A simple and accurate method for the estimation of yield stress by rotational
viscometry: application of the concept of infinite apparent viscosity
Hossein Kiani, Seyed Mohammad E. Mousavi, and Zeinab E. Mousavi
Department of Food science and Technology, Faculty of Agricultural Engineering and Technology, Campus
of Agriculture and Natural Resources, University of Tehran, Karaj, Iran ([email protected])
ABSTRACT
Yield stress is a very important engineering property of some food materials affecting the processing
requirements and the quality of the product. The estimation of the value of yield stress has been a challenging
issue and different methods have been employed for evaluation and calculation of that. In this research, the
infinite apparent viscosity of a dairy drink containing gellan and pectin was measured using a simple
rotational viscometery method and the data was used for the estimation of yield stress. Increasing
concentrations of gelllan gum (0, 0.03, and 0.05 %wt) with or without pectin (0.25 %wt) were added to a
fermented dairy drink and the rheological properties of the system were investigated. The Bingham yield
stresses were calculated as the intercepts of the flow diagrams and then the same data were used to illustrate
the curves of shear stress versus the inverse of apparent viscosity. An extrapolation of shear stress, σ, versus
the inverse of apparent viscosity, η-1, to find the intercept can give a value related to the yield stress, σ0. The
stability of distinct solid particles added to the drink was studied as a case study of yield stress application
and then the observations were compared to the mathematical prediction of the stability with regard to the
yield stress. The methods employed for calculation of yield stress were recognized to be potential methods
for prediction and engineering calculations due to their simplicity. Furthermore, the infinite viscosity method
can present a realistic definition of yield stress while can be easily calculated by simple rotational
viscometry. Further research is also needed to compare the data with the controlled-stress rheometer data to
demonstrate the validity of the methods with more accuracy.
Key words: Fluid gel; Rheology; Phase separation; Solid particle stability
NOTATION
σ
γ
K
n
σβ or σ0
ηpl
σc
ηc
Shear stress (Pa)
Shear rate (s-1)
Consistency Index (Pa.s0.5)
Flow Behavior Index
Yield stress (Pa)
Plastic Viscosity (Pa.s)
Casson's Yield stress (pa0.5)
Plasitic viscosity of Casson (Pa0.5.s0.5)
Fg
Fb
Fσ°
V
g
ρp
ρm
A
Gravitational force (N)
Buoyancy force (N)
Force related to the yield stress (N)
Volume (m3)
acceleration due to gravity (ms-2)
the density of the particle (kgm-3)
the density of the medium (kgm-3)
surface (m2)
INTRODUCTION
Yield stress is a very important engineering property of some food materials such as fluid gels affecting the
processing requirements and the quality of the product. Materials with yield stress are those exhibiting highly
shear-thinning rheological behavior and therefore behave close to solids at the static state but their rhological
behaviour become similar to those of fluids when they are a subject of shearing. This behaviour can be
utilized in different food products to prevent the sedimentation of dispersed particles in beverages.
A wide range of gelling biopolymers including agar, carrageenan, pectin, alginate, gellan and gelatine are
able to create yield stress in food products [5]. The value of yield stress and the properties of the final product
are dependent upon the nature of the biopolymer employed, as well as the processing parameters [1, 4, 5 7,
10].
Four well-known models can be employed for the rheological characterization of non-Newtonian fluid foods,
[6].
These models are the power law:
σ = kγ n
the Herschel–Bulkley model:
(1)
σ = σ o + kγ n
(2)
the Bingham model:
σ = σ 0 + η pl γ
(3)
and the Casson model:
σ 1 / 2 = σ C 1 / 2 + η C 1 / 2γ 1 / 2
(4)
Mullineux and Simmons (2008) [6] declared that in the case of processing equipment design or other similar
objectives, simplicity of the model is the most important factor for model selection and therefore the power
law model is preferred. However, if high accuracies are needed, low shear behaviors are taken to account
and/or yield stress is considered as an engineering factor that really is [9]; power law model could not be
suitable.
Several methods have been described to measure the static or dynamic yield stress employing characterized
apparatuses [8-9]. To calculate the Bingham yield stress, extrapolation of the selected linear part of the flow
curve is performed and possibly very low shear rates are not taken to account to make a linear region suited
with the Bingham model available [8-9]. This may be desirable because of the elimination of the possible
errors in low shear calculations. Whereas application of Herschel–Bulkley model results in a more accurate
fit if the instrument is well equipped, enabling calculation of very low shear behaviors. Another way to
estimate the value of yield stress is extrapolation of apparent viscosity versus shear stress curve to infinite
apparent viscosity.
In this study, gellan and gellan pectin-mixtures were added to a fermented dairy drink their effect on the
sedimentation behavior of added dried and grinded basil leaves were studied. Flow measurements were
employed to explore the rheological properties and two, including a noteworthy infinite apparent viscosity
method and a conventional Bingham approach, were employed to calculate the values of yield stress. An
approach toward the mathematical exhibition of the stability of the particles was provided considering yield
stresses calculated from the mentioned procedures.
MATERIALS & METHODS
Sample Preparation
Fresh skim milk (fat content by Gerber analysis ≈ 0.2 wt %) was provided from the dairy plant of the
department of Food Science and Technology, University of Tehran. Pasteurization was carried out by a water
bath at 90ºC for 15 minutes. Pasteurized milk was inoculated by a DVS type YF-3331 starter culture at 40ºC
(Chr. Hansen, Denmark). Incubation was performed until the pH reached 4.2-4.4. The total solids (TS)
content of the resulting yoghurt was determined by evaporation of ~1 g sample on a boiling bath, followed by
oven-drying at 102°C until constant weight was attained, yielding a value of TS = 9.6 wt %. Yoghurt was
firstly diluted (as partial dilution to TS = 7.5 wt %), mixed well (RW20 DZM Janke&Kunel mixer) and
homogenized (APV 1000 Lab homogenizer, Denmark) at 150 bars. NaCl (MERCK with 99.5% purity,
Germany) was also added before homogenization (0.5 % w/w of final product). Gellan gum (Deacylated
gellan, Kelcogel F, CpKelco, US) and gellan-high methoxy pectin (HMP; Provisco, PROVladd PEC 1902,
Switzerland) mixtures were added to the samples as stabilizing agents. Hydrocolloid mixtures were weighted
for each sample separately, dissolved in RO water and hydrated at 80ºC for 20 minutes in a shaking water
bath. Hot solutions were gently added to the partial diluted yoghurt kept at room temperature. Final total
solids content was set at 5% w/w. Gellan was added in different concentrations (0.01, 0.03, and 0.05 % w/w)
and a similar series of gellan concentrations with the coexistence of pectin (0.25 % w/w) was also prepared.
pH values of the final product that was 4-4.2 after mixing and processing was kept unchanged by storing the
samples at 5˚C until performing the measurements. Sample preparation was carried out in duplicates.
Homogenous spice (Sabzan Co., Tehran, Iran) particles were obtained by grinding and passing the leaves
from sieves. Particles passed through a 10-mesh sieve but remained on an 18-mesh sieve were collected and
the average diameter of the particles was considered to be 1.5 mm. The thickness of the spice leaves was
measured by using a magnetic instrument (Magna-Mike model 8000, USA). The density of the particles was
obtained by calculating the volume changes for the given weight of spices submerged in water. The spices
were blanched and stored in water to reach to the constant density before density measurement.
For basil particles the following data were obtained: ρp=1051.3 kgm-3, Particle radius = 1.5×10-3 m, Particle
thickness = 4.3×10-4 m , V = 3.038×10-12 m3 (as a cylinder), g = 9.8 ms-2 , A = 7.065×10-9.
Measurements and Analysis
Flow behavior analysis of the samples was carried out on a bob and cup HAAKE ROTOVISCO RV12
viscometer (Germany) at 20ºC. The NV geometry (doubled-gap coaxial-cylinder DIN 54 453) was employed.
The density of the fermented dairy drink samples were determined by a picnometer at 20 ºC.
The overall balance of all forces acting on a particle should be equal to zero if particle stability or motion
prevention is targeted. Four different forces act on a solid particle suspended in a liquid medium including
hydrodynamic force, intermolecular force, gravitational force and buoyancy force which the two later are
described as the following respectively:
F g = ρ pVg
(5)
F b = ρ mVg
(6)
For solid spice particles stabilized in a fermented dairy drink medium molecular forces and hydrodynamic
forces are not very important. In a drinking product, particles need to destroy the three-dimensional network,
if exist, to move and accordingly a proportional force will be required for this matter. This kind of force,
which is similar to hydrodynamic forces in its nature, is reasonably linked and in fact proportional to yield
stress and also to the particle surface exposed to medium which is described as:
F σ0 = σ 0 A
(7)
Microsoft Excel 2007 was used to investigate the data. Statistical analysis, if needed, was also carried out
using SAS 9.1 software (SAS Institute Inc., Cary, NC, USA).
RESULTS & DISCUSSION
Flow Behavior
The log-log plot of the apparent viscosity versus shear rate indicated that all samples exhibit non-Newtonian
shear-thinning behavior (Figure 1). Addition of gellan caused the viscosity of the fermented dairy drink to
increase noticeably.
Figure 1. Viscosity curves of fermented dairy drink with (filled symbols) or without (empty symbols) pectin, containing
0 (circle) and 0.05 % (triangle) of gellan.
The former particle size distribution and microscopy investigation [3] indicated that gellan-protein
interactions took place both permanently –resulting in formation of particle gels- and transiently –due to the
aggregative interactions of the particles- in the dilute fermented dairy drink. Both of the mentioned
phenomena along with gellan-gellan interactions (as proposed by Sworn [10]) resulted in the creation of
complex particle gels and the overlaps occurred between these particles affected the flow behavior of the
product significantly.
Yield Stress
Yield stress can be described as the required shear stress for initiation of flow in a plastic material resisting
against flow [9]. Yield stress can also be defined as a point in which the applied stress does not cause the
material to flow and shear rate is zero but just a little increase in stress results in the initiation of flow. An
experimental approach toward this definition leads to an approximation of yield stress called dynamic yield
stress described by Bingham, Herchel-Bulkey and Casson models. Another kind of description for yield
stress is that it is a point in which the resistance of the material against flow (viscosity) is infinite. An
extrapolation of apparent viscosity versus shear stress curve to infinite apparent viscosity can give a value
being related to the resistance and therefore to the yield stress.
Both of the mentioned approaches were employed to estimate the values of yield stress for the fluid gels of
fermented dairy drinks. Bingham yield stresses was estimated employing the flow diagrams of the samples
(Figure 2a). To estimate the infinite viscosity, an extrapolation of the inverse of apparent viscosity versus
shear stress curve was performed. The intercept of the linear trend line where the inverse of apparent
viscosity is zero and accordingly apparent viscosity is infinite was assumed to be the value of the yield stress
(Figure 2b). The range of the data employed for this procedure was the same as the data range used for flow
curve method.
a
b
Figure 2. Fitted curves of Bingham model (a) and curves of shear stress versus the inverse of apparent viscosity (b) and
the illustration of the magnitude of the yield stress for plain fermented dairy drink (○) and samples containing 0.25 % of
pectin (●), 0.05 % of gellan (∆) and 0.25-0.05 % of pectin-gellan (▲) as well as the corresponding equation of the fitted
curves upward respectively.
Gellan containing samples exhibited relatively high values of yield stress. Inversely, samples with no
hydrocolloid content had very low values for this property. The network created by gellan and molecules and
protein particles caused a high consistency but per application of a proportional shear stress, the network
broke suddenly, the particles lost their inter-particle connections and the product acquired the characteristics
of a fluid material. As it is shown in Figure 2b, compared to those samples with gellan gum, the yield stress
values detected for the stabilizer free sample was very low. In spite of a relatively high concentration applied,
samples with pectin alone showed low yield stress values. However, combined application of pectin and
gellan resulted in a major increase in the yield stress of the samples.
Due to the nature of the rotational measurements and destruction of the three-dimensional network of the
product during the measurements and on the other side the especial definition of the yield stress, reasonable
objections may be presented to the method. However, in practice, these methods are potential and applicable
for predictions and engineering calculations due to their simplicity. Furthermore, the infinite viscosity
method can present a realistic definition and prediction of yield stress and the values obtained from the flow
curve method were approximately higher the values observed for this method. The data obtained via such
methods could also be compared with the controlled-stress rheometer data to explore the validity of the
methods with more accuracy. Further research is also needed to investigate the transient interactions of the
particle gels, the way they are destroyed and the required forces to do so as well as reversibility of these
interactions. The importance and application of the yield stress values for stabilization and settlement
discussions will be presented in the following sections.
Solid Particle Stability
As an example for stabilization of solid particles in a fluid medium, solid spice particles of basil were added
to the fermented dairy drink samples including samples prepared with the presence of pectin (0.25% w/w),
gellan (0.05% w/w) and gellan-pectin mixture (0.05-0.25% w/w). For a stabilized sample of fermented dairy
drink (with 0.05 % w/w of gellan and 0.25% w/w of pectin): ρm = 1017.7 kgm-3 , σ0 = 0.7 pa; and ccording to
the equations (5), (6) and (7), it can be written:
F g = 3.13 ×1−8 N
F b = 3.03 × 10 −8 N
F σ 0 = 4.95 × 10 −9 N
Figure 3. Spice particles suspended in fermented dairy drink samples containing 0.25% pectin (A) 0.05% gellan (B) and
0.25-0.05% pectin-Gellan (C)
Fσ° should be greater than the absolute value of differences between Fg and Fb for stabilization purposes:
(
) (
)
−8
−8
F σ Ο ≥ F g − F b ≥ 3.13 × 10 − 3.03 × 10 ≥ 10 −9 N
Consequently it can be written:
σ Ο A ≥ 10 −9 ⇒ σ Ο ≥
10 −9
⇒ σ Ο ≥ 0.14 Pa
7.065 × 10 −9
Regarding to the calculations and the data obtained via viscometric measurements it could be predicted that
the particles would be suspended in the selected sample. Experimental measurements were in agreement with
the predicted results and particles remained suspended during all storage time (15 days) (Figure 3).
Pectin did not provide the required yield stress value and sedimentation of the particles occurred. Gellan
alone produced adequate yield stress values and particles were suspended but as it can be observed in Figure
3 serum separation was observed. Gellan-pectin mixture possessed the aim of this study; particles remained
suspended and phase separation did not occur.
CONCLUSION
A rheological approach toward the determination of yield stress was provided and fluid gels of gellan and
gellan-pectin in a fermented dairy drink were evaluated. Based on former investigations [2-3] it was revealed
that gellan, either alone or in combination with pectin, interacts with proteins, and creates particle gels.
Further particle-particle interactions and overlaps lead to formation of a through network of the particles and
as a result, a highly shear-thinning rheological behavior appears. This kind of rhelogical behavior is believed
to be a property of fluid gels. Yield stress is pivotal property of these systems and the estimation of the value
of yield stress has been a challenging issue of discussion. At the present study the methods of calculating the
yield stress values of the fluid gels via viscometric analysis were evaluated. An infinite viscosity method was
developed and adapted for this purpose and the results were compared with the values calculated according to
the Bingham model. Though an equal data range was used for interpolating and finding the values of yield
stress via both methods, the values calculated by Bingham method were significantly higher. The values
found by the infinite viscosity method seemed to be more accurate according to the definition and
understandability of this method. An example of yield stress application was also provided and the
importance of yield stress in particle stability was analyzed. Values obtained from both of the yield stress
estimation methods were in agreement with the experimental results of our case study. The case study was
the investigation of the stability of distinct solid particles added to the drink and evaluation of their stability
during the storage time compared with the mathematical prediction of the stability with regard to the yield
stress. The stability of the particles was anticipated by considering the overall balance of forces acting on the
particles and the results were shown to be in agreement with the experimental observation. Further
investigation is needed to verify the methods and evaluate their accuracy in more detail. Due to the
simplicity, reasonable results and also wide availability of the instrument applied, viscometric analysis of
fluid gels was recognized to be a potential method. In addition to the stabilization of added solid particles, it
was also demonstrated that the yielding behavior detected for fluid gels of gellan and gellan-pectin affect the
stability of colloidal protein particles of the dairy drink significantly and prohibit their sedimentation.
REFERENCES
[1] Frith W.J., Garijio X., Foster T.J. & Norton I.T. 2002. Microstructural origins of the rheology of fluid
gels. Gums and stabilizers for the food industry, 11, 95-103.
[2] Kiani H., Mousavi S.M., Emam-Djomeh Z. & Yarmand M.S. 2008. Effect of concentration and
hydration temperature of gellan gum on the stability and physical properties of acidified milk protein
solutions. Australian Journal of Dairy Technology, 63, 94-99.
[3] Kiani H., Mousavi S.M., Razavi H. & Morris, E.R. 2010. Effect of gellan, alone and in combination
with high-methoxy pectin, on the structure and stability of doogh, a yogurt-based Iranian drink. Food
Hydrocolloids, 24, 744-754.
[4] Martinez-Padilla L.P., Lopez-Araiza F. & Tecante A. 2004. Steady and oscillatory shear behavior of
fluid gels formed by binary mixtures of xanthan and gellan. Food Hydrocolloids, 18, 471–48.
[5] Moritaka H., Takeuchi M., Okoshi H. & Fukuba H. 2003. Particle and matrix gels of gellan gum effects
of filler particles on rheological properties of matrix gels. Food Hydrocolloids, 17, 577–583.
[6] Mullineux G. & Simmons M.J.H. 2008. Influence of rheological model on the processing of yoghurt.
Journal of Food Engineering, 84, 250–257.
[7] Nishinari K., Zhang H. & Ikeda, S. 2000. Hydrocolloid gels of polysaccharides and proteins. Current
Opinion in Colloid & Interface Science, 5, 195-201.
[8] Rao M.A. 1999. Rheology of fluid and semisolid foods: principles and applications. pp. 242-244, Aspen
Publishers Inc., Gaithersburg.
[9] Steffe J.F. 1996. Rheological methods in food process engineering. pp. 35-39, Freeman Press, East
Lansing.
[10] Sworn, G. 2000. Gellan gum. In: Phillips, G.O. & Williams P.A. (Eds.). Handbook of Hydrocolloids.
Woodhead Publishing Ltd., Cambridge, UK.