Describing the firmness, springiness and rubberiness of food1

gels using fractional calculus. Part I: Theoretical framework.2

T.J. Fabera,b,1,∗, A. Jaishankarc, G.H. McKinleyc3

aFrieslandCampina, PO box 1551, 3800 BN Amersfoort, The Netherlands4bPolymer Technology, Eindhoven University of Technology, PO Box 513 ,5600 MB Eindhoven, The Netherlands5

cDepartment of Mechanical Engineering, Massachusetts Institute of Technology,6

77 Massachusetts Avenue, Cambridge - MA 02139, USA7

Abstract8

Constitutive models for soft solids that quantitatively relate the state of the stress in the mate-

rial to the deformation history have the potential to be used in a structure-texture engineering

context, but successful examples are scarce. In the present work we define equations for the

firmness F, springiness S , and rubberiness R, of semi-soft food gels such as cheeses that ex-

hibit broad power-law stress relaxation over a wide range of timescales. The equations contain

only two material properties, which have their origin in the food microstructure: a fractional

exponent, which quantifies the frequency and temporal response and secondly a scale factor

or “quasi-property”, which sets the magnitude of the stress in the material. Together they form

a constitutive element, known as the ‘springpot’ or Scott Blair element which can accurately

capture the viscoelastic properties of food gels such as semi-hard cheeses. Using this model

it becomes possible to provide clear and unambiguous definitions of textural parameters such

as firmness, springiness and rubberiness, and to quantify their time- and interpendency. The

magnitude of the firmness and springiness are inversely related through the fractional con-

stitutive model. Our FSR-equations can be used in a texture engineering context to guide

effective product reformulation of soft-solid, hydrocolloidal gels.

Keywords: rational reformulation, food gels structure-texture engineering, constitutive9

model, fractional calculus, Scott Blair10

1. Introduction11

Product reformulation is a major theme in the food industry, which is driven by raw ma-12

terial costs, carbon footprint, or health incentives (Norton et al., 2006; Almeida-Rivera et al.,13

2007; Foegeding and Davis, 2011; Renard et al., 2006). However, when replacing or removing14

ingredients, sensory texture, shelf life and processability can be negatively affected (Renard15

et al., 2006). It is the task of the food product engineer to deliver solutions that overcome16

∗corresponding authorEmail address: timo.faber@frieslandcampina.com (T.J. Faber)

Preprint submitted to Food Hydrocolloids May 16, 2016

these trade-offs. Macro-scale food properties such as sensory texture are, to a large extent, de-17

termined by the microstructure of the food (Aguilera, 2005; Lucey et al., 2003). A wealth of18

engineering solutions are available to modify food microstructure based on insights from soft19

matter and colloidal science (Van der Sman and Van der Goot, 2009; Mezzenga et al., 2005;20

Stokes and Frith, 2008; Mezzenga and Fischer, 2013; Dickinson, 2012; Ubbink et al., 2008;21

Norton and Frith, 2001) as well as from process technology (Foegeding and Davis, 2011;22

Jousse, 2008; Almeida-Rivera et al., 2007). Transitioning between the micro- and the macro-23

scale and dealing with a multitude of constraints requires a rational and integrated product and24

process design approach (Norton et al., 2006; Almeida-Rivera et al., 2007), using models that25

can quantitatively relate food formulation and structure to processing and sensory texture.26

The fields of rheology and soft matter science have the potential to deliver such mod-27

els (Bourne, 2002a; Scott Blair, 1958; Mohsenin, 1970; Foegeding et al., 2003) especially28

when expressed in the form of a general frame-invariant constitutive relationship (Gallegos29

et al., 2004) for the essential rheological properties (Reiner, 1971) describing the material30

response to deformation. For non-food materials (e.g. in the polymer processing industry),31

application of rheological models to improve process throughput and product performance32

has now become standard; however, successful examples in the texture engineering field are33

scarce. Probable causes are the additional non-rheological factors that contribute to sensory34

judgment, such as phase transformations (Bourne, 1978), interaction with saliva (Kapsalis35

and Moskowitz, 1978; Choi et al., 2014) and repeated macroscopic fracture while the food36

material is orally processsed (Bourne, 1975; Chen, 2009).37

The firmness F, springiness S , and rubberiness R, of food gels, are examples of what38

Reiner (1971) called ‘technological properties’ rather than true material properties and these39

texture attributes are typically assessed solely by pressing and depressing the material by hand40

(Fig. 1) (Szczesniak and Bourne, 1969; Goh et al., 2003; Foegeding and Drake, 2007). Such41

manually-evaluated texture attributes can also be measured instrumentally using bulk rheolog-42

ical testing (Stokes et al., 2013) which makes these texture attributes excellent candidates for43

a quantitative structure-texture model which is based on analysis of the imposed kinematics44

using appropriate constitutive equations (or equations of state) for the material. These con-45

stitutive relationships should also then predict how well the product performs under handling46

conditions other than those of sensory texture assessment (Scott Blair and Coppen, 1940c),47

such as whether the product retains its intended shape when stacked and stored (Bourne, 1975)48

(Fig. 1), as well as provide insight into possible product reformulation strategies.49

Firmness is one of the most researched texture attributes for a wide variety of foods50

(Szczesniak and Bourne, 1969). The most common route to enhance the firmness of a food51

2

product is by transforming one or more of the hydrocolloidal food constituents into a gel.52

A second important function of gelation is to keep water and fat in the product, implying53

that firmness and composition are intrinsically related (Foegeding and Davis, 2011). Exam-54

ples of gelled foods are numerous and include yoghurt, cheese, starch puddings (Sala et al.,55

2008), hotdogs and sausages (Sala et al., 2008; Salvador et al., 2009), seafood analogues56

(Herranz et al., 2012) and tofu (Sato, 2003). When consumed, these food gels are masti-57

cated into smaller pieces and mixed with saliva to form a bolus that can be swallowed (Chen,58

2009; Hutchings and Lillford, 1988). The sensations arising when the product is touched59

and squeezed are good predictors for the sensations perceived in the first steps of mastication60

(Barrangou et al., 2006; Drake et al., 1999).61

Firmness (F) is determined while the food material is being pressed, whereas the texture62

attribute of springiness (S ) is measured after depressing the piece of food (Szczesniak, 1963a;63

van Vliet, 1991; Foegeding and Drake, 2007; Barrangou et al., 2006; Cakir et al., 2012). In64

former days, craftsmen in cheese production used to assess both the firmness as well as the65

‘spring’ of the cheese curds to determine if this precursor in the cheese-making process was66

ready to proceed to the next stage of processing (Scott Blair and Coppen, 1940a,b). For67

bread, springiness is a desired property (Matos et al., 2014) and this texture attribute is used68

as a measure for being fresh or stale (Purhagen et al., 2011). Springiness and rubberiness69

are two texture attributes that are often intertwined (Jowitt, 1974) and for both terms the70

synonym elasticity is sometimes used (Davis, 1937; Scott Blair, 1973; Hayakawa et al., 2014;71

Devezeaux de Lavergne et al., 2015; Brenner et al., 2015). Rubbery, springy, or elastic food72

gels are very popular in Japan (Brenner et al., 2015); however, they are also linked to eating73

difficulties (Funami et al., 2012; Hayakawa et al., 2014). In cheese, a Muenster type is allowed74

to be springy, whereas a Parmesan is not (Foegeding and Drake, 2007). ‘Rubbery’ has been75

used as a negative descriptor for cheese that has reduced fat content (Yates and Drake, 2007).76

George William Scott Blair, who is considered by many to be the founding father of food77

rheology (Szczesniak, 2002; Bourne, 2002b), had a special interest in the subjective aspects78

of firmness, which led to ground breaking panel studies (Scott Blair and Coppen, 1940c). He79

laid the foundations of a special branch of rheology called psychorheology (Scott Blair, 1947),80

which later evolved into the field of texture studies (Scott Blair, 1974). His central question81

was: What material property gives rise to key texture properties like the ‘body’ of dough82

and the ‘firmness’ of cheese? His approach was as simple as it was effective: ask panels to83

squeeze two graspable complex materials with known material constants and which are purely84

viscous (bitumen), purely elastic (rubber) or viscoelastic (cheese, unvulcanized rubber). He85

asked the panelist to judge which of the two materials they squeezed was softest with the86

3

restriction that the two objects were pressed at the same time, and for a well defined period.87

The conclusions were threefold. First human touch is very sensitive in discriminating between88

softer and firmer objects, even if one is purely elastic and the other purely viscous. Second,89

firmness is a time-dependent property and is judged dynamically, which is in line with the90

notion that the perception of sensory texture is the result of measuring a dynamic process91

rather than measuring a final state (Hutchings and Lillford, 1988; Devezeaux de Lavergne92

et al., 2015). Finally Scott Blair demonstrated that the time-dependence of the firmness for93

many of the complex materials he studied has the form of a power-law in time.94

Thus in rheological terms the firmness, springiness, and rubberiness of food gels can be95

considered as viscoelastic texture attributes. This suggests the use of constitutive models that96

correlate these attributes to true linear viscoelastic (LVE) material parameters characterizing97

the materials of interest. Various models have been fitted to material property data of cheese98

obtained from rheometry in the LVE region to correlate composition and processing condi-99

tions to cheese material properties. The most popular model is the generalized Maxwell model100

(Tschoegl, 1989), that has been applied by numerous authors (Hort and Le Grys, 2000; Sad-101

owska et al., 2009; Subramanian and Gunasekaran, 1997; Muliawan and Hatzikiriakos, 2007;102

Campanella and Peleg, 1997; Subramanian et al., 2006; Li Yuet Hee et al., 2008; Bertola et al.,103

2000; Venugopal and Muthukumarappan, 2001). To describe measured LVE behavior accu-104

rately over two decades of time, at least three relaxation modes (with six model parameters)105

are typically needed, from which it is hard to derive insight into structure-texture relations.106

Furthermore the fitted results are not valid outside the domain of measurement making ex-107

trapolation difficult and thus the extracted model parameters lack a strong physical underpin-108

ning (Kollmannsberger and Fabry, 2011; Jaishankar and McKinley, 2013) which is essential109

for a connection to the underpinning microstructure. One approach to reduce the number of110

parameters in the mechanical model is to fit a continuous spectrum of relaxation times to the111

material functions determined in the LVE regime (Winter and Mours, 1997; Del Nobile et al.,112

2007; Sadowska et al., 2009; Lorenzo et al., 2013). Again the spectral distribution parameters113

that are obtained lack physical meaning.114

When a creep / recovery test is performed to measure the temporal response of a food gel115

to deformation, the Burgers model is widely used to interpret the material properties that are116

measured (Scott Blair and Burnett, 1959; de Faria et al., 2013; Dolz et al., 2008; Harrington117

and Morris, 2009; Korus et al., 2009; Lorenzo et al., 2013; Meng et al., 2014; Moreira et al.,118

2011; Olivares et al., 2009). This four-parameter model is capable of describing viscoelas-119

tic deformation and the unrecoverable strain at the end of recovery (Scott Blair and Burnett,120

1959). It comprises of a spring and dashpot in series (Maxwell model) attached to a spring and121

4

dashpot in parallel (Kelvin-Voigt model). Scott Blair (1973) proposed to measure springiness122

S , by a dimensionless number composed from the characteristic relaxation time τr, and retar-123

dation time τret in the Burger model, S = τr/τret. However he also concluded that to capture124

more complex relaxation behaviour, the addition of an extra Kelvin-Voigt body is necessary,125

which makes the modeling increasing phenomenological in nature (Scott Blair and Burnett,126

1959).127

Various authors have independently recognized the power-law relaxation characteristics128

in food gels when performing Small Amplitude Oscillatory Shear (SAOS) tests and they have129

fitted emperical models of the form G′(ω) = Aωb and G′′(ω) = Cωd (where G′(ω) and G′′(ω)130

are the storage and loss modulus, respectively) to the measured frequency data. Here A, b,131

C and d are constants and ω the imposed oscillatory frequency. (Subramanian et al., 2006;132

Korus et al., 2009; Moreira et al., 2011; Ronda et al., 2013). Zhou and Mulvaney (1998) and133

Gabriele et al. (2001) further reduced the number of constitutive fitting parameters from four134

to two by fitting a similar model to the magnitude of the complex modulus |G∗(ω)|. In all these135

papers no attempt was made to represent these functional forms in the form of a constitutive136

model that could be applied in a predictive form to describe other deformations.137

Scott Blair (1947) was the first to focus on developing constitutive equations where the138

stress σ(t) is related to the strain γ(t) or strain rate γ(t) in the form of a power-law. He started139

with the Nutting equation (Nutting, 1921; Scott Blair and Coppen, 1942), Ψ = σβγ−1tk, where140

t is the time, β and k are constants and Ψ a “firmness intensity” (Scott Blair, 1943). From his141

experiments he concluded that the number of parameters in this equation could be reduced142

by writing the stress in terms of a fractional differential equation σ = Ψ(dβγ/dtβ). He called143

attention to two novel rheological concepts, the use of a fractional derivative with a fractional144

exponent (β), and the existence of a ‘quasi-property’ Ψ (Scott Blair et al., 1947). Scott Blair145

argued that such a quasi-property is intermediate between a shear modulus G and a viscosity146

η and is a measure of the dynamic relaxation processes within a material rather than of an147

equilibrium state. The magnitude of this quasi-property (here a “firmness intensity”) sets148

the scale of the stress in the material, whilst the fractional exponent quantifies the frequency149

and temporal response. The fractional constitutive framework was further developed in the150

1980’s by Bagley and Torvik (1983b,a) and Koeller (1984). The latter introduced a new151

mechanical element, the springpot, also referred to as the Scott Blair element (Mainardi and152

Spada, 2011), which is quantified by two material parameters; the quasi-property and the153

fractional exponent.154

Successful fitting of a fractional constitutive model to measured linear viscoelastic re-155

sponses can be found for red blood cells (Craiem and Magin, 2010), lung tissues (Suki et al.,156

5

1994) and arteries (Craiem and Armentano, 2007). A selective number of other examples can157

be found for describing food ingredients like gums (Jaishankar and McKinley, 2013) emul-158

sifiers (Ma and Barbosa-Canovas, 1996; Chen and Kim, 2004) and melon (Xu and Chen,159

2013). Bagley and Torvik (1983b) proved that at sufficiently high frequencies, the Rouse the-160

ory for polymer melts can also be compactly expressed in terms of an equivalent fractional161

constitutive equation containing a springpot with exponent β = 1/2. Several works (Heymans162

and Bauwens, 1994; Schiessel et al., 1995) have built constitutive models by hierarchically163

ordering springs and dashpots in self-similar structures, like those encountered in biological164

materials, and showed that the hierarchical model asymptotically approaches a springpot.165

The microstructural origins of the power-law relaxation commonly observed in complex166

materials can also be found in the distribution of lengths in dangling chains in imperfectly167

cross-linked elastomers (Curro and Pincus, 1983), the distribution of energy wells around168

the interacting elements that form soft glassy materials (Sollich et al., 1997; Jaishankar and169

McKinley, 2013) and in the continuous distribution of waiting times for a particle that diffuses170

in a heterogeneous structure (Bouchaud and Georges, 1990; Amblard et al., 1996). Tracking171

the trajectory of such a particle, as it diffuses through the heterogeneous structure, is a form172

of micro-rheology. Caggioni et al. (2007) used this technique in combination with shear173

rheology to characterize gellan gum micro gels. They showed that both bulk- and micro-174

rheological measurements displayed power-law frequency sweeps with identical slopes, and175

that this power-law slope changed when the microstructure was altered.176

In the remainder of the paper we build on the efforts of Davis and Scott Blair to link cheese177

texture attributes to essential rheological properties that can be extracted from rheometry and178

we proceed through the steps of building a structure-texture model based on a fractional con-179

stitutive framework. First we introduce the main mathematical relations that are required to180

construct this framework. We show that from these parameter values we can correctly pre-181

dict the evolution of both the relaxation modulus G(t) and creep compliance J(t). We next182

introduce exact definitions of the firmness F, springiness S , and rubberiness R, of food gels183

in terms of specific points on the creep / recovery curve. Using the tools of fractional calculus184

this allows us to derive expressions for the material’ firmness, springiness, and rubberiness in185

terms of the quasi-property and the power-law exponent that characterize the cheese. We show186

that we can quantitatively predict springiness and rubberiness from the measured /fitted creep187

curve which demonstrates that firmness, springiness, and rubberiness are interrelated and all188

governed by the two material parameters we determine in the linear viscoelastic regime. We189

finally outline how our equations can be used in the context of structure-texture design. In part190

II of this work we extend these texture engineering ideas to characterizing a full-fat, reduced-191

6

fat, and zero-fat cheese products over a wide range of temperatures and water / protein ratios.192

We also quantify the effect of large loadings that exceed the linear viscoelastic limit, common193

in oral processing (Goh et al., 2003), on the measured magnitude of firmness, springiness and194

rubberiness.195

2. Fractional constitutive framework196

To construct fractional constitutive equations in an identical fashion as classical constitu-197

tive models (e.g. the Maxwell model), Koeller (1984) defined a rheological element whose198

stress is proportional to the fractional derivative of the strain. He called this two-parameter199

viscoelastic element the ‘springpot’:200

σspingpot = Gdβγdtβ

(1)

This constitutive response is a generalization of the classical dashpot and spring: for β = 1,201

the springpot behaves as a dashpot, whereas for β = 0 it describes a linear elastic spring, as202

depicted in Figure 2(a). The front factor, denoted G, is equivalent to Scott Blair’s quasi-203

property (originally denoted Ψ) and has units of Pa sβ. The exponent β is a measure of the204

frequency or temporal dependence of the material response, as well as the relative degree of205

viscoelasticity, and, as such is related to the phase angle, tan(δ) in the material. The latter206

parameter is more commonly used to describe the solid- or liquid- like nature of semi-solid207

food gels such as cheese (Foegeding et al., 2011). Various material responses can be described208

with constitutive models derived from mechanical analogues that are constructed with the209

springpot, and comprehensive reviews can be found in Schiessel et al. (1995) and Friedrich210

et al. (1999) .211

The fractional constitutive framework builds on ideas from fractional calculus, that incor-212

porate integrals and derivatives of arbitrary order (Podlubny, 1999; Mainardi, 2010; Jaishankar213

and McKinley, 2014). For compactness these derivatives are called fractional derivatives and214

we give models, expressions and parameters derived from these derivatives the adjective ’frac-215

tional’ as well as materials that can be mechanically described by these entities.216

There are several alternative definitions for the fractional operator generically indicated by217

Eq. (1). To conform to the initial condition for appropriate rheological tests of γ|t=0 = 0, we218

use the Caputo derivative, which reformulates the fractional derivative into a fractional integral219

through integro-differentiation (Surguladze, 2002). The derivative in Eq.(1) with order 0 <220

7

β < 1 is then defined as :221

Gdβγdtβ≡

G

Γ(1 − β)

t∫0

(t − t′)−βγ(t′)dt′ (2)

The constitutive response of the springpot (Koeller, 1984; Torvik and Bagley, 1984) is thus222

equivalent to a Boltzmann memory integral with a power-law relaxation kernel, rather than223

the more familiar Maxwell-Debye exponential form (or sum of exponentials) commonly used224

in viscoelastic models (Ferry, 1980). The Caputo derivative has a Laplace transform, which225

for 0 < β < 1 is defined as (Podlubny, 1999; Surguladze, 2002):226

L

{dβ

dtβγ(t); s

}= sβγ(s) (3)

This enables us to derive analytical expressions for two standard linear viscoelastic shear227

material functions (Bird et al., 1987): the relaxation modulus G(t) and the creep compliance228

J(t). In a stress relaxation experiment, a sudden step in shear strain γ(t) = γ0H(t) is imposed229

(where H(t) is the Heaviside step function (Abramowitz and Stegun, 1964)), and the resulting230

evolution of the stress σ(t) in the material is measured. This provides the relaxation modulus231

G(t) ≡ σ(t)/γ0 for the material. By analogy the creep compliance, J(t) ≡ γ(t)/σ0 is obtained,232

by imposing a step in the shear stress acting on the sample, σ(t) = σ0H(t), and following the233

evolution of the strain γ(t) in the material over time.234

Other standard linear viscoelastic shear material functions, such as the storage and loss235

moduli {G′(ω), G′′(ω)}, are obtained by harmonic excitations in shear, also denoted as small236

amplitude oscillatory shear (SAOS) experiments. To find the appropriate fractional expression237

for these functions we use the Fourier transform of the Caputo derivative of the function γ(t),238

which is given by (Schiessel et al., 1995)239

F

{dβ

dtβγ(t);ω

}= (iω)βγ(ω) (4)

where γ(ω) = F {γ(t);ω}.240

Equations (2), (3) and (4) provide the mathematical framework to build fractional consti-241

tutive equations from mechanical analogues and solve them analytically. In this paper we will242

use fractional models consisting of a single springpot, (i.e. the canonical Scott Blair model243

(SB)) as well as two springpots arranged in series, which we call the Fractional Maxwell244

model (FMM). In the latter four-parameter model each element is described by a quasi-245

property (which we denote V or G) and an exponent (α or β), where we take α > β with-246

out loss of generality. This constitutive model is schematically depicted in Fig. 2(b). In the247

Maxwell model the stresses in each element are equal (σ = σ1 = σ2) and the strains are248

8

additive (γ = γ1 + γ2). With the constitutive relation for each springpot (Eq.(1) and Fig. 2(b))249

this gives the fractional Maxwell model (FMM):250

σ(t) +V

G

dα−βσ(t)dtα−β

= Vdαγ(t)

dtα(5)

where 0 ≤ β ≤ α ≤ 1.251

Materials that are described by a single power-law (e.g. Eq. (2)), have no characteristic252

timescale that can be identified. A discrete spectral representation of the power-law kernel253

in Eq. (2), requires an infinitely broad spectrum of time constants (Ng and McKinley, 2008).254

However for a material displaying two power-law regimes, as described by the FMM, we can255

unambiguously define one single characteristic timescale τc by the parameter combination:256

τc = (V/G)1/(α−β) (6)

This time corresponds to the characteristic time (or frequency ωc = 1/τc) at which the local257

power-law in the material response, transitions from one exponent (α) to the other (β). This258

expression also reduces to the expected value for the limit of a classical Maxwell model for259

which V → η, G→ G, α→ 1 and β→ 0, so that τc = η/G.260

3. Results261

3.1. Rheological definitions of firmness, springiness, and rubberiness262

There is no consensus on a unique definition of the firmness, springiness, and rubberiness263

of food gels such as cheese, nor its method of assessment (Foegeding and Drake, 2007), which264

hampers correlation of these texture attributes to essential material properties. The firmness F,265

can be judged while deforming a piece of cheese with the mouth (tongue and palate, incisors,266

front teeth or molars) or by hand (Fig. 1(a)). During the assessment, the cheese can be slightly267

pressed or completely bitten through (Drake, 2007). When choosing a method of assessment,268

panelists tend to make a choice that depends on the level of firmness itself (Szczesniak and269

Bourne, 1969).270

Correlative studies between panel test results and instrumental measurements do not pro-271

vide additional clarity on how firmness should be defined. It has been correlated to rheologi-272

cal properties covering the regimes of linear to non-linear rheology up to fracture (Foegeding273

et al., 2003; van Vliet, 1991). Since the development of the Texturometer (Friedman et al.,274

1963) and the entrance of the Instron Universal Testing Machine in texture measurements275

(Bourne, 1968), the uniaxial compression experiment has become very popular to assess firm-276

ness and is still deployed regularly (e.g. Liu et al. (2015); Cakir et al. (2012)). In this ex-277

periment (Fig. 1(b)), a cylindrical sample is crushed between two parallel plates either at a278

9

constant rate of displacement (Friedman et al., 1963; Bourne, 1968) or at constant strain rate279

(Luyten, 1988; Goh et al., 2003). In an alternative configuration, firmness is determined with280

a penetration test, where the upper plate is replaced by an indenter and the test material is left281

to rest in the cup or beaker in which it was prepared (e.g. Li Yuet Hee et al. (2008); Oliver282

et al. (2015); Bot et al. (2014)). Such indentation tests deform the material locally and are of283

specific relevance when the materials with the lowest firmness are too weak to hold their own284

weight. When using maximal force as a measurement for firmness, it appears that the relative285

firmness of two types of cheese depends both on the rate and the degree (or extent) of com-286

pression (Shama and Sherman, 1973; Bourne and Comstock, 1981). This gives an indication287

that when measured instrumentally, firmness has both a viscoelastic as well as a plastic (or288

flow) component.289

Firmness is often intertwined with hardness (Jowitt, 1974; Peleg, 1980; van Vliet, 1991).290

Jowitt (1974) prefers the term firm over hard and defines it as ’high resistance to deforma-291

tion’. Reiner and Scott Blair (1967) define hardness as a ’resistance to penetration’, i.e. the292

resistance to a specific type of deformation, which can be instrumentally quantified with the293

indentation test described above. They use ‘suppleness’ as an antonym for ‘hardness’ whereas294

‘soft’ is the antonym for ‘firm’ (Jowitt, 1974). Van Vliet (1991) proposes the use of firmness295

for recoverable viscoelastic deformations only and hardness for non-recoverable plastic de-296

formations. This is in line with Scott Blair’s findings that firmness is a time-dependent texture297

attribute.298

Two alternative studies indicate that firmness can indeed be measured with loading con-299

ditions that remain within the linear viscoelastic regime of the food material. The first is a300

series of comparative experiments on Monterey Jack and Mozarella cheese between an ex-301

pert panel and rheometry. Brown et al. (2003) showed that the maximum compliance, Jmax =302

max {J(t)}, determined in a 600 s creep test (Fig. 1(c)) in the linear viscoelastic region had303

a strong negative correlation with both hand- and mouth-evaluated firmness. Ewoldt (2013)304

demonstrated that by plotting the creep compliance J(t), the best discrimination between a305

soft and firm grade of therapy putty could be made. Ewoldt also showed that firmness is a306

sensory texture attribute that cannot only be measured with tactile senses but also by vision307

(Bourne, 2002b) as long as one probes the property over long time-scales. A soft material308

will lose shape rapidly over time, an undesirable feature for a cheese that needs to be sliced309

(Fig. 1(a)).310

In Fig. 3 we have summarized the findings from Scott Blair and Coppen (1940c); Scott Blair311

(1947), Brown et al. (2003) and Ewoldt (2013). In Fig. 3(a) we plot the compliance against312

time for a creep / recovery experiment (see also Fig. 1(c)) on two hypothetical materials.313

10

Firmness F, is defined as a resistance to creep, and can be expressed by the inverse of the314

maximum compliance J(t) measured at the end of the creep phase. The time at which we315

measure a texture attribute is generically called the time of observation to. For firmness this316

time is denoted as t f = to. Thus the firmness F is defined as317

F ≡ 1/max{J(t)} = 1/J(t f ) (7)

and has units of Pa. The datapoints that are used to calculate the firmness F, of materials318

A and B are indicated by the filled, colored square on the creep / recovery curve in Fig. 3.319

The corresponding time of observation t f is indicated by the hollow square on the time-axis.320

The two hypothetical materials A and B are equally firm at the time of observation t f = 10 s,321

however the two creep curves approach J(t f ) ≡ max {J(t)} differently. In Fig. 3(b) we have322

plotted the creep phase of the same experiment on a double logarithmic scale and extrapolated323

the curves beyond the time of observation t f (dashed line). It shows that the two materials each324

have the form of a power-law with different slopes β. The plot also emphasizes the importance325

of the time of observation in measuring firmness. If we had chosen a smaller observation time,326

material A would have been considered firmer than B and vice versa at longer times.327

The term ‘elastic’ is often used as a synonym for ‘springy’ (Davis, 1937; Devezeaux de328

Lavergne et al., 2015; Scott Blair, 1973). The word ‘elasticity’ was taken up in the list of329

primary parameters in the Texture Profile Analysis (TPA) (Szczesniak, 1963a). Davis (1937)330

pointed out that the use of ‘elasticity’ is not correct and that should be reserved for the as-331

sessment of a modulus by hand. More recently ‘elasticity’ has been replaced by ‘springiness’332

(Szczesniak, 2002) in Texture Profile Analysis and it is defined in TPA as ‘the rate at which a333

deformed material goes back to its undeformed condition after the deforming force has been334

removed’ (Szczesniak, 1963a). Davis and Blair argued that springiness is related to a time335

of relaxation (Davis, 1937; Scott Blair, 1973) and that the sensation involved is an amount336

of recovery instead of a rate. Based on the graphical definition of springiness in Friedman337

et al. (1963), van Vliet (1991) concluded that the springiness S indeed should be defined as338

an ‘extent’ to which a deformed material returns to its undeformed state. In their review of339

texture definitions, Foegeding and Drake (2007) identify authors that speak of ‘a rate at which340

sample “springs” back’, as well as those who speak of an ’extent of recovery’.341

Van Vliet (1991) combines both rate and extent of deformation in one definition by propos-342

ing ‘the tendency to instantaneously and completely recover from a large deformation after343

removal of the deforming force’, however he prefers to use the term rubberiness instead of344

springiness. Foegeding and Drake (2007) link rubberiness to springiness by defining it as ”the345

degree of rubberiness experienced when biting in a sample”. We propose that from a rheolog-346

11

ical perspective and based on the competing definitions above, it is justifiable and necessary347

to have two separate, distinct definitions for the springiness S and rubberiness R. Intuitively348

the word ‘springiness’ deals with sudden responses that are evaluated over a short period of349

time and thus the use of a rate is appropriate. As a consequence the term ‘rubberiness’ is used350

to relate to the total extent of strain that is recovered at the end of the creep / recovery exper-351

iment. In polymeric theories, rubbery materials are assumed to be purely elastic or perfectly352

cross-linked; in other words any applied deformation is reversed upon cessation of the appli-353

cation. By analogy, the term ‘rubberiness’ is used to relate to the amount of strain recovered354

at the end of a creep-recovery experiment.355

Our precise, rheological definitions for the springiness S and rubberiness R, follow from356

the location of the three markers on each creep / recovery curve in Fig. 3(a). For convenience357

we define the elapsed time of recovery as ∆t= t − t f (denoted by the dashed, horizontal black358

line in Fig. 3(a)). The springiness S is defined as the absolute secant rate of recovery just after359

the stress is released and is judged at a time of observation ts =t f + ∆ts. The springiness is360

thus unambiguously and mathematically defined as:361

S ≡

∣∣∣∣∣∣ J(ts) − J(t f )ts − t f

∣∣∣∣∣∣ =

∣∣∣J(t f + ∆ts) − J(t f )∣∣∣

∆ts(8)

and has units of 1 / Pa s, which is equal to the inverse of the units of viscosity. In practice one362

judges the springiness of a material such as cheese, by looking at the instantaneous response363

when the stress is released. It is thus logical to take ∆ts<< t f . We will use ∆ts= 0.1 s364

when we report values for springiness in the remainder of this paper, which is a time close365

to the response time of a modern rheometer at the time of publication. The location of the366

time of observation for measuring springiness is denoted on the time-axis is indicated by the367

hollow triangle in Fig. 3(a), which visually overlaps with the hollow square representing t f ,368

indicating that the elapsed time of recovery is very small. Note, that although the recovery369

time is short, the creep compliance of Material A (red triangle) has decreased considerably.370

This is reminiscent of a very springy material and is expressed by a secant (red dashed line)371

that is more steeply negatively sloped than the secant of material B (blue dashed line). Thus372

the materials A and B are equally firm, but not equally springy for a fixed time of observation373

t f .374

The rubberiness R, is defined as the extent to which a cheese returns to its original shape375

during the interval (t f , tr], where tr is the time we take for measuring rubberiness. If the strain376

is fully recovered at the time t = tr, then R = 1. If there is no strain recovery at t = tr , then377

the rubberiness R = 0. Thus R is mathematically defined as:378

12

R ≡J(t f ) − J(tr)

J(t f )= 1 −

J(t f + ∆tr)J(t f )

= 1 − F(t f )J(t f + ∆tr) (9)

which is a dimensionless quantity, and where ∆tr is the elapsed time of recovery for measuring379

rubberiness. In this paper we will use values of ∆tr= 10 seconds (hollow circle on the time axis380

in Fig. 3(a)) and ∆tr= 100 seconds, a time convenient to measure in the rheometer and which381

corresponds to practical times of judging the attribute. By choosing ∆tr>>∆ts we distinguish382

between the key features of the short and intermediate time response of power-law materials383

and make springiness and rubberiness two distinct attributes.384

To conclude, we want to address the suggestion that the definition of an antonym of a385

texture attribute (Jowitt, 1974) or the definition of an inverse of a rheological property (Reiner386

and Scott Blair, 1967) helps the comprehension of the differences between terms used. We387

propose to follow Jowitt (1974) and call a food gel with a low level of firmness ‘soft’. Foeged-388

ing and Drake (2007) defines ‘softness’ as easily ‘moldable’, we propose to use the latter term389

as an antonym for the rubberiness R. For the springiness S Jowitt (1974) proposes to use the390

antonym ‘plastic’, but this is a synonym for ‘moldable’ and has no sense of rate. We suggest391

the use of ‘squishy’ instead (Yates and Drake, 2007).392

In this section we have given precise rheological definitions of firmness, springiness, and393

rubberiness both in words as well as in terms of specific points on the creep compliance curve,394

J(t), of a creep / recovery experiment (Fig. 3). We will now use these rheological definitions to395

derive expressions for firmness, springiness, and rubberiness using the fractional constitutive396

framework. We subsequently refer to these definitions as the FSR-equations.397

3.2. Fractional equations for firmness, springiness, and rubberiness (FSR)398

For the analytical expressions of the creep compliance J(t), Laplace transformation (Eq. (3))399

of the constitutive equation for the springpot (Eq. (1)) is performed with a step shear stress as400

the input and then solving for the strain γ(t) by inverse Laplace transformation. For the Scott401

Blair model this gives the following expression for the compliance (Jaishankar and McKinley,402

2013):403

J(t) ≡γ(t)σ0

=1G

tβ

Γ(1 + β)(10)

and for the fractional Maxwell model404

J(t) =

(1V

tα

Γ(1 + α)+

1G

tβ

Γ(1 + β)

)(11)

As before V and Gare quasi-properties, and 0 ≤ β ≤ α ≤ 1 are the fractional exponents.405

For materials probed at strains within the linear viscoelastic limit which can be described406

by a single power-law, or springpot element, the compliance is given by Eq. (10). Using the407

13

definition of firmness in Eq. (7) the fractional equation for the firmness F of a Scott Blair408

element becomes409

F(t f ) =1

J(t f )= GΓ(1 + β)t−βf (12)

This expression for firmness is in line with Scott Blair’s observations (Scott Blair et al.,410

1947): it is a time-dependent texture attribute that depends on two material properties, the411

quasi-property G and the power-law exponent β, as well as on the time of observation t f . We412

have defined the firmness F, as being independent of the stress applied, which also corre-413

sponds to Scott Blair’s findings: although the stresses applied by panelists varied by a factor414

two, it did not affect their judgments of differences in firmness. Since the time of observation415

is essential for quantifying numerical values of the firmness, any measurement should always416

be provided together with the observation time t f and the characteristic power-law exponent417

β, to enable extrapolation to practical situations. The time dependency of the firmness F(t f ),418

favours a stress-controlled creep measurement (Fig. 1(c)) over the more common practice of419

reporting a modulus from a strain-rate-controlled compression experiment (Fig. 1(b)); the for-420

mer experiment naturally exposes time effects and allows us to interconnect the firmness to421

situations where stressess are applied for short times, such as sensory texture measurement,422

or for long times, such as in storing cheese (Fig. 1(a)) .423

The definition of the springiness S is given in Eq. (8). The point J(t f ) lies at the end of424

the creep curve and is given by the inverse of Eq. (12). The point J(ts) lies on the recovery425

curve and requires an appropriate fractional calculus expression for this part of the creep-426

recoil experiment, which is derived by Jaishankar and McKinley (2014) for the Scott Blair427

element:428

J(t) =tβ − (t − t f )β

GΓ(1 + β), for t > t f (13)

When we divide both nominator and numerator in Eq. (13) by tβf and substitute the equa-429

tion for firmness , Eq. (12), and our definition for the recovery time, ∆t= t - t f , we get430

J(t) =t−βf (tβ − ∆tβ)

F(t f ), for t > t f (14)

To derive an expression for the springiness S , we first define the elapsed recovery time at431

which we measure as ∆ts= ts − t f . The numerator of Eq. (8) is then expressed as432

J(ts) − J(t f ) =t−βf (tβs − ∆tβs )

F(t f )−

1F(t f )

, for ts > t f (15)

Substituting this in Eq. (8) and writing out ts = t f + ∆ts gives433

14

S =

∣∣∣(1 + ∆ts/t f )β − (∆ts/t f )β − 1∣∣∣

F(t f )∆ts(16)

The springiness equation reveals that this attribute is indeed inversely related to firmness.434

The time dependency of a springiness measurement is two-fold, it depends on the duration435

t f of the creep phase and the time ∆ts over which the secant rate is measured. We have436

depicted this two-fold time dependency in Fig. 4 where we have plotted springiness for the437

two hypothetical materials A and B from the previous section (Fig. 3). Recall that at a time438

t f = 10 s these two materials were equally firm, and that material A (with the smaller fractional439

power-law exponent) was the most springy. This is in line with what we read out from the440

solid lines in Fig. 4(a) when we take ∆ts= 0.1 seconds (for the elapsed time at which we441

measure springiness). However if we increase the value of the elapsed time ∆ts, we arrive at442

a point where we measure approximately equal springiness for both materials. For t f > 10 s443

even a crossover point appears as shown by the dashed lines for t f = 106 s. This crossover444

point shifts towards smaller time with increasing t f (data not shown). For small values of the445

elapsed recovery time ∆t = t− t f the springiness becomes independent of the total creep time446

t f . If we substitute the expression for firmness from Eq. (12) in Eq. (16) for times ∆ts that are447

small compared to the time of measuring firmness, t f , the latter time indeed cancels out:448

S ≈

∣∣∣∣∣∣∣∣ −(∆ts/t f )β

GΓ(1 + β)t−βf ∆ts

∣∣∣∣∣∣∣∣ =

∣∣∣∣∣∣ −∆tβ−1s

GΓ(1 + β)

∣∣∣∣∣∣ , for ∆ts << t f (17)

To derive an expression for the rubberiness R of the Scott Blair element, we begin with449

the definition given in Eq. (9). Substituting the time tr in Eq. (14) and subsequently defining450

this time in terms of elapsed recovery time, using tr = t f +∆tr gives:451

J(tr) =t−βf

((t f + ∆tr)β − ∆trβ

)F(t f )

, for t > t f (18)

When we substitute this equation for the compliance at the time at which we measure452

rubberiness, J(tr), into the mathematical definition of rubberiness, given by Eq. (9), we obtain453

the following expression:454

R = 1 − t−βf

((t f + ∆tr)β − ∆trβ

)(19)

It thus appears that rubberiness is not a function of the quasi-property G of the material455

directly, but of the two times ∆tr and t f , and the fractional exponent only. This two-fold time456

dependency is reflected in Fig. 4(b) where we have plotted the rubberiness R, for the same457

materials A and B as in Fig. 3 and Fig. 4(a). For rubberiness we do not observe a cross-over458

15

point of the two materials. However the plots of the rubberiness reveal another remarkable459

feature of this texture attribute: all power-law materials probed in the linear viscoelastic region460

will ultimately show a rubberiness approaching unity. Distinctions between the two materials461

only comes from differences in the fractional time exponent and the combination of values462

of t f and ∆tr chosen. In rheological terminology this means that the long term compliance463

of the recovery phase, modeled with a single springpot, will approach zero, so that all of the464

imposed strain is recovered elastically. Jaishankar and McKinley (2014) showed that this long465

term behaviour for J(t), Eq. (13), is approximated by466

J(t) ≈tβf

GΓ(β)

(tt f

)β−1

(20)

Since β < 1, the compliance thus monotonically approaches zero, which confirms the467

response in Fig. 4(b). The full elastic recovery of strain after a small applied deformation468

within the linear viscoelastic regime is a hallmark of protein gels (Leocmach et al., 2014).469

Our FSR-equations consist of only these material parameters and a specification of the470

time of measurement. This suggests that we may perform a a single firmness measurement471

(i.e. a creep test), determine the quasi-property G and fractional exponent β, and then obtain472

values for both springiness and rubberiness as long as we stay within the linear viscoelastic473

region; a separate recovery measurement is not required.474

Using fractional calculus, we have developed equations that enable the texture attributes of475

firmness, springiness, and rubberiness to be defined in terms of two constitutive material pa-476

rameters, which can be extended to four parameters if an even more accurate description over477

a much wider time scale is necessary. The FSR-equations show that in the linear viscoelastic478

regime these three texture attributes all time-dependent and coupled (to an extent that depends479

on the total experiment duration). From a single measurement of the linear viscoelastic creep480

compliance function J(t) we can accurately determine the magnitudes of the quasi-propertyG481

and fractional exponent β describing the material, and then predict values of the firmness, the482

springiness and the rubberiness. The firmness and springiness of a sample are anti-correlated483

through the magnitude of the quasi-property G, which sets the scale of the stresses in the484

material. By contrast, the rubberiness is not a function of the quasi-property, but only of the485

fractional exponent β. Note that if equal test times are used to measure firmness during creep,486

and to measure rubberiness during recovery, i.e. so that t f = ∆tr, Eq. (19) gives487

R = 2 − 2β (21)

and the rubberiness becomes time-independent.488

16

We emphasize that the FSR-equations (12),(16),(19) allow us to quantify the three texture489

attributes from any of the standard linear viscoelastic material functions, as we demonstrate490

in Fig. 5(a). Here we display a frequency sweep performed on zero-fat Gouda cheese, a soft-491

solid, hydrocolloidal gel, at a temperature of T = 10◦C and water/protein ratio of w/p = 1.8.492

The linear viscoelastic properties of the cheese show the typical power-law behaviour of a493

critical gel (Winter and Mours, 1997): a line of constant slope on a log-log plot over a wide494

range of frequencies for both the storage and loss modulus, {G′(ω),G′′(ω)}, with both curves495

nearly parallel. To retrieve the quasi-properties and exponents from these plots, we need an496

expression for the storage and loss modulus for the Scott Blair model. The complex modulus497

is obtained by Fourier transforming the constitutive equation for the springpot, Eq. (1), which498

results in:499

G∗(ω) = G(iω)β (22)

Following the procedure for separating out the real and the imaginary part, outlined by Friedrich500

et al. (1999) and Schiessel et al. (1995), one can readily find for the storage modulus501

G′(ω) = Gωβ cos (πβ/2) (23)

and for the loss modulus502

G′′(ω) = Gωβ sin (πβ/2) . (24)

The magnitude of the complex modulus, |G∗(ω)|, can be calculated from503

|G∗(ω)| =√(Gωβ cos (πβ/2)

)2+

(Gωβ sin (πβ/2)

)2= Gωβ. (25)

This set of equations shows that G′(ω) can be predicted from G′′(ω) and vice versa and that504

we can either fit equations (23), (24) or (25) to our dataset of G′(ω), G′′(ω), or |G∗(ω)| re-505

spectively. We choose Eq. (25) in combination with a least square optimisation procedure to506

obtain G and β, since it gives the least bias towards either the G′(ω) or G′′(ω) data points.507

The reconstituted curve of the SB model, using magnitudes of G = 4 × 105 Pa sβand β=0.18,508

is depicted by the solid lines in Fig. 5(a). It shows that the model gives a good fit for both509

the elastic and storage moduli, with only the two constitutive parameters, the quasi-property510

G and the fractional exponent β.511

The inset in Fig. 5(a) demonstrates that the phase angle is independent of the frequency512

ω, which is a second typical feature of critical gels. This corresponds to what the Scott Blair513

model predicts:514

tan(δ) =G′′(ω)G′(ω)

= tan (πβ/2) (26)

i.e. the phase angle is a function of the fractional exponent β only.515

17

In Fig. 5(b) we show the creep / recovery curve of a sample from the same zero-fat cheese516

(blue circles) as used in Fig. 5(a), at the same temperature of T = 10◦C. The dashed line517

denotes the prediction of the compliance J(t), which is obtained by substituting the acquired518

material parameters (from the SB fit to the data for the complex modulus depicted in Fig. 5(a),)519

into the SB model expressions for J(t) in the creep phase, Eq. (10), and in the recovery phase,520

Eq. (13), respectively. The hollow symbols in Fig. 5(b), and in the log-log representation521

of the creep phase in Fig. 5(c) and the recovery phase in Fig. 5(d), show that we accurately522

predict the evolution of J(t), from which we can then calculate the firmness F (circle), springi-523

ness S (triangle), and rubberiness R (circles) of zero-fat cheese. Both measured responses in524

Fig. 5(c) and Fig. 5(d) show some ‘creep ringing’ at very short times due to coupling of525

the elasticity with the moment of inertia of the instrument. Fig. 5(d) demonstrates that the526

compliance ultimately approaches zero, as predicted from Eq. (20).527

3.3. Texture engineering528

We have demonstrated how a framework based on fractional equations can be used to529

characterize and quantify the linear viscoelastic (LVE) response of hydrocolloidal gels. This530

framework is compact and effective and only one constitutive element, the two-parameter531

springpot or Scott Blair element (Eq. (1)) (Koeller, 1984), is required to describe the LVE532

material functions of cheese over a broad range of frequencies or times. The springpot is533

characterized by the quasi-property, G, and the fractional exponent, β, both introduced to534

rheology by Scott Blair et al. (1947). The quasi-property can be viewed as an ‘intermediate’535

material property that is appropriate for characterizing soft materials and which sets the scale536

of the stress in the material. It smoothly interpolates between an elastic shear modulus G537

and a Newtonian viscosity η. The fractional exponent quantifies the frequency and temporal538

response of the material and ranges from purely elastic (β = 0), to purely viscous (β = 1).539

Our measurements show that its magnitude is related to the underlying microstructure and540

composition of the material.541

From the descriptions of the manipulations and observations required to evaluate the firm-542

ness, springiness, and rubberiness of cheese from textural analysis (Szczesniak, 1963b; Davis,543

1937; Foegeding and Drake, 2007; van Vliet, 1991) we have deduced that the rheological544

analogues of these attributes are all defined by specific points on the creep / recovery curve545

(Fig. 3). We have demonstrated that the fractional constitutive framework allows us to ob-546

tain quantitative descriptions of this rheological experiment that can be expressed explicitly547

in terms of the quasi-property and the fractional exponent of the model. By substituting the548

appropriate timescales of observations we are able to predict the textural measures of firmness549

18

F, springiness S , and rubberiness R, from the material properties G and β. We refer to our550

predictive equations based on essential material properties as the FSR-equations (Eq. (12),551

(16), (19) respectively). They guide the effective reformulation of semi-soft food gels such as552

cheese, while keeping the product at a desired level of firmness, springiness, and rubberiness.553

In Fig. 6 we sketch out one example of our FSR-equations being used as a design tool.554

The markers in Fig. 6(a) represent two sets of material parameters {G, β}A and {G, β}B . These555

sets describe the hypothetical materials A and B from Fig. 3 and are mapped onto firmness556

contours, which are calculated with Eq. (12) and plotted on a logarithmic scale. In Fig. 6(a)557

we take the same time of observation t f = 10 s as in Fig. 3, which results in the materials558

A and B being equally firm, i.e. FA = FB. The markers in Fig. 6(a) thus lie on the same559

iso-firmness line, and we take the magnitude of F = 6 × 104 Pa at t f = 10 s represented by560

these markers, as our desired level of firmness.561

When we increase the time of observation in Fig. 6(b) from t f = 10 s to t f = 106 s, the iso-562

firmness lines rotate counterclockwise, from a more horizontal to more vertical orientation.563

This tells us that at short times, the magnitude of the quasi-property G dominates firmness564

assessment, whereas at long times of observation it is dominated by the fractional exponent β565

of the material. This results in material A being the firmer of the two materials at t f = 106 s.566

It has the lowest quasi-property but also the lowest fractional exponent giving FA = 6 × 103567

Pa as compared to FB = 2× 101 Pa at t f = 106 s. If retaining shape after long times of loading568

(e.g. during storage) is a product requirement, material A is preferred over material B.569

In Fig. 6(c) we map these same two materials A and B onto a springiness contour plot. This570

figure shows that a more elastic material with a low value of the quasi-property is the most571

springy. Fig. 6(d) demonstrates that there is no significant effect of the observation time t f on572

the magnitude of springiness assessed in the material, provided that we use small values for the573

elapsed recovery time ∆ts � t f . This corresponds to our analysis of the springiness equation574

in section 3.2, the bulk of the elastic strain energy stored in the material is accumulated (and575

returned) at short time scales. For values of the fractional exponent β > 0.8, the springiness576

rapidly drops and the material becomes very ’squishy’. If low springiness is preferred over577

retaining shape, than we chose material B. For rubberiness, similar plots to those shown in578

Fig. 6(a)-(d) can also be created. If we take equal times of observations for firmness and579

rubberiness, t f = ∆tr, then equation (21) applies and all iso-rubberiness lines are vertical: the580

magnitude of the rubberiness is independent of the magnitude of the quasi-property G. The581

most rubbery material will be material A, which has the lowest fractional exponent.582

Design plots such as Fig. 6 reflect graphical representations of how rheology can, and583

should, be used in the context of structure-texture engineering. The foundations for this ap-584

19

proach were laid more than 60 years ago by Davis (1937) and Scott Blair (1947; 1959), two585

pioneers in food rheology. They both combined well-defined deformations of food-stuffs with586

a central question: which essential material property gives rise to a specific texture attribute.587

We favour their approach over the more undirected approach of statistically correlating large588

amounts of rheological data to the results from Quantitative Descriptive Analysis (QDA) of589

multiple texture attributes. Our reservations to the latter approach are for two reasons. First590

the deformations and observations in QDA that lead to the texture judgment are ill-defined591

and may vary from panel to panel. This a priori weakens correlations with the measurements592

obtained from the carefully designed rheological experiment. Our second argument to favour593

the Scott Blair / Davis approach, is that material properties are intrinsic properties whereas594

texture attributes are extrinsic in nature (Reiner, 1971). This essential difference is clearly595

demonstrated by the plots in Fig. 6. Whereas the coordinates of the markers that indicate the596

material properties of materials A and B do not change from plot (a)-(d), the grade and orien-597

tation of the contours that represent texture vary, depending on the attribute of interest and the598

timescale of observation chosen. The extrinsic nature of such sensory texture attributes makes599

it very hard, if not impossible, to make an unambiguous connection from a panel judgment to600

material models representing the food microstructure, when a purely statistical model is used601

as an intermediate. Once dynamic and quantitative relations between texture and the essential602

material properties are established (such as the FSR-equations developed in this study), one603

can build on the long track record of quantitative rheological testing for probing structure-604

property relations.605

4. Conclusion606

We have created a structure-texture engineering model which can help guide the ratio-607

nal reformulation of semi-soft food gels while maintaining reference levels of firmness F,608

springiness S , and rubberiness R. The central features of the model are rheological definitions609

of these texture attributes in the linear viscoelastic regime. Firmness is defined as the inverse610

of the compliance at the end of the creep regime, springiness as the secant rate of change of the611

compliance during the first 0.1 seconds of recovery, and rubberiness as the relative amount of612

strain recovered at the end of the creep recovery period. The firmness, springiness, and rubber-613

iness (FSR) equations are constructed using a fractional constitutive framework, which can be614

visualized pictorially in terms of a mechanical element known as a springpot (Koeller, 1984).615

This framework compactly and effectively captures the predominant power-law relaxation and616

retardation of complex multi-scale materials such as cheese.617

20

The FSR-equations contain only two material parameters, the quasi-property G, and the618

fractional exponent β (Scott Blair, 1947), and the resulting stress or strain in the material619

depends on an experimental time of observation to. These two material parameters can be620

extracted from measurements of any of the standard linear viscoelastic material functions:621

the complex modulus, G∗(ω)= G′(ω) + i G′′(ω), the relaxation modulus, G(t), or the creep622

compliance, J(t).623

Our FSR-equations are derived from linear viscoelastic theory and predict time-dependent624

responses for the firmness (as demonstrated by Scott Blair et al. (1947)) and also for the625

springiness and rubberiness assessed for a material. Firmness is linearly related to the quasi-626

propertyG, whereas springiness is inversely related toG. This rationalizes the anti-correlation627

between the two texture attributes commonly found experimentally. We have chosen to mea-628

sure springiness at very short times of observation after the imposed shear stress is released,629

which results in a texture attribute that is independent of the time of observation for measur-630

ing firmness, t f . Rubberiness is not a function of the quasi-property (i.e. the magnitude of the631

stress in the material), but only of the fractional exponent β, t f , and the elapsed recovery time632

for measuring rubberiness, ∆tr. For the specific case where both times are equal, t f = ∆tr, time633

cancels out and rubberiness is merely a function of the fractional exponent β that characterizes634

the timescale of the material response.635

The viscoelastic nature of the firmness implies that for specific pairs of power-law ma-636

terials, the experimental conditions of creep and recovery can be chosen such that different637

materials appear to be equally firm. For values of the creep times that are common in judging638

cheese texture, one material can be assessed as the firmest, whereas for creep times relevant639

to storage the same material may be regarded as the softest. If one waits for the materials to640

recover from the imposed stress for long times, both power-law materials will fully recover641

from the applied stress and will be judged to be completely rubbery, however at shorter times642

the sample will appear to have different rubberiness. For all three examples of viscoelastic643

time-dependency outlined above, our FSR-equations can calculate if and when these transi-644

tions occur, provided the material parameters G and β of each material are known. The food645

engineer must be keenly aware of the temporal dependencies in these textural attributes before646

performing a rheological test.647

Although there has been a long standing promise for rheology to deliver quantitative648

structure-texture relations for foods, successful examples are rare. In this paper we have im-649

plicitly formulated three prerequisites for a successful structure-texture model for food gels,650

based on shear rheometry. The first is selecting texture attributes that can be measured with651

bulk rheology and thus are assessed at first touch or bite (before mastication disrupts sample652

21

homogeneity). This holds for all three of the attributes firmness F, springiness S , and rub-653

beriness R, defined here. The second is to have rheological definitions that leave no room654

for selective interpretation and can be directly evaluated from well-defined rheological exper-655

iments. We have done so by defining realistic observation times and by revisiting the early656

creep / recovery experiments performed by Davis (1937) and argue that J(t) should be the657

standard rheological curve for measuring and defining firmness, springiness, and rubberiness.658

The third prerequisite is to have a constitutive model which accurately describes the mate-659

rials response to a deformation with as few material parameters as possible and from which660

the creep / recovery curve can be reconstituted. The model parameters should be intrinsic661

material constants and thus be obtainable from an independent rheological experiment, e.g.662

small amplitude oscillatory shear (SAOS). Fractional constitutive equations, such as the Scott663

Blair and fractional Maxwell model, based on the ideas of fractional calculus meet these re-664

quirements. The key to a successful structure-rheology-texture model is to determine material665

parameters that predict complex, time-dependent (and often subjectively-defined) properties666

such as texture which are founded in the material structure; Scott Blair’s (1947) original ideas667

of quasi-properties and fractional exponents do just that.668

In Part II of this paper (Faber et al., 2016), we will show that the FSR-equations ((12), (16),669

(19)) will also quantify the firmness, springiness, and rubberiness in the viscoelastic regime670

from a measurement of the relaxation modulus G(t). We also address the over-prediction by671

the Scott Blair model of the rate of recovery of cheese subjected to loadings that exceeds the672

linear viscoelastic limit and which therefore result in irreversible plastic flow events occurring.673

Acknowledgments674

The authors thank Prof. H. Meijer, Dr. L. Van Breemen, and Dr. H. Luyten for guidance,675

and Anne van de Pijpekamp for technical assistance. We also thank FrieslandCampina for676

funding this research.677

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List of Figures977

1 Three methods of assessing the firmness, springiness, and rubberiness of cheese:978

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in simple shear to quantify firmness springiness, and rubberiness in the linear983

viscoelastic limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34984

2 (a) Schematic representation of the springpot, a mechanical element defined985

by the quasi-property G and the power-law exponent β, which we call the986

fractional exponent (adapted from Jaishankar and McKinley (2013)). For a987

material characterized by a single power-law only one element is needed to988

capture its linear viscoelastic response, we call this the Scott Blair model (SB).989

The arrows demonstrate how the springpot interpolates between a Newtonian990

dashpot with viscosity η (β = 1) and a Hookean spring with modulus G991

(β = 0). The corresponding constitutive equations are given below each992

element. (b) Mechanical analogue of the Fractional Maxwell Model (FMM)993

constructed from two springpots. The FMM has four material parameters:994

two quasi-properties (G and V) and two fractional exponents (β and α). . . . 35995

3 (a) Creep / recovery curves of two hypothetical materials A (red) and B (blue)996

with identical firmness and different springiness and rubberiness. Specific val-997

ues of the compliance for each material indicated with filled markers are used998

to calculate firmness F, (squares), springiness S , (triangles), and rubberiness999

R, (circles). Their corresponding times of observations, t f , ts and tr are plotted1000

as hollow markers on the time-axis. We define the elapsed time of recovery as1001

∆t = t− t f (black dashed line). Firmness is the inverse of the maximal creep1002

compliance max {J(t)} and thus has units of Pa. Springiness has units of (Pa1003

s)−1 and is defined as the secant rate of recovery just after the stress is removed1004

at t = t f + ∆ts, where ∆ts << t f . For each material, the dashed, colored, se-1005

cant line goes through the correspondingly colored filled square and triangle.1006

The absolute magnitude of the slope of the secant is equal to the springiness1007

S . Rubberiness is defined as the relative extent to which the compliance re-1008

covers from the stress applied. If J(t) = 0 Pa at t = tr then the rubberiness1009

R = 1. We have chosen tr such that ∆tr >> ∆ts. Material A (red) is more1010

springy and more rubbery than material B (blue). (b) The same experiment as1011

in (a) but now plotted on a log-log scale, without the recovery phase and with1012

the creep phase extrapolated (dashed line). The plot shows that A and B are1013

power-law materials: J(t) ∝ tβ, where β is the power-law exponent. This plot1014

emphasizes the importance of controlling and reporting time when assessing1015

firmness: only at t f are the two materials equally firm, before or after this1016

point the material with the lowest compliance is the firmest. In the text we1017

also demonstrate that the magnitude of the springiness and firmness are also1018

dependent on the time of observation chosen. . . . . . . . . . . . . . . . . . 361019

32

4 Effect of measurement time on the values of (a) springiness and (b) rubberi-1020

ness. The colors represent the two hypothetical materials A (red) and B (blue)1021

from Fig. 3. The solid and dashed line represent a time t f for measuring firm-1022

ness of t f = 10 s and t f = 106 s respectively. The springiness S and rubberiness1023

R, are functions of both t f , and the elapsed time of recovery ∆t at which we1024

calculate S and R. The markers denote the elapsed recovery times we have1025

chosen to calculate springiness, ∆ts= 0.1 s (triangles in (a)) and rubberiness,1026

∆tr= 10 s (circles in (b) ). Our choice of ∆ts<< t f makes the springiness S1027

independent of the time for measuring firmness t f . Our choice of ∆ts<< ∆tr1028

makes springiness and rubberiness two distinct properties. . . . . . . . . . . 371029

5 (a) Determining the quasi-property G and fractional exponent β of zero-fat1030

cheese with a water/protein ratio w/p= 1.8 and a temperature of T = 10◦C.1031

The material parameters are obtained by fitting the Scott Blair model (SB,1032

black line) for the complex modulus, Eq. (25) to the storage and loss mod-1033

ulus measurements {G′(ω),G′′(ω)} (filled and unfilled symbols respectively).1034

The SB model gives a good fit demonstrating that the relaxation behaviour1035

of cheese is well described by a single power-law over frequencies 1 ≤ ω ≤1036

100 rad s−1. Inset: phase angle as a function of the frequency. (b) Creep /1037

recovery experiment (σ0 = 100 Pa) of zero-fat cheese at T = 10◦C. Dashed1038

line: prediction of the compliance J(t) during creep and recovery using the1039

Scott Blair element, Eq. (10) and Eq. (13), with t f = 10 s and ∆tr = 10 s, and1040

the material properties from the fit in (a). The hollow symbols are the specific1041

points from the predicted creep / recovery curve which are used to calculate1042

the firmness F, (square), springiness S , (triangle), and rubberiness R, (circle).1043

(c,d) Same experiment and predictions as in (b) plotted on a log-log scale and1044

with the creep phase (c) and the recovery phase (d) separated. (d) The com-1045

pliance ultimately approaches zero, as predicted from Eq. (20). These plots1046

demonstrate that our equations correctly predict the firmness, springiness, and1047

rubberiness of power-law materials such as cheese, from the two constitutive1048

parameters G and β describing the material. . . . . . . . . . . . . . . . . . . 381049

6 Contour plots of the magnitude of the firmness F, Eq. (12), and the springi-1050

ness S , Eq. (16), of power-law materials such as cheese. The magnitude of1051

the firmness and springiness are functions of the quasi-property G and frac-1052

tional exponent β describing the material. The times of observation t f and ∆ts1053

at which the firmness and springiness are measured are indicated above each1054

plot. The scale bars and contour lines are spaced logarithmiccally. The ma-1055

terial properties of the hypothetical materials A and B from Fig. 3 and Fig. 41056

are mapped onto each contour plot as red and blue markers respectively. (a)1057

At t f = 10 s, materials A and B are equally firm and the iso-firmness lines are1058

close to horizontal, meaning the firmness F is dominated by the magnitude of1059

the quasi-property G. (b) At t f = 106 s, the iso-firmness lines are inclined to1060

be almost vertical and the magnitude of the fractional exponent β dominates1061

the magnitude of the firmness F. For times t f>10 s, material A is firmer than1062

material B, which corresponds to what we have depicted in Fig. 3(b). (c,d)1063

Springiness is hardly affected by the value of t f . All iso-springiness lines1064

slope downwards as the exponent β increases. For materials with equal frac-1065

tional exponent β, the material with the smallest value of the quasi-property1066

G is the most springy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391067

33

At rest, t<0 At rest, t<0

Shear stress suddenly removed

Constant shear stress applied

(b) Uniaxial compression (c) Simple Shear (a) Human senses

σt = σ0

σ = 0

Creep, 0 < t < tcreep

Recovery, tcreep < t < trecover

γ = 0

γt= xt / y0

γt = (xtcreep-xt) / y0

σt = 0x

x

y0

y0

y0

x0

Touch - hands

Sight - eyes

Constant neg. strain rate appliedLoad, 0 < t < tload

y

εt = -ln(x/x0)

σt = NF / (πy2) x

Constant pos. strain rate appliedUnload, tload < t < tunload

ε=ε0• •

y0

x0

ε=0, NF=0•

y

εt = -ln(x/x0) = ε0t σt = -NF / (πy2)

•

x

ε=-ε0••

Touch - mouth

Figure 1: Three methods of assessing the firmness, springiness, and rubberiness of cheese: (a) with human sensesusing touch (mouth or hands) or sight (eyes) or (b,c) with rheological measurements. (b) Monotonic uniaxial com-pression is the most commonly used test protocol but involves complex kinematics: non-linear viscoelastoplasticityand fracture. In this paper we use (c) creep / recovery in simple shear to quantify firmness springiness, and rubberi-ness in the linear viscoelastic limit.

34

1 0 ,

,

1 0 ,

,

(a) (b)

Figure 2: (a) Schematic representation of the springpot, a mechanical element defined by the quasi-property G andthe power-law exponent β, which we call the fractional exponent (adapted from Jaishankar and McKinley (2013)).For a material characterized by a single power-law only one element is needed to capture its linear viscoelasticresponse, we call this the Scott Blair model (SB). The arrows demonstrate how the springpot interpolates betweena Newtonian dashpot with viscosity η (β = 1) and a Hookean spring with modulus G (β = 0). The correspondingconstitutive equations are given below each element. (b) Mechanical analogue of the Fractional Maxwell Model(FMM) constructed from two springpots. The FMM has four material parameters: two quasi-properties (G and V)and two fractional exponents (β and α).

35

0 2 4 6 8 10 12 14 16 18 20 220

0.20.40.60.8

11.21.41.61.8

2x 10

−5

∆t = t − tf

S

time, t [s]

com

plia

nce,

J(t

) [P

a−1 ]

B

B

A

A

10−2

10−1

100

101

102

10−6

10−5

10−4

β=0.2

β=0.7

tf

time, t [s]

com

plia

nce,

J(t

) [P

a−1 ]

A

B

(a) (b)

Figure 3: (a) Creep / recovery curves of two hypothetical materials A (red) and B (blue) with identical firmnessand different springiness and rubberiness. Specific values of the compliance for each material indicated with filledmarkers are used to calculate firmness F, (squares), springiness S , (triangles), and rubberiness R, (circles). Theircorresponding times of observations, t f , ts and tr are plotted as hollow markers on the time-axis. We define theelapsed time of recovery as ∆t = t− t f (black dashed line). Firmness is the inverse of the maximal creep compliancemax {J(t)} and thus has units of Pa. Springiness has units of (Pa s)−1 and is defined as the secant rate of recovery justafter the stress is removed at t = t f + ∆ts, where ∆ts << t f . For each material, the dashed, colored, secant line goesthrough the correspondingly colored filled square and triangle. The absolute magnitude of the slope of the secant isequal to the springiness S . Rubberiness is defined as the relative extent to which the compliance recovers from thestress applied. If J(t) = 0 Pa at t = tr then the rubberiness R = 1. We have chosen tr such that ∆tr >> ∆ts. Material A(red) is more springy and more rubbery than material B (blue). (b) The same experiment as in (a) but now plotted ona log-log scale, without the recovery phase and with the creep phase extrapolated (dashed line). The plot shows that Aand B are power-law materials: J(t) ∝ tβ, where β is the power-law exponent. This plot emphasizes the importance ofcontrolling and reporting time when assessing firmness: only at t f are the two materials equally firm, before or afterthis point the material with the lowest compliance is the firmest. In the text we also demonstrate that the magnitudeof the springiness and firmness are also dependent on the time of observation chosen.

36

10−3

10−2

10−1

100

101

102

103

104

10−9

10−8

10−7

10−6

10−5

10−4

10−3

recovery time, ∆t [s]

sprin

gine

ss, S

[1/(

Pa

s)] A

B

10−3

10−2

10−1

100

101

102

103

104

00.10.20.30.40.50.60.70.80.9

1

recovery time, ∆t [s]

rubb

erin

ess,

R [−

]

A

AB

B

(a) Springiness (b) Rubberiness

Figure 4: Effect of measurement time on the values of (a) springiness and (b) rubberiness. The colors representthe two hypothetical materials A (red) and B (blue) from Fig. 3. The solid and dashed line represent a time t f formeasuring firmness of t f = 10 s and t f = 106 s respectively. The springiness S and rubberiness R, are functions ofboth t f , and the elapsed time of recovery ∆t at which we calculate S and R. The markers denote the elapsed recoverytimes we have chosen to calculate springiness, ∆ts= 0.1 s (triangles in (a)) and rubberiness, ∆tr= 10 s (circles in (b)). Our choice of ∆ts<< t f makes the springiness S independent of the time for measuring firmness t f . Our choice of∆ts<< ∆tr makes springiness and rubberiness two distinct properties.

37

100

101

102

103

104

105

frequency, ω [rad/s]

str

ora

ge

& lo

ss m

od

ulu

s,

G’ &

G’’

[Pa

]

zero fat

fit10

010

110

20

0.2

0.4

0.6

ω [rad/s]

tan

δ [

−]

0 5 10 15 200

1

2

3

4

5x 10

−5

time , t [s]

com

plia

nce, J(t

) [1

/Pa]

prediction

(a) (b)

10−2

10−1

100

101

10−6

10−5

10−4

time, t [s]

co

mp

lian

ce

, J(t

) [1

/Pa

]

creep

11 12 15 2010

−6

10−5

10−4

time , t [s]

com

plia

nce, J(t

) [1

/Pa]

recovery

(c) (d)

Figure 5: (a) Determining the quasi-property G and fractional exponent β of zero-fat cheese with a water/proteinratio w/p= 1.8 and a temperature of T = 10◦C. The material parameters are obtained by fitting the Scott Blair model(SB, black line) for the complex modulus, Eq. (25) to the storage and loss modulus measurements {G′(ω),G′′(ω)}(filled and unfilled symbols respectively). The SB model gives a good fit demonstrating that the relaxation behaviourof cheese is well described by a single power-law over frequencies 1 ≤ ω ≤ 100 rad s−1. Inset: phase angle as afunction of the frequency. (b) Creep / recovery experiment (σ0 = 100 Pa) of zero-fat cheese at T = 10◦C. Dashedline: prediction of the compliance J(t) during creep and recovery using the Scott Blair element, Eq. (10) and Eq. (13),with t f = 10 s and ∆tr = 10 s, and the material properties from the fit in (a). The hollow symbols are the specificpoints from the predicted creep / recovery curve which are used to calculate the firmness F, (square), springinessS , (triangle), and rubberiness R, (circle). (c,d) Same experiment and predictions as in (b) plotted on a log-log scaleand with the creep phase (c) and the recovery phase (d) separated. (d) The compliance ultimately approaches zero,as predicted from Eq. (20). These plots demonstrate that our equations correctly predict the firmness, springiness,and rubberiness of power-law materials such as cheese, from the two constitutive parameters G and β describing thematerial.

38

0 0.2 0.4 0.6 0.8 110

3

104

105

fractional exponent, β [−]

quas

i pro

pert

y,

[Pa

sβ]

log[Firmness], F [Pa]

−2

0

2

4

0 0.2 0.4 0.6 0.8 110

3

104

105

fractional exponent, β [−]

quas

i pro

pert

y,

[Pa

sβ]

log[Firmness], F [Pa]

−2

0

2

4

(a) Firmness, t f = 10 s (b) Firmness, t f = 106 s

0 0.2 0.4 0.6 0.8 110

3

104

105

fractional exponent, β [−]

quas

i pro

pert

y,

[Pa

sβ]

log[Springiness], S [1/(Pa s)]

−8

−7

−6

−5

−4

0 0.2 0.4 0.6 0.8 110

3

104

105

fractional exponent, β [−]

quas

i pro

pert

y,

[Pa

sβ]

log[Springiness], S [1/(Pa s)]

−8

−7

−6

−5

−4

(c) Springiness, t f = 10 s , ∆ts= 0.1 s (d) Springiness, t f = 106 s , ∆ts= 0.1 s

Figure 6: Contour plots of the magnitude of the firmness F, Eq. (12), and the springiness S , Eq. (16), of power-law materials such as cheese. The magnitude of the firmness and springiness are functions of the quasi-property Gand fractional exponent β describing the material. The times of observation t f and ∆ts at which the firmness andspringiness are measured are indicated above each plot. The scale bars and contour lines are spaced logarithmiccally.The material properties of the hypothetical materials A and B from Fig. 3 and Fig. 4 are mapped onto each contourplot as red and blue markers respectively. (a) At t f = 10 s, materials A and B are equally firm and the iso-firmness linesare close to horizontal, meaning the firmness F is dominated by the magnitude of the quasi-propertyG. (b) At t f = 106

s, the iso-firmness lines are inclined to be almost vertical and the magnitude of the fractional exponent β dominatesthe magnitude of the firmness F. For times t f>10 s, material A is firmer than material B, which corresponds to whatwe have depicted in Fig. 3(b). (c,d) Springiness is hardly affected by the value of t f . All iso-springiness lines slopedownwards as the exponent β increases. For materials with equal fractional exponent β, the material with the smallestvalue of the quasi-property G is the most springy.

39