate geometry.Cheeses were tested at four temperatures – 10, 15, 20 and 25C– which was controlled using an induction heating device thatwas integrated into the equipment. The shear moduli ofcheeses at 1 Hz were analyzed in this study. The phase angleswere less than 20° of all samples, indicating that the G* ofcheeses are mainly represented by the G′. Therefore, the fol-lowing analyses were only conducted on the G′ of cheeses at1 Hz.

Volume Fraction of Fat in Cheddar Cheese

The cheese composition data were obtained from Rogerset al. (2010). The fat content based on % w/w can be con-

verted to volume fraction based on % v/v using the densitiesof cheese and milk fat (Luyten 1988; Luyten and van Vliet1990). The density of Cheddar cheese was calculated from the% w/w and densities of its major components: fat, protein andwater (Sahin and Sumun 2006). General temperature depen-dences of densities of fat, protein and water can be presentedas follows (Sahin and Sumun 2006):

ρ fat T= −925 6 0 4175. . (17)

ρprotein T= −1330 0 5184. (18)

ρwater T T= + × − × ×− −997 2 3 144 10 3 757 103 3 2. . . (19)

where densities (r) are in kg/m3 and temperatures (T) are in Cand vary between -40 and 150C. The densities of milk fat at10, 15, 20 and 25C were estimated using Eq. 17 and the esti-mated data are close to the reported data (Walstra et al. 2006).The estimated density of full-fat Cheddar cheese (33.0% w/wfat) at 25C is 1,086 kg/m3 and within the literature range of1,071–1,090 kg/m3 (Mayes and Radford 1983; Min et al.2010). The volume fraction of fat was calculated from % w/wof fat and the estimated densities of milk fat and Cheddarcheese (Table 1).

Pal’s model 1 and 2Guth’s model (1945) Eq. 4

Pal s model 1 and 2(2002) Eq. 10 and 11Simple model

Eq. 1

Involving a particle Palierne’s model (1990) E 9

qModified Palierne’s model

1 and 2 (Pal, 2008)

Einstein’s

Interaction term (1990) Eq. 9

K ’ d l Modified Kerner’s

I

Eq. 16, 14 and 15

Einstein smodel

(1911) Eq 3

Kerner’s model(1956) Eq. 12

Modified Kerner smodel 1 and 2

(Lewis and Nielsen 1970)

Involvit

(1911) Eq.3

Involving a maximum Van der Poel’smodel (1958)

(Lewis and Nielsen, 1970)Eq. 13, 14 and 15

ing a Pthe gel

volume fraction of the filling particles φmax

model (1958)Eq. 12 Involving a maximum

volume fraction of the

Poisson

l matrix

U d

Eilers’ model Mooney’s

model

filling particles φmax

n’s ratix v

m

Uemura andTakayanagi’s

d l (1966)(1941) Eq. 7model

(1951) Eq. 6

tio of

model (1966)Eq. 12

Involving the shear modulus of the filling particles Gf

FIG. 3. DEVELOPMENT OF THE THEORETICALMODELSThe Kerner’s model, the van der Poel’s model,and the Uemura and Takayanagi’s model(Eq. 12) have the same formula as thePalierne’s model (Eq. 9) if assuming thePoisson’s ratio of the matrix to be 0.5.

TABLE 1. VOLUME FRACTION OF FAT IN CHEDDAR CHEESE

Fat content (% w/w)

Fat content (% v/v)

10C 15C 20C 25C

3.0 4.03 4.04 4.04 4.058.5 10.8 10.8 10.8 10.815.5 19.5 19.6 19.6 19.620.3 25.1 25.1 25.1 25.123.0 27.9 27.9 27.9 28.028.8 34.3 34.3 34.3 34.333.0 39.1 39.1 39.1 39.2



Statistical Analysis

The G′ of cheeses was analyzed using the general linear modelprocedure of the SAS statistical software package (Version9.1; SAS Institute, Inc., Cary, NC). Analysis of variance(ANOVA) was conducted with means separation to deter-mine differences between treatments. Significant differenceswere established at P < 0.05.

Modeling the Experimental Data

Before modeling, the cheese G′ was classified into six groupsaccording to the Tukey’s honestly significant difference test(Tukey’s HSD) on cheese age (Fig. 4). The Tukey’s HSD testwas conducted for the cheese G′ at each temperature indi-vidually. The same group of data was fitted to the theoreticalequations using the nonlinear regression program of the SASstatistical software package (Version 9.1; SAS Institute, Inc.).Twelve models were used to fit the G′ of cheeses at 10C, whileonly the modified Kerner’s models 1 and 2 (Eqs. 13–15) andthe modified Palierne’s models 1 and 2 (Eqs. 14–16) wereused to fit the G′ of cheeses at four temperatures. For nonlin-

ear regression, a pseudo-R2 can be calculated to evaluate thegoodness-of-fit for the model (Eq. 20).

Pseudo- Residual Total CorrectedR SS SS2 1= − / (20)

where the SSResidual and SSTotal Corrected are the residual sum ofsquares and the corrected total sum of squares (Cameron andTrivedi 2005).

RESULTS

Statistical Analysis

The effects of temperature, cheese age and fat content on theG′ of cheeses were analyzed using ANOVA. Table 2 shows theresults when all data were analyzed jointly. Significant differ-ences (P < 0.05) exist for G′ because of temperature, age andfat content. Because temperature caused significant effects(P < 0.05) on the cheese G′, data for cheeses at four tempera-tures were analyzed individually (Table 3). The G′ of cheesesat 10 or 15C was not significantly affected by age. However,the cheese age caused significant effects (P < 0.05) on the G′of cheeses at 20 and 25C. Aging of cheese at 8C did not lead to

Cheese age (weeks)

0 5 10 15 20 25

Che

ese

G' (

kPa)

0

200

400

600

800

1000

Cheese age (weeks)

0 5 10 15 20 25

Che

ese

G' (

kPa)

0

200

400

600

800

Cheese age (weeks)

0 5 10 15 20 25

Che

ese

G' (

kPa)

0

100

200

300

400

500

Cheese age (weeks)

0 5 10 15 20 25

Che

ese

G' (

kPa)

0

50

100

150

200

250

300

3.0% 8.5% 16% 20% 23% 29% 33%

10C Group (2~24 weeks)

20C Group 1(2~12 weeks)

15C Group (2~24 weeks)

20C Group 2 (12~24 weeks)



a

a

a

a a

b

b

b

b b

cc

c

cd d

e efef

fg g

BA

DC

FIG. 4. THE CHANGE OF CHEESE STORAGEMODULI (G′) WITH CHEESE AGEFour figures correspond to four temperatures:10C (A), 15C (B), 20C (C) and 25C (D). Thecheese age causes no significant difference inthe cheese G′ marked with the same lowerletter in each figure. The Tukey’s HonestlySignificant Difference test was conducted forthe cheese G′ at each temperature individually.�, 3.0%; �, 8.5%; �, 16%; �, 20%; ,23%; �, 29%; �, 33%.



major changes in the cheese microstructure and image analy-sis showed minimal changes in the sizes and shape factors offat globules (Rogers et al. 2010). The fat phase is mainly solidand dominates the cheese G′ at 10 and 15C; therefore, theminimal changes of fat globules with aging would lead toinsignificant change of cheese G′. The contributions of fatglobules to cheese G′ decrease as temperature increases andprotein phase becomes dominant. The cheese G′ is deter-mined by protein phase at 20 and 25C and decreases withaging because of the changes in protein network. The Tukey’sHSD was conducted for the cheese G′ at each temperatureindividually. The cheese G′ was classified into the same groupwhen the cheese age caused no significant differences(P < 0.05) (Fig. 4). This allowed for several data points (i.e.,aging times) at each fat level. In the following analysis, themodels were established for the six groups of data with no sig-nificant differences caused by the aging time (Fig. 4): cheesesof 2–24 weeks at 10C, cheeses of 2–24 weeks at 15C, cheeses of2–8 weeks at 20C, cheeses of 12–24 weeks at 20C, cheeses of2–12 weeks at 25C, cheeses of 12–24 weeks at 25C. Changes inthe fat content caused significant effects (P < 0.05) on the G′of cheeses at 10, 15 and 20C, but not for the cheeses at 25C(Table 3). This again reflects the effect of solid fat content onthe cheese rheological properties at 10, 15 and 20C.

Modeling the Rheological Propertiesof Cheeses

The cheese G′ values at 10C were fit to the 12 theoreticalmodels (Fig. 5). The parameters in each model and the corre-

sponding pseudo-R2 are listed in Table 4. The models takinginto accounts the filler storage modulus (G′ of fat particle[Gf′]), except for the Pal’s model 1, show higher pseudo-R2

than the four viscosity models – Einstein’s model, Guth’smodel, Mooney’s model and Eilers’ model (Table 4).

The viscosity models – Einstein’s model, Guth’s modeland Eilers’ model (fmax = 1 or 0.74) – generate good fits withpseudo-R2 > 0.5; however, the Mooney’s model at fmax = 1and Eilers’ model at fmax = 0.64 overestimate the reinforcingeffect of fat globules to cheese G′ and generate poor fits withpseudo-R2 < 0.5 (Table 4; Fig. 5B). The Mooney’s models atfmax = 0.64 and 0.74 failed to fit experiment data with nega-tive pseudo-R2 and were not shown in Fig. 5B. These viscos-ity models were compared in Fig. 6A, assuming a G′ of gelmatrix (Gm′) = 142 kPa for the casein gel matrix, where themaximum packing volume fraction of the fillers wasassumed to be unity (fmax = 1) in the Mooney’s model andthe Eilers’ models. The estimated Gm′ was calculated usingthe following method. A second-order polynomial relation-ship (R2 = 1) was established between the G* and the caseinto water ratio (C : W) for a rennet casein gel based on thedata from Zhou and Mulvaney (1998), where the G* are376, 107 and 43.1 kPa for the rennet casein gels with C : Wof 0.89, 0.58 and 0.38 at 10C. The protein to water ratios forCheddar cheeses in this study were between 0.60 and 0.71(Rogers et al. 2010). The average is 0.64 and corresponds toa G* of 142 kPa, which is used as an approximate for Gm′. Allthe four models predict similar results when ff is less than10%; however, the effects of filler interactions become sig-nificant at a higher ff (Fig. 6A). The Einstein model does nottake into account the interactions among fillers and under-predicts the composite gel G′ when ff reaches a high value(>20%) (Fig. 6A). The filler interactions have been consid-ered in the equation development in Mooney (1951) andEilers (1941); therefore, the two models predict higherstorage moduli than the Einstein’s model even at fmax = 1(Fig. 6A). The effect of the fmax on the curve shape of theMooney’s and Eilers’ models is shown in Fig. 6B,C. The fmax

depends on the packing geometry of the fillers, which istypically 0.64 for a random packing or 0.74 for a hexagonalclose packing of uniform spheres (Pal 2002). It is equal toone if the fillers can take all the space of a composite gel asmore and more of them are added to the system. Therefore,a lower fmax value suggests that the fillers can influence eachother more easily or have stronger interactions, and corre-sponds to a higher effective volume fraction of fillers (yff)and a more significant enhancement of the fillers to thecomposite gel G′ (Fig. 6B,C). Although Guth’s, Mooney’sand Eilers’ models take into account the filler interactions,they generate poorer fit with the experiment data of CheeseG′ than the Einstein model. In addition, a lower fmax value inthe Mooney’s and Eilers’ models decreases the pseudo-R2

value of the fits (Table 4). This could be because of either

TABLE 2. TEST OF FACTOR EFFECTS ON THE CHEESE RHEOLOGICALPROPERTY, USING ANALYSIS OF VARIANCE

Source of variance F value P value

Temperature 459 <0.0001Cheese age 12.1 <0.0001Fat content 55.6 <0.0001

All data were analyzed jointly.

TABLE 3. TEST OF FACTOR EFFECTS ON THE CHEESE RHEOLOGICALPROPERTY USING ANALYSIS OF VARIANCE

Source of variance Temperature (C) F value P value

Cheese age 10 2.13 0.098715 1.03 0.40520 4.97 0.002925 8.63 <0.0001

Fat content 10 33.4 <0.000115 9.59 <0.000120 5.04 0.000925 2.36 0.0515

Data of different temperatures were analyzed separately.



the interaction among the fat globules in the cheese matrixis weak or critical factors are missing in these viscositymodels.

The viscosity models do not involve the G′ of fillers (Gf′)and have been improved by several groups (Kerner 1956; vander Poel 1958; Uemura and Takayanagi 1966; Palierne 1990).The theoretical models including Gf′ were applied to fit thecheese G′ in Fig. 5C–G. The simple model generates a linearrelationship between the cheese G′ and ff with a pseudo-R2 of0.663 (Table 4; Fig. 5C); however, this isothermal model takesno account of the material structure. The Palierne’s model,which is established for a particle-filled gel, fits the cheese datavery well (pseudo-R2 = 0.730) but predicts an extremelylarger Gf′ (Table 4; Fig. 5C). The Pal’s model 1 (fmax = 1) takesinto account the filler interactions and predicts a relativelymore realistic Gf′; however, the fit is not good (pseudo-R2 = 0.423) (Table 4; Fig. 5C). The Pal’s model 1 at fmax = 0.64and 0.74 failed to fit experiment data with negative pseudo-R2

and is not shown in Fig. 5. After assuming a fmax of 0.64, thefour theoretical models –the modified Kerner’s models 1 and2 and the modified Palierne’s models 1 and 2 – predict morerealistic G′ values of the fat globules and show good fits

(pseudo-R2 > 0.7) (Table 4; Fig. 5D–G). The G′ values of themilk fat at 1 Hz are around 1,700 kPa at 10C (Rohm andWeidinger 1993). However, the Pal’s model 2 (fmax = 1) stillsuggests a very high G′ of fat globule (Table 4; Fig. 5C). Ahigher fmax value leads to a higher pseudo-R2 value for thePal’s model 2, but a slightly lower pseudo-R2 value for themodified Kerner’s models 1 and 2 and the modified Palierne’smodels 1 and 2 (Table 4), although there is no significantchange in their curves at different fmax in Fig. 5D–G. Note thatthe modified Kerner’s model 2 and the modified Palierne’smodel 2 generate the same curve as the Palierne’s model whenfmax = 1 (Fig. 5C,E,G; Table 4). The six models along with thePalierne’s model were compared in Fig. 7 assumingGm′ = 142 kPa and Gf′ = 1,688 kPa. The Gf′ = 1,688 kPa for fatglobule is estimated using the storage modulus of butter at10C from Rohm and Weidinger (1993). A lower fmax valuecorresponds to a more significant enhancement of the fillersto the composite gel storage modulus because of the samereason as discussed for the viscosity models in Fig. 6B,C(Fig. 7A–F). Although the six models are developed from thePalierne’s model or the Kerner’s model, which are equivalentto each other when assuming a Poisson’s ratio of 0.5, only the

Fat % v/v

0 10 20 30 40

Einstein's modelGuth's modelMooney's model (φmax=1)

Eilers' model (φmax=1)

Eilers' model (φmax=0.74)

Eilers' model (φmax=0.64)

Fat % v/v

0 10 20 30 40Simple modelPalierne's modelPal's model 1 (φmax=1)

Pal's model 2 (φmax=1)

Fat % v/v

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Che

ese

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1000φmax=1

φmax=0.74

φmax=0.64

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φmax=1

φmax=0.74

φmax=0.64

Fat % v/v

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φmax=1

φmax=0.74

φmax=0.64

Fat % v/v

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φmax=1

φmax=0.74

φmax=0.64

Fat % v/v

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Che

ese

G' (

kPa)

0

200

400

600

800

1000Figure B

Figure CB CA

D E F GModified Kerner's

model 1Modified Kerner's

model 2Modified Palierne's

model 1Modified Palierne's

model 2

FIG. 5. MODELING THE CHEESE STORAGE MODULI (G′) AT 10C USING VARIOUS THEORETICAL MODELSThe parameters in these models are listed in Table 4. The cheeses of different ages were fitted using the same model as the cheese age caused nosignificant change of the cheese G′ at 10C. Figure 5A shows the G′ of cheeses of different ages (�, 2 weeks; , 4 weeks; �, 8 weeks; �, 12 weeks;�, 24 weeks). The theoretical models which do not involve the maximum volume fraction (fmax) or with a fmax = 1 were compared in Fig. 5B,C. Themodified theoretical models, which involve the effect of fmax, were compared in Fig. 5D–G, with the solid, dotted and dashed lines corresponding tofmax of 0.64, 0.74 and 1. The Kerner’s model, van der Poel’s model, and Uemura and Trkayanagi’s model have the same formula as the Palierne’smodel (shown in Fig. 5C) if assuming the Poisson’s ratio of the matrix to be 0.5.



TABLE 4. THE PARAMETERS IN THE THEORETICAL MODELS FOR THE CHEESES AT 10C

Figure Model fmax M = Gf/Gm Gm (kPa) Gf (kPa) Pseudo-R2

Models with one adjustable parameters Gm

Fig. 5B Einstein’s model — — 275 — 0.621Guth’s model — — 162 — 0.561

Models with two adjustable parameters Gm and fmax

Not shown Mooney’s model 0.64 — 70.6 — -1.62Mooney’s model 0.74 — 105 — -0.527

Fig. 5B Mooney’s model 1 — 152 — 0.329Eilers’ model 0.64 — 150 — 0.289Eilers’ model 0.74 — 174 — 0.536Eilers’ model 1 — 204 — 0.698

Models with two adjustable parameters Gm and Gf

Fig. 5C Simple model — 5.17 218 1,130 0.663Palierne’s model — • 234 • 0.730

Models with three adjustable parameters Gm, Gf and fmax (the Poisson’s ratio is assumed to be 0.5 in the modified Kerner’s model 1 and 2)Not shown Pal’s model 1 0.64 7.61 186 1,410 -0.677

Pal’s model 1 0.74 10.2 190 1,940 -0.046Fig. 5C Pal’s model 1 1 21.3 191 4,070 0.423

Not shown Pal’s model 2 0.64 27.0 194 5,250 0.494Pal’s model 2 0.74 52.0 193 10,100 0.579Fig. 5C Pal’s model 2 1 • 196 • 0.671

Fig. 5D Modified Kerner’s model 1 0.64 6.42 255 1,640 0.766Modified Kerner’s model 1 0.74 8.76 249 2,190 0.759Modified Kerner’s model 1 1 17.9 241 4,310 0.746

Fig. 5E Modified Kerner’s model 2 0.64 11.2 245 2,740 0.752Modified Kerner’s model 2 0.74 20.5 240 4,910 0.744Modified Kerner’s model 2 1 • 234 • 0.730

Fig. 5F Modified Palierne’s model 1 0.64 4.33 261 1,130 0.769Modified Palierne’s model 1 0.74 5.76 254 1,460 0.761Modified Palierne’s model 1 1 11.2 244 2,750 0.747

Fig. 5G Modified Palierne’s model 2 0.64 7.23 249 1,800 0.754Modified Palierne’s model 2 0.74 12.8 243 3,110 0.746Modified Palierne’s model 2 1 • 234 • 0.730

Fat % v/v

0 10 20 30 40

Che

ese

G' (

kPa)

0

200

400

600

800

1000Einstein's modelGuth's modelMooney's model (φmax=1)

Eilers' model (φmax=1)

Fat % v/v

0 10 20 30 40

φmax=1

φmax=0.74

φmax=0.64

Fat % v/v

0 10 20 30 40

φmax=1

φmax=0.74

φmax=0.64

A B CMooney’s

model Eilers’ model

FIG. 6. THEORETICAL MODELS FOR SUSPENSION VISCOSITYAn assumptions of Gm = 142 kPa is used for all models. The maximum volume fraction (fmax) is equal to 1 for the Mooney’s model and the Eilers’model in Fig. 6A. The effects of fmax on the G′ were compared in Fig. 6B,C for the Mooney’s model and the Eilers’ model.



modified Kerner’s model 2 and the modified Palierne’s model2 predict the same result as the Palierne’s model at fmax = 1(Fig. 7D,F), while the others suggest stronger enhancingeffects of the fillers to the composite gel storage modulus(Fig. 7A–C,E). The Pal’s model 1, the modified Kerner’smodel 1 and the modified Palierne’s model 1 (Fig. 7A,C andE) predict more significant reinforcement of the fillers to thecomposite gel storage modulus than the Pal’s model 2, themodified Kerner’s model 2 and the modified Palierne’s model

2 (Fig. 7B,D and F), respectively, when compared at the samefmax. The differences among the six models can be attributedto various mathematical assumptions for the filler interactionterm in the equation development. First, the Pal’s models 1and 2 were developed by integrating a differential equation, inwhich the incremental increase in the modulus as a result ofadding more fillers was calculated using the Palierne’s model;however, the other four models were simply modified fromthe Palierne’s model by replacing the actual volume fraction

Fat % v/v

0 10 20 30 40

Che

ese

G' (

kPa)

0

200

400

600

800

1000

φmax=1

φmax=0.74

φmax=0.64

Palierne's model

Fat % v/v

0 10 20 30 40

φmax=1

φmax=0.74

φmax=0.64

Palierne's model

Fat % v/v

0 10 20 30 40

Che

ese

G' (

kPa)

0

200

400

600

800

1000φmax=1

φmax=0.74

φmax=0.64

Palierne's model

Fat % v/v

0 10 20 30 40

φmax=1 (Palierne's model)

φmax=0.74

φmax=0.64

Fat % v/v

0 10 20 30 40

Che

ese

G' (

kPa)

0

200

400

600

800

1000φmax=1

φmax=0.74

φmax=0.64

Palierne's model

Fat % v/v

0 10 20 30 40

φmax=1 (Palierne's model)

φmax=0.74

φmax=0.64

A

Modified Kerner’s model 1

Modified Kerner’s model 2

B

C

D

E

F

Pal’s model 1

Modified Palierne’s model 1

Modified Palierne’s model 2

Pal’s model 2

FIG. 7. THE EFFECT OF THE MAXIMUMVOLUME FRACTION (fmax) ON THE G′PREDICTED FROM THE THEORETICAL MODELSAssumptions of Gf = 1,688 kPa andGm = 142 kPa are used for all models. Thecurve calculated from the Palierne’s model isoverlapped with the curves calculated from themodified Kerner’s model 2 and the modifiedPalierne’s model 2 at fmax = 1 in Fig. 7D,F.



of fillers (ff) with an effective volume fraction of fillers (yff)to account for the filler interactions (Lewis and Nielsen 1970;Pal 2002, 2008). Second, the way of replacing ff with yff is dis-similar in different models, e.g., the modified Kerner’s models1 and 2 and the modified Palierne’s models 1 and 2 (Eqs. 13,16) (Lewis and Nielsen 1970; Pal 2008). Third, the yff wasestablished based on the mathematical boundary conditionsand thereby took various forms in different studies, e.g.,Eqs. (14) and (15) (Mooney 1951; Lewis and Nielsen 1970;Pal 2002, 2008). Therefore, the filler interaction term or yff

generates dissimilar calculations of the enhancement of thefillers to the gel stiffness in different models; for example, themodification because of this term is more significant in themodified Kerner’s model 1 and the modified Palierne’s model1 (Fig. 7C,E) than in the modified Kerner’s model 2 and themodified Palierne’s model 2 (Fig. 7D,F), corresponding togreater deviations from the Palierne’s model.

When the models are selected simply based on the highestpseudo-R2 values, the modified Kerner’s model 1 and 2 andthe modified Palierne’s models 1 and 2 show the best fit(Table 4). As a lower fmax value generates a slightly higherpseudo-R2 value for the four models (Table 4), the fmax wasassumed to be 0.64 for the following analysis.

The G′ of cheeses at different temperatures were classifiedinto six groups according to the Tukey’s HSD test (Fig. 4) andfitted to the modified Kerner’s models 1 and 2 and the modi-fied Palierne’s models 1 and 2, assuming fmax = 0.64 and aPoisson’s ratio of 0.5 (Fig. 8; Table 5). The four modelspredict similar results of Gf′ and the shear modulus of proteingel matrix (Gm′) except for the modified Kerner’s model 2,which predicts relatively higher values of Gf′ at 10 and 15C(Table 5). The modified Kerner’s model 2 (Fig. 7D) alwayspredicts less significant enhancing effect of the fillers to the gelstorage modulus than the other three (Fig. 7C,E and F);therefore, this underpredicting nature was compensated byusing a higher Gf′. Generally speaking, the four modelsdescribe the variation of cheese G′ verse fat content very well(Fig. 8) and generate the Gf′ and Gm′ showing the same chang-ing tendency with temperature or aging time (Table 5).

With increasing temperature, the cheese G′ graduallydecreaseswhencomparedatthesamefatcontent(Fig. 8).Notethe difference in the scale of cheese G′ at various temperatures.Thepseudo-R2 drops toa lowervaluewith increasingtempera-ture for all models (Table 5). This is because the change ofcheese G′ with fat content becomes insignificant as tempera-ture increases, corresponding to an increase of the P value inTable 3. The cheese G′ increased with increasing fat content at10C, indicating a reinforcing effect of fat globules to the cheesetexture (Fig. 8A,B,G,H). This reinforcing effect diminisheswith increasing temperature, resulting in a slightly decreasingtendency of the cheese G′ with increasing fat content at 25C(Fig. 8E,F,K,L). These observations are coincident with theparameters in the theoretical models (Table 5). The Gf′ is

higher than the Gm′ at T � 20C and both decrease withincreasing temperatures (Table 5). The Gf′ decreases fasterthantheGm′with increasingtemperature(Table 5),suggestingthat the melting of fat globules is faster than the softeningof the protein gel matrix in this temperature range (10–25C).Because the triacylglycerols in milkfat have melting tempera-tures between -40 and 40C, milkfat shows a broad meltingrange (Vithanage et al. 2009). The solid content in milkfatdecreases greatly from 10 to 25C, leading to a decrease of Gf′(Wright et al. 2001; Vithanage et al. 2009). Therefore, the fatglobules transform from relatively rigid fillers at 10C todeformable fillers at 25C, corresponding to a less significantstrengthening effect on the cheese G′. The Gf′ estimated fromthe theoretical models is within the range of milk butter G′reported in other studies,which varies with the lipid composi-tion (Fig. 9).The Gf′ and the Gm′ are close to each other at 25C,in agreement with the insignificant (P = 0.0515, Table 3)changes of cheese G′ with fat content at this temperature.

The decrease of Gm′ with increasing temperature is becauseof the decrease in the total number and/or strength of bonds,e.g., casein–casein interactions, in the cheese matrix (Luceyet al. 2003). The hydrophobic interactions tend to increase instrength with temperature but may reduce in total numberbecause of less contact areas between casein particles, leadingto a possible net reduce in overall strength (Lucey et al. 2003).Other factors, such as hydrogen bonding and the electrostaticinteractions among molecules, and the solubility of Ca salts,also change with temperature and influence the gel structure(Lucey et al. 2003). In the present study, the temperatureeffects on the protein gel matrix and the filler fat particleswere distinguished from each other. Results suggest that thedecrease of cheese G′ is a result not only of the melting of fatsbut also of the weakening of protein gel matrix with increas-ing temperature, while the effect of fat melting is more signifi-cant in this temperature range (10–25C).

The aging effect on cheese G′ can be detected at 20 and 25C(Table 3; Fig. 4).When modeled for the cheese G′ at 20 or 25C,the Gf′ does not change much but the Gm′ slightly decreases asthe cheeses are aged longer, suggesting that the protein gelmatrix becomes weaker (Table 5). The cheese G′ follows thesame decreasing trend as Gm′ with the cheese age, although thedifference is not large (Fig. 4). This again suggests that thecheeseG′ isdeterminedbytheproteingelmatrixat20and25C.The decrease of cheese rheological properties with cheese agehas been observed in other studies and attributed to a reduc-tion of intact caseins as a result of proteolysis (Creamer andOlson 1982; Lawrence et al. 1987; Guinee et al. 2000; Everettand Olson 2003). This effect mainly decreased the storagemodulus of the protein gel matrix and generates a consequentsoftening effect on the cheese G′.

A fmax of 0.64 is assumed in the models applied for thecheese G′ at different temperatures (Fig. 8; Table 5), whichsuggests a random packing geometry of the fat globules as



spheres in cheeses. Rogers et al. (2010) analyzed the micro-structure images of cheeses containing different amounts offat, which were observed at room temperature. The averageshape factor of fat globule gradually shifts from 0.85 to 0.65 asthe fat content increases from 3 to 33%, showing that theshape of fat globule deviates from a perfect sphere, which hasa shape factor of 1 (Rogers et al. 2010). Therefore, the packinggeometry of fat globules may not be a random packing ofspheres at room temperature and possibly corresponds to ahigher fmax. However, as the G′ of cheeses does not change sig-nificantly with fat content at room temperature, no furtherwork was conducted to establish a model with a higher fmax

for the cheeses at room temperature.

DISCUSSION

In a particle-filled gel, the fillers can be classified as either“active” or “inactive” according to their interactions with thegel matrix (Ring and Stainsby 1982; van Vliet 1988; Luytenand van Vliet 1990; Dickinson and Chen 1999). Active fillershave a strong interaction with the gel matrix and can reinforcethe gel strength if the filler is stiffer than the gel matrix; whileinactive fillers have little affinity with the gel matrix andcannot strengthen the gel (Ring and Stainsby 1982; van Vliet1988; Luyten and van Vliet 1990). For inactive fillers, the stiff-ness of a composite gel decreases with increasing the volumefraction of fillers (van Vliet 1988; Luyten and van Vliet 1990;

Fat % v/v

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Fat % v/v

0 10 20 30 40

Che

ese

G' (

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800

1000

Che

ese

G' (

kPa)

0

100

200

300

400

500

Fat % v/v

0 10 20 30 40

Che

ese

G' (

kPa)

0

100

200

300

Fat % v/v

0 10 20 30 40

A B

C D

E F

G H

I J

K L

Modified Kerner’s models Modified Palierne’s models

10C 10C15C 15C

20C 20C20C 20C

25C 25C25C 25C

2 week 4 weeks 8 weeks 12 weeks 24 weeks

FIG. 8. MODELING THE CHEESE STORAGE MODULI (G′) USING THE MODIFIED KERNER’S MODELS (A–F) AND MODIFIED PALIERNE’S MODELS (G–L)The parameters in the models are listed in Table 5. The fmax value is assumed 0.64. The dot and solid lines correspond to the modified Kerner’s model 1and 2, respectively, in Fig. A–F, and the modified Palierne’s model 1 and 2, respectively, in Fig. 8G–L. �, 2 weeks; , 4 weeks; �, 8 weeks; �, 12weeks; �, 24 weeks.



Dickinson and Chen 1999). The rheological property of acomposite gel filled with inactive particles is similar to thetheoretical behavior of a spherical foam structure, where thefillers essentially act as aqueous holes entrapped in the gelmatrix (van Vliet 1988; Jampen et al. 2001). During smalldeformation, the mechanical forces exerted on the gel canonly be passed onto the intermediate aqueous layers aroundthe noninteracting fillers, that is, only the intermediateaqueous layers are deformed (Fig. 10A; van Vliet 1988; Luytenand van Vliet 1990). This structure is similar to the gel filledwith aqueous holes (Fig. 10B). The theoretical models, whichare established base on the assumption that the fillers are per-

fectly adhesive to the gel matrix, may predict an effective fillerstorage modulus of zero for inactive fillers (a zero storagemodulus of aqueous holes) (van Vliet 1988; Luyten and vanVliet 1990; Jampen et al. 2001). Active fillers, as part of thecomposite gel, can be deformed during small deformation ofthe gel and contribute to the overall stiffness (Fig. 10C; vanVliet 1988; Luyten and van Vliet 1990). Along with the defor-mations of fillers, the interfaces between active fillers and gelmatrix are also deformed and show stiffness because of inter-facial tension (van Vliet 1988; Luyten and van Vliet 1990;Taylor 1934). In this study, the storage modulus of cheeseincreased with fat content at 10, 15 and 20C, suggesting active

TABLE 5. THE PARAMETERS IN THE MODIFIED KERNER’S MODELS 1 AND 2 AND THE MODIFIED PALIERNE’S MODELS 1 AND 2 (fmax = 0.64; vm = 0.5)

Model Figure Temperature (C) Age (weeks) M = Gf/Gm Gm (kPa) Gf (kPa) Pseudo-R2

Modified Kerner’s model 1 Fig. 8A 10 2–24 6.42 255 1,640 0.766Fig. 8B 15 2–24 4.46 220 982 0.666Fig. 8C 20 2–12 1.63 196 318 0.162Fig. 8D 20 12–24 1.79 164 292 0.440Fig. 8E 25 2–8 0.719 166 120 0.0740Fig. 8F 25 12–24 0.851 125 106 0.0506

Modified Kerner’s model 2 Fig. 8A 10 2–24 11.2 245 2,740 0.752Fig. 8B 15 2–24 6.07 214 1,301 0.649Fig. 8C 20 2–12 1.68 196 329 0.148Fig. 8D 20 12–24 1.90 163 309 0.422Fig. 8E 25 2–8 0.685 167 115 0.0809Fig. 8F 25 12–24 0.831 125 104 0.0555

Modified Palierne’s model 1 Fig. 8G 10 2–24 4.33 261 1,130 0.769Fig. 8H 15 2–24 3.25 225 733 0.672Fig. 8I 20 2–12 1.48 197 290 0.178Fig. 8J 20 12–24 1.58 165 261 0.456Fig. 8 K 25 2–8 0.808 164 132 0.0592Fig. 8 L 25 12–24 0.897 124 111 0.0416

Modified Palierne’s model 2 Fig. 8G 10 2–24 7.23 249 1,800 0.754Fig. 8H 15 2–24 4.55 218 992 0.653Fig. 8I 20 2–12 1.57 196 308 0.158Fig. 8J 20 12–24 1.73 164 283 0.432Fig. 8 K 25 2–8 0.749 166 124 0.0717Fig. 8 L 25 12–24 0.865 125 108 0.0503

Temperature (oC)

5 10 15 20 25 30

Gf'

(kP

a)

101

102

103

104

105

Modified Kerner's model 1Modified Kerner's model 2Modified Palierne's model 1Modified Palierne's model 2Mainland Butter at 1 Hz a

Pam's Butter at 1 Hz a

HMT Butter at 1 Hz b

Market Butter at 1 Hz b

Austria Butter at 1 Hz c

FIG. 9. COMPARISON OF THE FAT GLOBULARSTORAGE MODULI (Gf′) FROM DIFFERENTMODELS AND FROM OTHER STUDIESA. Data is from Vithanage et al. (2009); B. Datais from Shukla et al. (1994); C. Data is fromRohm and Weidinger (1993). Butters wereobtained from a variety of sources in thesestudies. HMT Butter refers to high meltingtriglyceride butter, which is enriched with highmelting triglycerides and has a higher meltingtemperature range than normal butter.



fillers. The reinforcing effect of fat globules on cheese rheol-ogy was also found in Gouda cheese at 14C (Luyten 1988;Luyten and van Vliet 1990; Visser 1991). Therefore, the shearmodulus of fillers (Gf) can be predicted as a sum of fat globuleshear modulus (Gfat) and the shear modulus as a result ofinterface deformation (Ginterface). The Ginterface can be estimatedusing Taylor’s method as Ginterface = 2g/R, where g is the interfa-cial tension and R is the radius of fat globules (Taylor 1934;van Vliet 1988). For natural milk fat globules, g is in the rangeof 1–2 mN/m (Walstra and Jenness 1984) and R is 1–2 mm(Fig. 1; Rogers et al. 2010). This yields a Ginterface of 1–4 kPa,which is quite small compared with the range of cheesestorage moduli (>100 kPa) in this study and can be neglected.Therefore, the major contribution of fat globule to the cheesestiffness is Gfat, which strongly depends on the temperature;that is, Gf @ Gfat. This is in accordance with the literature dataof Gfat (Fig. 9). According to the model prediction, the fatglobules tend to reinforce the cheese G′ at 10, 15 and 20Cwhen Gf′ > Gm′ and do not significantly change the cheese G′at 25C when Gf′ @ Gm′. The compression modulus of Gouda

cheese was found independent of the fat content at 20C(Luyten 1988; Luyten and van Vliet 1990; Visser 1991). If thetemperature is high enough that the fat globules can trans-form to a completely liquid state, they may have zero storagemodulus as liquid droplets (neglecting the storage modulusas a result of interface deformation). However, the contribu-tion of interface deformation to the storage modulus of liquiddroplets can significantly influence the storage modulus ofthe composite system if the matrix storage modulus is lowand comparable with this value (van Vliet 1988; Everett andOlson 2003; Rosa et al. 2006). The fat globules may have alower Gf′ than Gm′ and show a weakening effect on the cheeseG′ at a higher temperature (at least >25C). The decrease ofcheese G′ with increasing fat content at T > 25C has beenobserved in other studies (Luyten 1988; Luyten and van Vliet1990; Visser 1991; Ustunol et al. 1995; Venugopal and Muthu-kumarappan 2003; Brighenti et al. 2008).

The interaction between filler and gel matrix mainlydepends on the surface properties of the filler, especially thenature of the stabilizing layer (van Vliet 1988; Luyten and

Small deformationNo deformation

A A Small deformation of the Intermediate

aqueous layersGel filled with

intermediate aqueous layersq y

non-interactingfillers

No deformation of the non-interacting fillers

Non-interactingfillers

B B Small deformation of the aqueous holes

water waterGel filled with

water waterAqueous holesaqueous holes

water water

C C Small deformation of the interacting fillers

InteractingGel filled with

interacting fillersSmall deformation of the interfaces between the

fillers interacting fillers and the gel matrix

FIG. 10. SCHEMATIC REPRESENTATION OF GELS FILLED WITH NONINTERACTING FILLERS (A), AQUEOUS HOLES (B) OR INTERACTING FILLERS (C)UNDER SMALL DEFORMATIONThe figures are modified from van Vliet (1988) and Luyten and van Vliet (1990).



van Vliet 1990; McClements et al. 1993; Cho et al. 1999;Dickinson and Chen 1999; Michalski et al. 2002). For the fatparticles dispersed in a protein gel matrix, protein-stabilizinglayers usually lead to active fillers while small molecular weightsurfactant-stabilizing layers generally correspond to inactivefillers (vanVliet 1988; Luyten and vanVliet 1990; McClementset al. 1993; Cho et al. 1999; Dickinson and Chen 1999; Michal-ski et al. 2002). The fat globules coated with small molecularweight surfactant-stabilizing layers do not interact with theprotein gel network, and thereby play roles as structure break-ers (McClements et al. 1993; Dickinson and Hong 1995).Evenwith protein-stabilizing layers,the fat globules coated with dif-ferent proteins can have dissimilar influences on the rheologi-cal properties of cheeses or milk gels (Visser 1991; Cho et al.1999; Everett and Olson 2000, 2003). Cho et al. (1999) foundthat the fat globules covered by casein have a greater enhance-mentontheG′of acidmilkgel thanthosecoveredbywheypro-teins, which was attributed to more interactions between thecasein-stabilizing layers with the casein-based gel network.However, Everett and Olson (2003) compared the fractureproperties of cheeses containing fat globules stabilized withnative membrane materials, caseins and whey proteins. Theyfound that the as2-casein corresponded to the weakest fracturestrain of the cheese.These cheeses also had elongated and clus-tered fat globules observed in the microstructure (Everett andOlson 2003).This was explained as the as2-casein contains a lotof phosphate groups and exposes these hydrophilic groups onthe surface of fat globules as the hydrophobic portions areburied within the fat globules, therefore decreasing the inter-actions between the fat globule surface and the casein matrix(Everett and Olson 2003).

Generally speaking, dissimilar effects on the gel rheologicalproperties of fat globules coated with different materials aremainly because of the filler–matrix interactions. However, itshould be recalled that coating materials can influence theinterfacial tension and thereby affect the stiffness as a result ofinterface deformation (Ginterface = 2g/R) as discussed before. Ifthe connection between the filler and the matrix is weak, theconnection may break when a load is applied, changing theactive fillers to be inactive ones. Sato and Furukawa (1963)introduced an adhesion parameter to the theory for the tensilefracture of a composite gel, taking into account the strength ofthe filler–matrix interactions. However, the shear modulus ofa composite gel is usually measured in the linear viscoelasticregion, in which the small deformation is supposed to causeno detrimental effect on the gel structure, i.e., the adhesionbetween the filler and the matrix. Therefore, the fillers areassumed to be perfectly adhesive to the gel matrix and becomeparts of the composite gel in the theoretical models (Kerner1956; van der Poel 1958; Uemura and Takayanagi 1966; Lewisand Nielsen 1970; Palierne 1990; Pal 2002, 2008). In this study,the interaction between fat particles and the protein gel matrixis assumed to be completely adhesion and have equal strength

among samples. The native fat globules should have the samecoating layers as the cheeses were produced from similar pro-cessing conditions except for different fat content (Rogerset al. 2010). Although coalesced fat particles were observed,especially at high fat content,each fat particle,either natural orcoalesced, is assumed to be an individual particle interactingwith the casein gel matrix (active filler).

The stiffness of a composite gel filled with active particlesdepends on the concentration and the physical properties ofthe fillers (Tolstoguzov and Braudo 1983; van Vliet 1988). Inmost mathematical models, the concentration and the G′ ofthe fillers were considered while the other factors, e.g., theshape and size of the fillers, were not. Some of the physicalproperties of the fillers will be discussed as follows.

The effects of shape, size and size distribution of fillers onthe rheological properties of a composite gel have been inves-tigated by several groups (Dickie 1976; Richardson et al. 1981;Rosa et al. 2006). The shape of fillers can alter their reinforceeffects in an order of rods > plates > cubes > spheres, whilethe effects of particle size and size distribution are not verysignificant compared with the shape effect (Dickie 1976;Richardson et al. 1981; Rosa et al. 2006; Manski et al. 2007).The shape, size and size distribution of particles are not ofteninvolved in the mathematical models, in which the fillers areassumed to be spheres of the same size and homogeneouslydispersed (Kerner 1956; van der Poel 1958). The microstruc-ture of Cheddar cheese shows fat globules of larger size andless spherical shape with increasing fat content, correspond-ing to increasing fat globule area and decreasing shape factorin images (Fig. 1; Rogers et al. 2010). Although a few fat glob-ules showed nonspherical shape because of coalescence, espe-cially in cheese with 33% fat, the average shape factor of fatglobules ranged from 0.60 to 0.90 (Rogers et al. 2010), sug-gesting that the fat globules can be better described as spheri-cal shapes rather than rods or plates. In addition, themaximum volume fraction of fat globules is closer to 0.64than a higher value (0.74 or 1), according the goodness-of-fitof the selected mathematical models – the modified Kerner’smodels 1 and 2 and the modified Palierne’s models 1 and 2(Table 4), suggesting a random packing geometry of spheres.

McClements et al. (1993) evaluated the fracture strength ofemulsion gels containing oil droplets of different sizes andfound that the gel strength decreased with increasing oildroplet size when the mean droplet diameter is <1 mm andreached a lower level, which did not change much withdroplet size, when the mean droplet diameter is >1 mm. Theysuggested that the fillers may disrupt the gel network if theyare larger than the pore size of the gels (McClements et al.1993). This effect is only significant when the filler diametersare small enough to compare with the pore size of the gels(<1 mm for whey protein isolate gels in McClements et al.1993). The mean diameter of fat globules, as calculated fromthe mean area of fat globules in the two dimensional micro-



scopic images of cheese intersections, is 2–4 mm and muchlarger than the pore size of the gels (Fig. 1; Rogers et al. 2010).This disrupting effect should not be significant amongcheeses of different fat contents. Xiong et al. (1991) andMichalski et al. (2002) found that the smaller fat globulesshowed greater enhancement to the milk gel fracture proper-ties and shear modulus than the larger globules at the same fatcontent, because of an increase of the globule numbers andmore surface area per volume for filler–matrix interactionswith size decreasing. They explained this as the active fat glob-ules are part of the gel network by forming connections withcaseins; therefore, an increase of fat globule number canincrease the density of cross-link in the composite gels, result-ing in a stronger gel texture (Xiong et al. 1991; Michalski et al.2002). However, this effect cannot be quantified based on thecurrent mathematical models. Although the size of fat par-ticles in cheeses gradually grows with increasing fat content(Rogers et al. 2010), the object (fat particle) number per areaimage is at the same level in cheeses of 8–33% fat contentexcept for the cheese of 3% fat content, which shows less par-ticles than others (data not shown). The size of fat globulescan also influence their stiffness by changing the shearmodulus because of interface deformation (Ginterface = 2g/R).This effect is more significant for liquid oil droplet (van Vliet1988) than for solid fat globules (Everett and Olson 2003).

Although the effects of fat content on the cheese rheologyhave been investigated, the application of theoretical modelsin this area can provide new insights and theoretical supportfor previous studies (Guinee et al. 2000; Lucey et al. 2003;Rogers et al. 2010). The contributions from the protein gelmatrix and fat particles to cheese rheology can be distin-guished in the mathematical models, allowing precise analysisof the changes of the two parts at various conditions, such astemperature and aging time. The temperature effect on thecheese rheology was proposed to be mainly because ofmelting of fat globules at T < 40C, while the aging effect wasattributed to the proteolysis of casein gel matrix (Guinee et al.2000; Lucey et al. 2003). These hypotheses are supported bythe estimated stiffness of fat globules and gel matrix at varioustemperature and aging time. The cheese G′ was strongly rein-forced by the Gf′ at 10 and 15C but dominated by the Gm′ at 20and 25C, while the Gm′ decreases as aging cheeses for a longertime, leading to a decrease of the cheese G′ at 20 and 25C(Table 5). The rheological properties of cheeses can also bealtered by the surface materials on fat particles as observed inmodel gels (van Vliet 1988; Cho et al. 1999; Everett and Olson2003). A quantitative analysis may attribute the effect ofsurface materials (or the filler–matrix interaction) to theeffective stiffness of fillers. Therefore, it is possible to predict aproper combination of the fillers (fat or other particles), thegel matrix (casein gel) and the coating materials (proteinsand/or surfactants) for a desirable composite gel texture(cheese) at a certain condition, e.g., refrigerating temperature

or room temperature. This provides an approach to screen fatreplacers to achieve a target cheese texture. However, someimportant factors are still absent in the current theoreticalmodels, e.g., shape and size of the fillers. A further develop-ment of the mathematical models based on either the theo-retical aspect or the empirical aspect is necessary for betterunderstanding of the role of fat globules in cheese texture.

CONCLUSIONS

The relationship between the fat content and the cheese G′ arewell described using the theoretical models established forsuspensions and particle-filled gels, among which the twomodified Kerner’s models (Lewis and Nielsen 1970) and thetwo modified Palierne’s models (Pal 2008) best described thedata. A maximum filler volume fraction of 0.64 leads to betterfit than 0.74 in the four theoretical models, suggesting arandom packing geometry of fat globules in cheeses. The esti-mated Gf′ is much larger than that of the Gm′ at 10C; however,it decreased in a faster rate than Gm′ with increasing tempera-ture and reached an almost equal value as Gm′ at 25C. There-fore, the strengthening effect of the fat content to the cheeseG′ becomes less significant as temperature increases becausethe fat globules transform from relatively rigid fillers to softfillers. The estimated Gf′ is always larger than zero, suggestingthat the fat globule behaves as an active filler in the cheeses.

ACKNOWLEDGMENTS

Paper No. FSR10-20 of the Journal Series of the Departmentof Food, Bioprocessing and Nutrition Sciences, North Caro-lina State University, Raleigh, NC 27695-7624. Support fromthe North Carolina Agricultural Research Service, DairyManagement Inc. and the Southeast Dairy Foods ResearchCenter are gratefully acknowledged. The use of trade namesin this publication neither implies endorsement by the NorthCarolina Agricultural Research Service of the productsnamed nor criticize similar ones not mentioned. This projectwas part of a collaboration with Utah State University and wewould like to thank the Department of Nutrition and FoodSciences at Utah State for the production and aging of thecheeses. The technical support of Paige Luck is gratefullyacknowledged.

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