Article Experimental Comparison of Static Rheological Properties of Non-Newtonian Food Fluids with Dynamic Viscoelasticity

Nihon Reoroji Gakkaishi Vol.46, No.1, 1~12 (Journal of the Society of Rheology, Japan) ©2018 The Society of Rheology, Japan Article Experimental Comparison of Static Rheological Properties of Non-Newtonian Food Fluids with Dynamic Viscoelasticity

† Kanichi Suzuki and Yoshio Hagura

Graduate School of Biosphere Science, Hiroshima University, 1-4-4 Kagamiyama, Higashi-Hiroshima, 739-8528, Japan (Received : July 28, 2017)

Correlations between static rheological properties (G: shear modulus, μ: viscosity, μapp: apparent viscosity) of four food fluids (thickening agent solution, yoghurt, tomato puree and mayonnaise) measured employing a newly developed non-rotational concentric cylinder (NRCC) method and dynamic viscoelastic properties (|G*|: complex shear modulus, G′:

storage shear modulus, G″: loss shear modulus, |η*|: complex viscosity, tanδ: loss tangent, μapph: apparent viscosity) mea-

sured by a conventional apparatus (HAAKE MARS III) were studied. μapp and μapph for each sample fluid agreed well.

μapp≒|η*| obeying the Cox–Merz rule was obtained for the thickening agent solution, μapp < |η*| for mayonnaise and

μapp << | η*| for tomato puree and yoghurt. μ was lower than μapp or μapph for all samples, and reached μapp with an increasing shear rate. Plotting G and |G*| against the shear rate and angular velocity respectively in the same figure revealed G < |G*| for tomato puree, G≒|G*| for yoghurt and G > |G*| for mayonnaise and thickening agent solution. The applicability of two-element models in analyzing the viscoelastic behavior of sample fluids was investigated by comparing two character-

istic times τM = 1/(ωtanδ) and τK = tanδ/ω, corresponding to Maxwell and Kelvin–Voigt models respectively and evaluated

from measurements of tanδ, with τ = µ/G and τapp = µapp /Gapp calculated from rheological properties measured by the NRCC

method. Here Gapp (= G(μ/μapp) is an apparent shear modulus newly defined to evaluate the shear-rate dependency of the

elasticity of Non-Newtonian fluids. Results show τK agreed well with τapp, suggesting the possibility of applying the Kelvin–Voigt model to the viscoelastic behavior of sample fluids. Key Words: Non-Newtonian / Food fluid / Viscoelasticity / Non-rotational concentric cylinder method / Apparent shear modulus

1. INTRODUCTION shear rate, whereas the flow properties are usually measured for a high shear rate with large deformation. Dilute solutions and suspensions in fluid foods such as The dynamic viscoelastic behaviors of materials have fruit juice, milk and vegetable oil behave as Newtonian flu- been explained theoretically using two two-element models. ids1). However, most fluid foods are Non-Newtonian fluids One is the series model (Maxwell model) of the viscosity and that have complicated flow properties and commonly behave shear modulus while the other is the parallel model (Kelvin– as viscoelastic materials2). The flow properties of Non- Voigt model) of the viscosity and shear modulus2, 3). For Non- Newtonian fluids are usually studied by examining the shear- Newtonian fluids, however, it is difficult to verify the estima- rate dependency of apparent viscosity. Meanwhile, the visco- tions of viscosity and shear modulus from dynamic elastic property of Non-Newtonian fluids has been analyzed viscoelastic properties. Furthermore, it is difficult to estimate conventionally employing a dynamic testing method, be- the dynamic viscoelastic behavior from flow properties mea- cause a static method of testing viscoelasticity is difficult to sured employing a static experimental method. The reason is apply to fluids owing to their lack of a fixed form. However, that there is almost no information on the viscosity or shear the relationship between the flow properties and dynamic vis- modulus applicable to the analysis of the dynamic viscoelas- coelasticity of Non-Newtonian fluids has not yet been clari- tic properties of fluids, because no reliable experimental ap- fied. One reason is that the dynamic viscoelastic properties paratus with which to measure the viscosity (not apparent obtained from experimental results in a linear region with viscosity) or the shear modulus for fluid materials has yet small deformation are basically useful for only a very low been developed. The validity of estimations of the viscosity and shear modulus for Non-Newtonian fluids from dynamic † Corresponding author. viscoelastic measurements made in experiments has there- E-mail: [email protected] Tel: +81-82-422-7339, Fax: +81-82-422-7339 fore not been verified, even if it would be shown that either of

1 Nihon Reoroji Gakkaishi Vol.46 2018 the two-element models for dynamic viscoelasticity is adapt- properties of Non-Newtonian fluids by varying the plunger able to the Non-Newtonian fluid studied. (or cup) speed. Furthermore, it is expected that the method Against the background of the rheological study de- can be used to study the physical properties of complex-com- scribed above, a new experimental method of measuring the ponent fluids containing solid materials that established dy- static rheological properties of fluid materials, including namic experimental apparatuses have difficulty in measur- Non-Newtonian fluids, was proposed recently. The method, ing11). However, the validity or usefulness of measurements called the non-rotational concentric cylinder (NRCC) meth- of the viscosity and shear modulus of Non-Newtonian fluids od, measures static viscoelastic properties (i.e., viscosity, made using the NRCC method has not been verified, because shear modulus and apparent viscosity) of a sample fluid si- no reliable data on the values of viscosity and shear modulus multaneously during a very short movement of a plunger or of Non-Newtonian fluids have been obtained for comparison. cylinder (cup) along the vertical axis at constant speed4). The The present study investigated the relationship between time needed for one measurement including the computer rheological properties including static viscoelastic values for calculation is several seconds. Few methods of measuring the four Non-Newtonian food fluids (i.e., a thickening agent rheological properties of fluids from the change in shear solution, plain yoghurt, tomato puree and mayonnaise) mea- stress on a cylindrical plunger surface during fluid flow sured employing the NRCC method and dynamic viscoelas- through a concentric annular space have been proposed. The tic properties measured by a conventional dynamic experi- penetroviscometer evaluates relative viscosity by comparing mental apparatus under the same shear-rate conditions. The the time–shear stress curve of a fluid sample with that of a dynamic viscoelastic properties measured were the complex standard liquid of known viscosity5). The back extrusion shear modulus, complex viscosity, storage shear modulus, method measures the viscosity of Newtonian fluids and the loss shear modulus and loss tangent. The shear-rate or angu- apparent viscosity of Non-Newtonian fluids from the change lar velocity dependency of these values was compared with in shear stress on a plunger surface by applying the theory for that of the viscosity, shear modulus and apparent viscosity of fluid flow through an annular 6-10)channel . These methods the sample fluids. The paper discusses two basic rheological evaluate viscosity or apparent viscosity by measuring the in- components of the dynamic viscoelastic property (i.e., vis- crease in shear stress on a plunger surface when a fluid is cosity and shear modulus) and the applicability of two-ele- made to flow upward through an annular space by forcing ment viscoelastic models to the food fluids studied as well as down a plunger a comparatively long distance into the fluid the meaning and usefulness of the viscosity and shear modu- in a cylindrical tube. However, none of these methods esti- lus measured with the NRCC method. mate the shear modulus of Non-Newtonian fluids. Meanwhile, the NRCC method measures static rheological properties, in- 2. MATERIALS AND METHODS cluding the viscosity, shear modulus and apparent viscosity of a sample fluid by analyzing theoretically the response of 2.1 Materials shear stress on the surface of a plunger when it moves over a Three food fluids and a thickening agent were purchased short distance at constant speed. For Newtonian fluids, the from a local market. The thickening agent (Thickness Yell, shear modulus is measured as zero. The plunger is immersed Wakodo, Japan) was used to make a Non-Newtonian solution in a sample fluid in a cylinder or cup to the prefixed depth in of which the main ingredients were dextrin and polysaccha- advance of measurement. The plunger (or cup) usually moves ride thickener. The solution was prepared by dissolving the a short distance of about 0.5–1.5 mm that depends on the flu- agent in water at 60 °C at a concentration of 2.5 g/100 g-wa- idity of the sample and the speed of the plunger. This distance ter with gentle hand stirring using a spatula. Having dissolved through which the plunger moves is determined such that adequately, the solution was kept standing at room tempera- there are enough measurement points for analysis of the ture for 1 day prior to measurement to ensure the stability of change in shear stress; the number of measurement points in rheological properties. Yoghurt (Meiji Bulgaria Yoghurt, unit time is inversely proportional to the plunger speed. The Meiji Co., Ltd., Japan) was stored in a refrigerator at 2–3 °C NRCC method is thus able to measure the rheological prop- for 1 day before measurement. Tomato puree (Kagome Co., erties consecutively without large deformation of the sample Ltd., Japan) and mayonnaise (Kewpie Co., Japan) were used fluid in contrast to other viscometers, such as the rotational without treatments. In advance of the rheological experi- viscometer or cone–plate apparatus. The method can also ments on samples, the accuracy of the viscosity measurement evaluate the shear-rate dependency of the rheological for the NRCC method was calibrated using deionized water

2 SUZUKI • HAGURA : Experimental Comparison of Static Rheological Properties of Non-Newtonian Food Fluids with Dynamic Viscoelasticity

−3 13) (viscosity: 0.89 × 10 Pa·s) and six viscosity standard liq- between the cup bottom and the plunger bottom, Lb, was uids (Brookfield Engineering Lab. Inc., USA) for which the 16.5 mm. The rheometer used in this study measured the viscosity ranged from 4.7 × 10−3 to 1.008 × 102 Pa·s. shear stress on the plunger from the sample fluid in a cup while the cup on the stage moved upward a very short dis- 2.2 Measurements tance (instead of the plunger moving downward). The stage Dynamic viscoelastic properties (i.e., complex shear returned to the initial position automatically after each run of modulus, |G*|, storage shear modulus, G′, loss shear modu- the measurement at a preset speed. The moving distance of lus, G″, complex viscosity, |η*|, and loss tangent, tanδ) and ap- the stage approximately ranged 0.5–1.5 mm according to the parent viscosity, µapph, were measured using a HAAKE MARS measuring speed or shear rate. Average values of rheological III rheometer (Thermo Fisher Scientific Inc., Germany) with properties were obtained from seven measurements for each plate/plate geometry (D = 20 mm, gap of 0.5 mm). sample fluid. The values of angular velocity, ω, for the dy- Measurements were conducted two or three times under the namic viscoelastic measurement using the HAAKE MARS same conditions using a new sample for each measurement. III were set in a nearly equal range of the shear rate, dγ/dt, for The NRCC method (NRCC Visco-Pro, Sun Scientific Co., the NRCC method. All measurements were made at 25 °C. Ltd. Japan) was used with a CR-3000EX rheometer (Sun Scientific Co., Ltd. Japan) to measure static viscoelastic 3. THEORY AND CHARACTERISTICS properties; i.e., viscosity, µ, shear modulus, G, and apparent OF THE NRCC METHOD viscosity, µapp. Figure 1(a) is a schematic diagram of the ap- paratus for the NRCC method. The inner diameter of the out- This section outlines the theory and measuring method er cylinder (cup) was 50 mm. Three plungers having a diam- of the NRCC method. A detailed explanation of the NRCC 4) eter of 49 mm (ratio of the plunger diameter to cup diameter method was given in the literature . The plunger (with radius κ = 0.98), 45 mm (κ = 0.90) and 35 mm (κ = 0.70) were used Ri) is dipped in advance of each measurement a distance Lo in in the calibration test of viscosity depending on the viscosity the sample fluid in a cup (with radius Ro) as shown in of water and the viscosity standard liquids. The plunger hav- Fig. 1(b). The initial distance between the plunger bottom ing a diameter of 35 mm was used for the rheological mea- and the cup bottom is Lb. From this initial spatial arrange- surement of food fluids, because of the viscous nature of the ment, the plunger is moved downward or the cup is moved samples. This meant that the gap between the cup and plung- upward a short distance ΔL at constant speed Vp. During the er was 7.5 mm. The initial immersion distance of the plunger period of cup movement, t = ΔL/Vp, the immersion distance

of the plunger in the sample fluid increases from Lo to L ow- in the sample fluids, Lo, was 49 mm, and the initial distance ing to the upward flow of the fluid due to the cup movement:

2 L = Lo + {Vp t/(1 – κ )} (κ = Ri/Ro). (1)

The NRCC method analyzes the change in the force acting on the plunger during the plunger movement over a short dis- tance.

3.1 Newtonian fluid The theory of the proposed method is based on the theo- ry for fluid flow through an annulus12). For a Newtonian fluid

with no elastic property, the total force, Fv, acting on the

plunger is the sum of the force Fs acting on the side wall and

the force Fp acting on the bottom area:

2 Fv = Fs + Fp = 2π Ri Lσi – π Ri ΔP. (2)

The basic equation and boundary conditions are (see the

Fig. 1 Schematic diagram of the apparatus for the NRCC method. Nomenclature list for symbols)

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Fig. 2 Theoretically estimated force–time curves obtained using the NRCC method for a Newtonian fluid (a), an elastic ma- terial (b) and a non-Newtonian fluid (c).

d (r σr ) / dr = r ΔP / L. (3) The resulting equation for Fv is thus simplified as

2 ur = 0 at r = Ro, ur = Vp at r = Ri. (4) Fv = – 2πμVp αL = – 2πμVp α[Lo + {Vp t/(1 – κ )}], (12)

Newton’s law of viscosity is where α is a geometric constant of the apparatus expressed as

2 2 2 – (dur / dr ) = σr / μ. (5) α = (1 +κ )/{(1 +κ )ln κ + (1 – κ )}. (13)

ur and σr can be derived theoretically from Eqs. (3) and (4) The value of α depends only on κ, and the absolute value of α combined with Eq. (5): increases with increasing κ. Fv changes with time as shown in Fig. 2(a). If it is possible to measure the force at the start of 2 2 ur = {Vp ln(r/Ro)/ ln κ} + [Ro ΔP{1 – (r/Ro) plunger movement (theoretically, t = 0), the force Fv can be 2 + (κ – 1) ln(r/Ro)/ ln κ}/(4μL)], (6) simplified toF vo, expressed as

σr = {rΔP/(2L)} Fvo = – 2πμVp αLo. (14) 2 2 – [μVp + {Ro ΔP(κ – 1) /4L}]/ (r ln κ). (7) The viscosity of the Newtonian fluid can be evaluated from

The shear stress at the plunger wall, σi, is obtained by Eq. (14) as well as from Eq. (12). A minute difference in the substituting Ri = κRo into r: change in the measured force acting on the plunger from the theoretical curve due to the effect of mechanical motion of

σi = {κRoΔP/(2L)} the apparatus was corrected by NRCC Visco-Pro analysis 2 2 – [μVp + {Ro ΔP(κ – 1)/4L}]/(κRo ln κ). (8) software in the present study.

The average upward flow rate of the sample,u av, is 3.2 Elastic material When the sample is a perfect elastic material with 2 2 Ro uav = [1/ {π R {(1 – κ )}}∫Ri 2π r urdr. (9) Young’s modulus E (E = 3G, where G is the shear modulus), the force acting on the plunger wall is expressed as

The relationship between uav and Vp is expressed as

Fes = 2πLo σr = 2πLo G (dZ/dr), (15) 2 2 uav = Vp κ / (κ – 1). (10) where Z is a relative shear distance in the axial direction be-

ΔP can be derived from equations (6), (9) and (10): tween the plunger and sample. Fes is derived by integrating r

from Ri to Ro, integrating Z from 0 to Z and combining with 2 2 2 2 ΔP = 4μLVp /[Ro {(1 + κ ) ln κ + (1 – κ )}]. (11) Z = Vp t/ (1 –κ ) as

4 SUZUKI • HAGURA : Experimental Comparison of Static Rheological Properties of Non-Newtonian Food Fluids with Dynamic Viscoelasticity

2 Fes = – [2πLoVp t G/{(1 – κ ) ln κ}]. (16) 1) The tangent, Ft, of the force acting on the plunger at t → 0 is estimated from the regression curve of the mea- The compressible force on the bottom area of the plunger, sured force–time values.

Fec, is expressed by postulating E = 3G: 2) The viscosity, μ, is calculated from the intercept, Fvo, of the tangent using Eq. (14). 2 Fec = 3π(κ Ro) Vp t G/Lb. (17) 3) The shear modulus, G, is evaluated from the difference

between estimated Ft and Fv acting on the plunger at any

The total force acting on the plunger, Fe, is thus the sum of point in time, t, by substituting the viscosity, μ, into Eqs.

Fes, and Fec: (12) and (20). As stated in section 1, it is not possible to verify the viscosity

Fe = KVpG t, (18) and shear modulus evaluated following the above procedure, because there are no standard Non-Newtonian fluids with where K is a constant that depends on the geometry of the which to confirm the accuracy of the measurements made us- apparatus, expressed as ing the NRCC method.

2 2 K = [{3π(κRo) /Lb } – 2πLo/{(1 – κ )ln κ}]. (19) 3.4 Apparent viscosity of a Non-Newtonian fluid When the sample is a Non-Newtonian fluid, the gradient The force acting on the plunger in the case of an elastic mate- of the tangent of the measured force curve decreases sharply rial is proportional to time as shown in Fig. 2(b). The shear immediately after the measurement owing to the effect of modulus, G, can be obtained from the gradient, KVpG. The stress relaxation, and reaches a nearly constant value in short practical applicability of Eq. (19) has been confirmed using time. By postulating that the total force is due to the viscous agar gels as an elastic model sample4). force only, the NRCC method estimates the apparent viscos-

ity, μapp, from the total force acting on the plunger at the arbi-

3.3 Non-Newtonian fluid trary time, teq, at which the gradient of the tangent reaches a To estimate the viscosity and shear modulus of a Non- constant value: Newtonian or viscoelastic fluid, it is necessary to evaluate the 2 force due to the viscous nature and the force due to the elastic Ftn = – 2πμappVp α[Lo + {Vp teq/(1 – κ )}]. (21) nature of the fluid separately by dividing the total force acting on the plunger into these two forces. The measured force act- The present study evaluates μapp from the force acting on the ing on the plunger for a Non-Newtonian fluid is commonly plunger at the end time of plunger movement by assuming plotted as a curved line owing to the effect of stress relaxation the end time satisfies the condition fort eq. 4, 12) as shown in Fig. 2(c). The gradient of the tangent of the curve The shear rate at the plunger wall, (dγ/dt)Ri, is decreases with time. It is therefore impossible to simply com- 2 2 bine Fv and Fe when evaluating the viscoelastic properties of (dγ/dt)Ri = – (1 – κ ) Vp α/{(1 + κ )Ri}. (22) a Non-Newtonian fluid. However, the force acting on the plunger at the starting time of plunger movement (theoreti- 3.5 Accuracy of the NRCC method for viscosity cally, t → 0) is considered to consist of the forces due to vis- measurement cosity and elasticity for the Non-Newtonian fluid without the Figure 3 shows the calibration results of the viscosity effect of stress relaxation. Thus, the tangent, Ft, of the force– measurement for the NRCC method using Newtonian fluids time curve at t → 0 was derived from the combined force of of known viscosity. The samples were deionized water and

Fv (Eq. (12)) and Fe (Eq. (18)) by differentiating their com- six viscosity standard liquids (Brookfield Engineering Lab. bined equation with respect to time: Inc., USA). The viscosities of samples at 25 °C ranged from 0.89 × 10−3 Pa·s to 1.008 × 102 Pa·s. The largest and smallest 2 2 Ft = Fvo – 2πμVp αt/(1 – κ ) + KVpG t. (20) root-mean-square errors of measurements were 0.094 and 0.0084, respectively. The results demonstrate that the NRCC When the sample is a Newtonian fluid with no elastic nature, method can measure a wide range of viscosities of Newtonian Eq. (20) reduces to Eq. (12). The NRCC method evaluates the fluids with practical accuracy within the measurement errors viscosity and shear modulus of the sample fluid as follows. given above.

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Fig. 3 Calibration results of the viscosity for the NRCC method obtained using deionized water and six viscosity standard liquids (Brookfield Eng. Lab., USA) at 25 °C.

4. RESULTS AND DISCUSSION decreased gradually with an increase in the shear rate. The viscosity approached the apparent viscosity in the higher 4.1 Comparison of the viscosity, μ, and apparent shear-rate region as shown in Fig. 5. The results indicate that

viscosity, μapp, obtained by the NRCC method the viscosity of a shear-thinning Non-Newtonian fluid is not

with the apparent viscosity, μapph, obtained by constant but rather decreases with an increase in the shear the HAAKE MARS III rate as does the apparent viscosity, though the shear-rate de- Figure 4 shows two examples of measured force-time pendency is weaker than that of the apparent viscosity. It was curves and analyzed results by the NRCC method for a supposed that the inner structure of mayonnaise or thickening Newtonian fluid and a Non-Newtonian fluid. Figure 4(a) was agent solution changed more at a bending point of the shear- the result of a viscosity standard liquid (μ = 1.0 × 10−1 Pa·s at rate dependency of the viscosity than in the shear-rate region 25 °C, Brookfield Eng. Lab., USA) and Fig. 4(b) was of to- below the bending point. It is also considered that the shear mato puree (Kagome Co., Ltd., Japan). Both experimental rate dependency of the viscosity with the bending point might curves agreed satisfactorily with the theoretically estimated relate to the flow behavior of the sample with a yield stress. force-time curves shown in Fig. 2(a) and Fig. 2(c). When the But, this supposition should be confirmed by further study. shear modulus, G, was measured as zero, the apparent viscos- Meanwhile, the viscosities of tomato puree and yoghurt were ity was not shown in the analyzed results for the reason of similar to apparent viscosities, though the shear-rate depen- that the analysis software classified the sample as a Newtonian dencies of the viscosity were slightly weaker than those of fluid. the apparent viscosities. It was assumed from the results that Experimental results of the viscosity, μ, and apparent the change in the inner structure of these food fluids occurs viscosity, μapp, of four food fluids measured employing the gradually, corresponding to the entire range of the shear rate NRCC method and the apparent viscosity measured using the studied. conventional apparatus, HAAKE MARS III, μapph, are shown in Fig. 5. The results reveal that the NRCC method can mea- 4.2 Comparison of the apparent viscosity, μapp, sure the apparent viscosity of Non-Newtonian fluids, because obtained by the NRCC method with the com- almost the same values of μapp and μapph were measured for plex viscosity, |η*|, obtained by the HAAKE each sample fluid. The viscosity was lower than the apparent MARS III viscosity for all samples, showing the difference in shear-rate Figure 6 compares the apparent viscosity, μapp, of the dependencies. The viscosities of mayonnaise and thickening sample fluids measured employing the NRCC method with agent solution were considerably low compared with the ap- the complex viscosity, |η*|, measured by the HAAKE MARS parent viscosities in the low-shear-rate region. The shear-rate III. μapp and |η*| are plotted against the shear rate, dγ/dt, and dependency of viscosities was rather small. However, the dif- angular velocity, ω = 2πf, respectively, where γ is the shear ference between the viscosity and apparent viscosity strain, t is time and f is the frequency. According to the

6 SUZUKI • HAGURA : Experimental Comparison of Static Rheological Properties of Non-Newtonian Food Fluids with Dynamic Viscoelasticity

14) Cox–Merz rule , values of μapp correspond to |η*| for poly- 4.3 Comparison of the shear modulus, G, ob- 15, 16) mer solutions and dispersions , though the former values tained by the NRCC method with the dynamic are several times lower than the values of |η*| for weak gel shear moduli, |G*|, G′, G″, obtained by the 17) systems . Among the food fluids studied, the values of μapp HAAKE MARS III almost agreed with |η*|, indicating the Cox–Merz rule could The static shear modulus, G, measured employing the be adopted for the thickening agent solution of which the NRCC method and the dynamic shear moduli (|G*|: complex main ingredients were dextrin and polysaccharide thickener. shear modulus, G′: storage shear modulus, G″: loss shear

Values of μapp for mayonnaise were slightly lower than |η*|. modulus) measured by the HAAKE MARS III are plotted in The remaining two fluids showed weak gel-like behavior, be- Fig. 7 against the shear rate, dγ/dt, for G and angular cause the values of μapp were several times lower than the frequency, ω, for |G*|, G′ and G″. In the range of angular values of |η*|. Differences between |η*| and μapp for the four frequency studied, the dynamic shear moduli of each fluid sample fluids decreased in the order tomato puree > yo- increased with ω. The values of |G*| and G′ were almost the ghurt > mayonnaise > thickening agent solution. same. This result indicates that energy loss in each fluid due to viscous flow was rather small. Meanwhile, the shear

Fig. 4 Examples of experimental force-time curves and analyzed results by the NRCC method for a Newtonian fluid (a) and a non-Newtonian fluid (b). (a): Viscosity standard liquid (μ = 1.0 × 10−1 Pa·s at 25 °C , Brookfield Eng. Lab., USA) (b): Tomato puree (Kagome Co., Ltd., Japan)

Fig. 5 Comparison of the viscosity, μ, and apparent viscosity, μapp, of four food fluids measured employing the

NRCC method with the apparent viscosity, μapph, measured by the HAAKE MARS III.

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Fig. 6 Experimental results of the apparent viscosity, μapp, of the sample fluids measured employing the NRCC

method and the complex viscosity, |η*|, measured by the HAAKE MARS III. Values of μapp and |η*| are plotted against the shear rate, dγ/dt, and angular frequency, ω, respectively.

Fig. 7 Experimental results of the static shear modulus, G, measured employing the NRCC method with the dy- namic shear moduli (|G*|: complex shear modulus, G′: storage shear modulus, G″: loss shear modulus) measured by the HAAKE MARS III. Values of G and the dynamic shear moduli are plotted against the shear rate, dγ/dt, and angular frequency, ω, respectively.

8 SUZUKI • HAGURA : Experimental Comparison of Static Rheological Properties of Non-Newtonian Food Fluids with Dynamic Viscoelasticity

Fig. 8 Comparison of the apparent shear modulus, Gapp, against dγ/dt with the complex shear modulus, |G*|, against ω for sample fluids.

modulus, G, of tomato puree, yoghurt and mayonnaise de- Gapp = G (µ/µapp). (23) creased with an increase in the shear rate. However, the G values of the thickening agent solution were almost unaffect- Changes in Gapp are then compared with the dynamic shear ed by the shear rate. Figure 7 shows that G < |G*| for tomato moduli measured by the HAAKE MARS III. Although the puree, G |G*| for yoghurt, and G > |G*| for mayonnaise apparent viscosity of a Non-Newtonian fluid is higher than and thickening agent solution. These results suggest that vis- the static viscosity owing to the effect of elasticity, the appar- ≒ coelastic (Non-Newtonian) fluids for which the Cox–Merz ent shear modulus, Gapp, is assumed to be smaller than the rule can be adopted as for polymer solutions and dispersions static shear modulus, G, at a rate of µ/µapp because of a vis- might have G > |G*|. However, this inference should be con- cous effect. When µ and µapp are similar, as for tomato puree, firmed by further investigation, because the results ofthe Gapp might be close to G. Figure 8 compares the apparent present study are only for four samples. shear modulus, Gapp, and the complex shear modulus, |G*|, of each sample against the shear rate, dγ/dt, and angular veloci-

4.4 Proposal of the apparent shear modulus, Gapp, ty, ω. Among the four samples, there was a large difference

and comparison of Gapp with the dynamic between Gapp and |G*| for tomato puree but little difference shear modulus, |G*|, obtained by the HAAKE for the other three samples especially in the low shear rate or

MARS III angular velocity region. Gapp for mayonnaise and thickening Static rheological properties of Non-Newtonian fluids agent solution increased with increasing dγ/dt or ω. The pres- have been discussed mainly in terms of the measurable ap- ent study obtained a relationship between the apparent shear parent viscosity. Several flow models describing the shear- modulus and complex shear modulus that was similar to the rate dependency of the apparent viscosity and the existence relation between the apparent viscosity and the complex vis- 1–3) or absence of a yield value have been proposed . For Non- cosity. Gapp was almost one order of magnitude smaller than Newtonian fluids, however, the static shear modulus and ap- |G*| for tomato puree but close to |G*| for the other three parent viscosity might be affected by the shear rate because samples. These results suggest that the static rheological their inner structures change with the shear rate. The present properties, G and µ, measured by the NRCC method and the study thus defines an apparent shear modulus that indicates apparent shear modulus, Gapp, defined in this study are useful the shear-rate dependency of the shear modulus of Non- to the discussion of the viscoelastic behavior of Non- Newtonian (viscoelastic) fluids as Newtonian fluids. However, for the limited samples studied,

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Fig. 9 Comparison of characteristic times τM = 1 / (ω tanδ) for the Maxwell model and τK = tanδ/ω for the

Kelvin–Voigt model with characteristic times τ = µ/G and τapp = µapp /Gapp calculated from the values of µ,

G and Gapp employing the NRCC method.

the present study could not clarify for what kind of Non- constant or slightly increased with ω (data not shown). The

Newtonian fluid do Gapp and |G*| coincide. result indicates that τ for each sample was not constant, and varied with the shear rate or angular velocity owing to the 4.5 Application of two-element models to the change in viscosity and elasticity. analysis of the viscoelastic behavior of sam- The present study thus evaluated two characteristic

ples times, τM and τK, corresponding to τ for the Maxwell model Dynamic viscoelastic behavior has been analyzed theo- and Kelvin–Voigt model respectively, from the dynamic retically by postulating that materials to be studied have in- measurements of tanδ: trinsic values of viscosity, µ, and shear modulus, G. The con- tributions of µ and G are usually estimated from the relation τM = 1 / (ωtanδ) for the Maxwell model, (24) of the loss tangent (tanδ) with angular velocity, ω. As one of the two-element models, the Maxwell model leads theoreti- τK = tanδ/ω for the Kelvin–Voigt model. (25) cally to tanδ = G″/G′ = 1/(ωτ), where τ = µ/G. In the case of 2) the Kelvin–Voigt model, we might obtain tanδ = G″/G′ = ωτ . τM and τK were then compared with the characteristic times

Therefore, if the relaxation time or characteristic time, τ, is a τ = µ/G and τapp = µapp /Gapp calculated from µ, G and Gapp em- constant, it is expected that tanδ will decrease in reverse pro- ploying the NRCC method as shown in Fig. 9. τM and τK are portion to ω for materials of which the viscoelastic behavior plotted against angular velocity, ω, while τ and τapp are plotted follows the Maxwell model. Meanwhile, tanδ would increase against the shear rate, dγ/dt. Figure 9 clearly shows that τM in proportion to ω for materials that obey the Kelvin–Voigt was about one order of magnitude larger than τK, though both model. For solid materials, it might be possible to assume τM and τK decreased exponentially with ω. Meanwhile, for all that τ is a constant, because there is no or extremely little samples, τapp and its gradient closely match τK and its gradient change in the inner structure. For Non-Newtonian fluids, in the case of the Kelvin–Voigt model. The result suggests however, it is disputable to postulate that τ can be a constant, that the viscoelastic behavior of the sample fluids in this because the inner structure is readily affected by deformation study can be expressed by the Kelvin–Voigt model. τ was and material flow. The present study found that tanδ for all shorter than the other three characteristic times, and the de- samples measured by the HAAKE MARS III was almost creasing gradient was lower than that of the other three times.

10 SUZUKI • HAGURA : Experimental Comparison of Static Rheological Properties of Non-Newtonian Food Fluids with Dynamic Viscoelasticity

Figure 9 also suggests that it is possible to estimate the dy- model and as tanδ = ωτ for the Kelvin–Voigt model, where namic viscoelastic behavior of Non-Newtonian fluids from τ = µ /G. Therefore, in the case of a constant τ value, it was the static rheological properties measured employing the expected that tanδ increases in reverse proportion to angular

NRCC method by adopting τapp = µapp /Gapp instead of τ = µ/G velocity, ω, for the Maxwell model behavior and that tanδ is in the theoretical models. Although these results show the proportional to ω for the Kelvin–Voigt model. However, the possibility of inferring the viscoelastic behavior of Non- experimental results did not agree with the theoretical expec- Newtonian fluids using two-element models, it is necessary tation. Therefore, two relaxation or characteristic times to accumulate more experimental results for many kinds of τM = 1/(ωtanδ) and τK = tanδ/ω corresponding to the Maxwell viscoelastic fluids and verify the general applicability of the model and Kelvin–Voigt model respectively were evaluated results obtained in this study. from the measurements of tanδ. Those values were compared

with τ = µ/G and τapp = µapp /Gapp calculated from the rheolog- 5. CONCLUSION ical properties measured employing the NRCC method. The results show that τK was in good agreement with τapp. The Correlations between static rheological properties (G: main findings of the study are as follows. shear modulus, μ: viscosity, μapp: apparent viscosity) of four 1) It might be possible to apply the Kelvin–Voigt model to Non-Newtonian food fluids (thickening agent solution, yo- the viscoelastic behavior of sample fluids. ghurt, tomato puree and mayonnaise) measured employing a 2) The characteristic time, τ, for Non-Newtonian fluids can newly developed NRCC method and dynamic viscoelastic be expressed appropriately as τapp = µapp/Gapp instead of properties (|G*|: complex shear modulus, G′: storage shear τ = µ/G because the viscoelastic properties of Non- modulus, G″: loss shear modulus, |η*|: complex viscosity, Newtonian fluids change with the shear rate or angular tanδ: loss tangent, μapph: apparent viscosity) measured em- velocity. ploying a conventional dynamic apparatus (HAAKE MARS 3) The dynamic viscoelastic behaviors of Non-Newtonian

III) were investigated. μapp and μapph for each sample fluid fluids are possibly predictable from the static rheologi- agreed well with each other. In terms of the relationship be- cal properties measured using the NRCC method and a tween μapp and |η*|, μapp |η*| obeying the Cox–Merz rule characteristic time of τapp = µapp/Gapp. was obtained for thickening agent solution, μapp < |η*| was 4) The results suggest that static values of viscosity, µ, and ≒ obtained for mayonnaise, and μapp << |η*| was obtained for shear modulus, G, measured employing the NRCC tomato puree and yoghurt. μapp of mayonnaise was slightly method might have meaning for physical properties. lower than |η*|. However, the remaining two fluids showed To verify the generality of the results, it is necessary to inves- weak gel-like behavior because μapp was several times lower tigate experimentally and theoretically the relationships of than |η*|. μ was lower than μapp or μapph for all samples, and static rheological properties and dynamic viscoelastic behav- reached μapp with an increase in the shear rate. A plot of G and iors for different Non-Newtonian fluids. |G*| against the shear rate, dγ/dt, and angular velocity, ω, re- spectively in the same figure showed that G was nearly con- NOMENCLATURE stant or decreased slightly as dγ/dt increased, though |G*| in- creased with ω. A comparison of G with |G*| showed that E: Young’s modulus, Pa G < |G*| for tomato puree, G |G*| for yoghurt and G > |G*| f: frequency, 1/s for mayonnaise and thickening agent solution. The present F : total elastic force acting on the plunger, N ≒ e study newly defined an apparent shear modulus asG app= G(μ/ Fec: compressive force acting on the bottom area of the plung-

μapp) to evaluate the shear-rate dependency of elasticity for er, N

Non-Newtonian fluids by following the definition of the ap- Fes: elastic shear force acting on the side wall of the plunger, N parent viscosity, μapp, which is an overall indicator of fluidity Fp: viscous force acting on the bottom area of the plunger, N including the effect of changes in structure and elasticity with Fs: viscous shear force acting on the side wall of the plunger, N the shear rate. Compared with G, Gapp was nearer to |G*| for Ft: tangent of the combined force of Fv and Fe at t = 0, N all samples without tomato puree. The applicability of two-el- Ftn: tangent of the combined force of Fv and Fe at t = teq, N ement models to the analysis of the viscoelastic behavior of Fv: total viscous force acting on the plunger, N sample fluids was investigated. The loss tangent, tanδ, is ex- Fvo: total viscous force acting on the plunger at t = 0, N presses theoretically as tanδ = G″/G′ = 1/(ωτ) for the Maxwell G: shear modulus, Pa

11 Nihon Reoroji Gakkaishi Vol.46 2018

G′: storage shear modulus, Pa τapp: characteristic time defined asτ app = µapp /Gapp, s

G″: loss shear modulus, Pa τM: characteristic time for the Maxwell model (= 1/(ωtanδ)), s

|G*|: complex shear modulus, Pa τK: chatacteristic time for the Kelvin–Voigt model (= tanδ/ω), s

Gapp: apparent shear modulus defined in this study, Pa ω: angular frequency (= 2 πf ), rad/s K: constant used in Eq. (19), m L: dipped distance of the plunger in the sample fluid during REFERENCES measurement, m 1) Steffe JF, “Rheological Methods in Food Process

Lb: distance between the plunger bottom and cup bottom, m Engineering”, (1992), Freeman Press, East Lansing,

Lo: initial dipped distance of the plunger in the sample fluid, m Michigan. ΔL: moving distance of the plunger, m 2) Rao MA, Steffe JF, “Viscoelastic Properties of Foods”, ΔP: pressure drop for distance L, Pa (1992), Elsevier Applied Science, London. r: distance from the center of the coaxial cylinder system, m 3) Ferguson J, Kemblowski Z, “Applied Fluid Rheology”, (1991), Elsevier Science Publishers, Essex, England. Ri: radius of the plunger (= κRo), m 4) Suzuki K, Nippon Shokuhin Kagaku Kougaku Kaishi, 46, 657 Ro: radius of the cup, m (1999) (in Japanese). t: moving time of the plunger (=ΔL/ Vp), s 5) Bikerman JJ, J Colloid Sci, 3, 75 (1948). teq: arbitrary time at which the gradient of the tangent reaches 6) Morgan RG, Suter DA, Sweat VE, American Society of a constant value, s Agricultural Engineers, Paper No. 79- 6001 (1979). u : flow rate at distancer , m/s r 7) Osorio FA, Steffe JF, J Texture Studies, 18, 43 (1987). uav: average flow rate of the sample, m/s 8) Osorio FA, Steffe JF, Rheologica Acta, 30, 549 (1991). Vp: moving speed of the plunger, m/s 9) Brusewitz GH, Yu H, J Food Eng, 27, 259 (1996). Z: relative shear distance in the axial direction between the 10) NASARUDDIN F, CHIN N, YUSOF Y, Internat. J. Food plunger and sample, m Properties, 15 (3), 495 (2012). α: geometric constant, – 11) Keawkaika S, Hagura Y, Suzuki K, Food Sci. Technol. Res, γ: shear strain, – 16, 23 (2010). δ: phase shift, rad 12) Bird BB, Stewart WE, Lightfoot EN, “Transport Phenomena”, tanδ: loss tangent, – (1960), John Wiley & Sons, New York. 13) Lange NA, “Handbook of Chemistry”, 10th ed., (1967), σi: shear stress at the plunger side wall (r = Ri), Pa McGraw-Hill, New York. σr: shear stress at distance r from the center of the coaxial 14) Cox W, Merz E, Journal of Polymer Science 28, 619 (1958). cylinder system, Pa 15) Smith R, Glasscock J, Korea-Australia Rheol. J,16(4), 169 κ : ratio of Ri to Ro (= Ri /Ro), – (2004). μ: viscosity, Pa·s 16) Winter HH, Rheoogical Acta, 48, 241 (2009). μ : apparent viscosity, Pa·s app 17) Gunasekarana S, Ak MM, Trends in Food Sci. & Tech, 11, 115 |η*|: complex viscosity, Pa·s (2000). τ: characteristic time defined asτ = µ/G, s

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