• extrusion;
  • food;
  • kinetics;
  • mechanochemistry;
  • shear;
  • starch


  1. Top of page
  2. Abstract
  3. Introduction
  4. Kinetic Models for Mechanochemical Reactions
  5. Mechanochemical Kinetics in Food Processing
  6. Discussion
  7. Conclusion
  8. References

Abstract:  In some food processing operations, raw materials sustain shearing and heating simultaneously. These thermomechanical treatments will induce chemical, physicochemical, and biochemical changes, such as starch degradation and gelatinization, protein denaturalization, loss of nutritious components, and inactivation of enzymes and their inhibitors. In this article, only kinetic aspects of these changes are reviewed. The Basedow and Zhurkov models are commonly used to describe shear effects on the rate constants of mechanochemical reactions. After a brief description of these 2 models, their applications in food are reviewed deeply, with emphasis on the activation energy reduction in the Zhurkov model and the shear activation energy in the Basedow model. Since the changes occurring in food systems often involve various interactions at the atomic, molecular, and supramolecular levels, they were described only phenomenonlogically by the mechanochemical kinetic models. The limitation, opportunity, and extension of the mechanochemical approach to modeling food kinetics are discussed.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Kinetic Models for Mechanochemical Reactions
  5. Mechanochemical Kinetics in Food Processing
  6. Discussion
  7. Conclusion
  8. References

Mechanochemistry is defined as a branch of chemistry in which chemical phenomena, such as chemical reactions, induced by mechanical actions such as shear and extensional deformations are studied (Sohma 1989). During the course of chemical reactions, reactant molecules need to surmount an energy barrier to allow their transformation into products. The energy needed for this process is usually provided by heat, light, pressure, or electrical potential, which act either by changing the distribution of the reactants on their ground-state potential energy surface or by moving them onto an excited-state potential energy surface and thereby facilitate movement over the energy barrier. A fundamentally different way of initiating or accelerating a reaction is the use of force to deform reacting molecules along a specific direction of the reaction coordinate. Shear-induced molecular degradation is the most common mechanochemical reaction and has been well documented in polymer research (Sohma 1989; Beyer and Clausen-Schaumann 2005; Hickenboth and others 2007; Caruso and others 2009).

Food raw materials are usually sheared and heated simultaneously in some processing operations such as extrusion, ball milling, and high-pressure homogenization. These thermomechanical treatments will induce physical, chemical, physicochemical, and biochemical changes. The physical changes involve the modification of rheological and thermophysical properties and size reduction of starch caused by tribomechanical treatment (Herceg and others 2010) and ball milling (Wu and Miao 2008). The chemical changes involve gluten degradation during dough mixing (Macritchie 1975), free-radical formation (Schaich and Rebello 1999), and intermolecular disulfide cross-linking (Fischer 2004) in wheat flour protein after extrusion; degradation of modified starch during high-pressure homogenization (Modig and others 2006), of starch during extrusion (Yam and others 1994). The physicochemical changes include starch swelling activated by colloid mill treatment (Padokhin and others 2006), starch gelatinization, and conversion by extrusion (Lai and Kokini 1991), aggregation of gluten during mixing (Morel and others 2002). The biochemical changes are exemplified by the shear-induced inactivation of trypsin inhibitor (TI) (van den Hout and others 1998) and α-amylase (van der Veen and others 2004).

The desirable and undesirable changes as stated above will determine the final quality of foods. Kinetics plays an important part in the modeling of food quality. During thermomechanical processing, both thermal and mechanical effects contribute to the rate of a reaction, and the relationship between the reaction rates and the thermal and mechanical effects is necessary. This topic is dealt with in mechanochemical kinetics and has been deeply studied in polymer degradation. In the last 20 y, this mechanochemical kinetic approach has been used to model the thermomechanical processing of foods, mainly for extrusion cooking. Its applications and results will be reviewed in this work.

Kinetic Models for Mechanochemical Reactions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Kinetic Models for Mechanochemical Reactions
  5. Mechanochemical Kinetics in Food Processing
  6. Discussion
  7. Conclusion
  8. References

The loss of a food component or its properties (such as the degree of starch gelatinization) can be described by the following general rate law:

  • image(1)

where C donates the concentration of a component or its properties, t reaction time, k reaction rate constant, and n reaction order.

Temperature is known to have a significant effect on reaction rates. Typically, temperature dependence of rate constants obeys the well-known Arrhenius relation:

  • image(2)

where kT is thermal-induced rate constant, kT0 preexponential factor, ET activation energy for thermal-induced reaction, R the universal gas constant, and T temperature.

As for mechaniochemical reactions, mechanical effects also play an important role in determining rate constants. The Basedow and Zhurkov models were commonly used in food literature to describe the rate constants as a function of mechanical effects.

The Basedow model

Analogously with the development of the Arrhenius rate equation from a Boltzmann distribution of thermal energy, Eq. (3) has been developed from a Boltzmann distribution of mechanical energy to model polymer degradation in a shear field (Goodman and Bestul 1955):

  • image(3)

where kS is the reaction rate constant for the shear-induced degradation and kS0 is a constant analogous to kT0 in the Arrhenius equation. ES is shear activation energy. J is the average rate of shear energy input. The factor a relates J to the average amount of mechanical energy temporarily stored in a mole of chain bonds of the sheared system.

The shear energy (aJ) can also be expressed as the product of τ and V, where τ is the shear stress macroscopically applied to the system, and V is activation volume of the sheared system. Thus, Eq. (3) was rewritten by Basedow and others (1979) as:

  • image(4)

The term V in Eq. (4) is an empirical material-related parameter without direct correspondence to real space. Since the quantity V is not known, relative values of activation energies (ES/V) are only obtainable. The term τV can be interpreted as a characteristic energy, which acts as the counterpart of RT in the Arrhenius equation.

The Zhurkov model

Bueche (1960) and Zhurkov and Koresukov (1974) independently applied the rate process theory to studies of polymer degradation. They assumed a 2-state model in which the initial and the final states are taken as unbroken and broken bond states, respectively, and the rate from the initial state to the final is estimated on the basis of the rate process theory. It has been inferred that the rate of bond scission under stress in polymeric materials is a thermomechanically activated process. According to this model, the thermally activated energy barrier for chain scission is reduced due to the local tensile force acting on the bond. The rate constant for chain scission obeyed in this case a modified Arrhenius equation:

  • image(5)

where E0 is bond dissociation energy in the absence of stress, ΔE activation energy reduction. ΔE will be some function of the tensile stress (ψ) that is acting on the molecular chain along its axis, f(ψ). The crucial question is what is the definite expression of f(ψ)?

Infrared measurements on the frequency shift in stressed polymer suggested a linear decrease of bond dissociation energy with the local stress (Wool 1980), therefore f(ψ) = βψ, with the constant β identified as the activation volume for bond scission. Quadratic relationships between ΔE and ψ have also been proposed (von Schmeling 1970). However, the linear relation is most popular for its simplest form. For a very complicated reaction process, a power relationship with exponent between 1 and 2, or even less than 1 may exist (van der Veen and others 2004).

The work can also be expressed as the product of force and distance. Bueche (1960) indicated that shearing action caused tensile stress on the bonds and the reduced E could be expressed as , where F is the applied force on the bond and δ is a distance approximately equal to the distance the bond would stretch before breaking. The tensile force F is ready to be measured with the help of single-molecule technique, so the Bueche relation is often used in single-molecule force spectroscopy (SMFS). More elaborate relationships between ΔE and F have been developed by Bell (1978) and Evans and Ritchie (1997). As no fundamental laws are known that would dictate any particular form of such a relationship (Reimann 2002), the most systematic approach to quantitative analysis of mechanochemical kinetics may be to represent the ensemble-averaged rate constant for a reactant distorted to force F as a Taylor expansion around the rate constant of the strain-free reaction (Huang and Boulatov 2010).

However, a word of caution should be given here. Term τ in Eq. (4) represents shear stress macroscopically applied to bulk materials, while term ψ in Zhurkov theory donates tensile stress locally exerted on chemical bonds. ψ is different from τ in direction and magnitude. However, it is often assumed in food research that the microscopic tension is equal to the bulk shear stress, then βψ= Vτ. Equation (5) becomes:

  • image(6)

In this case, the activation volume V acts just as an adjustable parameter. All the approximations (ΔE ≈ βψ, ψ≈τ) are lumped into this parameter. Its value must be determined experimentally.

Mechanochemical Kinetics in Food Processing

  1. Top of page
  2. Abstract
  3. Introduction
  4. Kinetic Models for Mechanochemical Reactions
  5. Mechanochemical Kinetics in Food Processing
  6. Discussion
  7. Conclusion
  8. References

In the following, we will discuss the application of the Basedow and Zhurkov models in food processing and the approaches to determining model parameters. The resulting values of all kinetic parameters are not the major concern of this article. Instead, we focus on the estimates of activation energy and its reduction in the Zhurkov model and these of shear-induced activation energy in the Basedow model.

Application of the Zhurkov model

The Zhurkov model was used by Cai and Diosady (1993) to describe the effects of shear stress on the rate constants of starch gelatinization during extrusion. The shear stress term in Eq. (6) was calculated from apparent viscosity (η) and shear rate (inline image) using

  • image(7)

The rheological model developed by Harper and others (1971) for cooked cereal dough was used by Cai and Diosady (1993) to calculate the apparent viscosity of the extruded wheat starch:

  • image(8)

where W donates moisture content. For a constant screw configuration, inline image is proportional to screw speed (Yacu 1985):

  • image(9)

where G, geometrical coefficient, is a constant for a given screw configuration; N is screw speed.

Substituting Eqs. (8) and (9) in Eq. (7) yields

  • image(10)

Making inline image Eq. (10) is simplified to

  • image(11)

The value of inline image can be calculated for a given extrusion operation where material temperature, moisture content, and screw speed are known.

Substituting Eq. (11) in Eq. (6) gives

  • image(12)

While determining the model parameters in Eq. (12), term VG0.51 is regarded as one parameter. The results given by Cai and Diosady (1993) were E0= 45636.9 J/kg-mol, and VG0.51= 1.784 × 10−3 m3/kg-mol. Here, the unit of activation energy is joule per kilogram-mole (a unit used in earlier literature = 1000 J/g mol, now the unit is joule per mole). This showed that the authors should take the general gas constant R as 8314 J/kg-mol. After estimating residence time t from Figure 3 in the original article, we calculate starch gelatinization degree (Dg) using Eq. (8) in the article with R = 8314 J/kg-mol and the parameter values given by the authors. The resulting Dg is 1. Even taking R = 83.14 J/kg-mol, still Dg= 1. While taking R = 0.8314 J/kg-mol, Dg= 0. Only when taking R = 8.314 J/kg-mol, is the resulting Dg on the order 0.1, which is a reasonable value for gelatinization degree. The general gas constant R should be 8.314 J/g-mol. So, we inferred that the authors may have made a mistake on the unit of R. If this is the case, the resulting E and VG0.51 should be 45.6369 kJ/mol and 1.784 × 10−3 m3/mol, respectively. We calculated inline image from temperature, moisture content, and screw speed, to be 35134 to 162226 Pa. Based on the above inference, the activation energy reduction and its percentage of E0 (45.6369 J/mol) were calculated to be 0.063 to 0.289 kJ/mol and 0.14% to 0.64%, respectively.

Ilo and Berghofer (1998, 2003) used the same procedure as that of Cai and Diosady (1993) to determine the model parameters of thiamin destruction and amino acid loss during extrusion cooking of maize grits, except that they used the rheological model developed by themselves, inline image (Ilo and others 1996). The resulting activation energy reduction was

  • image(13)

Ilo and Berghofer (1998, 2003) found that VG0.15 is 0.06568 m3/mol for thiamin destruction and 0.01164, 0.004989, and 0.010809 m3/mol for the loss of lysine, arginine, and cystine, respectively.

We estimated the material temperatures in the report of Ilo and Berghofer (2003) using the regression model developed by Ilo and others (1996), who used the same extruder and processing conditions as Ilo and Berghofer (2003) did. Then, inline image was evaluated using inline image. The shear-induced activation energy reduction for lysine, arginine, and cystine loss was calculated, 1.89 to 3.18, 0.81 to 1.36, and 1.75 to 2.95 kJ/mol, respectively. As a percentages of E0 (124.71, 67.60, and 74.33 kJ/mol for lysine, arginine, and cystine, respectively), the values are 1.5% to 2.5%, 1.2% to 2.0%, and 2.4% to 3.9%, respectively. The material temperature and its corresponding feed moisture and screw speed were not detailed in the report of Ilo and Berghofer (1998). The central levels of N, T, and M (73 rpm, 170 °C, and 13%) were used to calculate inline image. The activation energy reduction calculated using Eq. (13) and its percentage of E0 (99.41 kJ/mol) are 16.09 kJ/mol and 16.19%, respectively.

Van den Hout and others (1998) used the Zhurkov model to express the effect of shear force in an extruder on the inactivation of TI; however, they calculated ΔE with a different assumption. They proposed that the activation energy for the inactivation of a TI molecule is reduced by an amount equal to the work done in moving a segment of the molecule with a cross-sectional area S over a distance x with respect to the other parts of the molecule:

  • image(14)

where Na is the Avogadro constant, S =π d20/4, x can be approximated by τ d0/ Me, Me and d0 are the elastic modulus and diameter of the molecule, respectively. Equation (14) becomes

  • image(15)

Me and d0 were estimated as 106 N-m−2 and 3.5 × 10−9 m, respectively. The viscosity model of Morgan (1979) was used to calculate the shear stress. Shear rates in the screw channel, clearance, and die were calculated separately. Then, ΔEs in the 3 regions were obtained and ranged from 0.1 kJ/mol in the screw channel to 0.7 kJ/mol in the extruder die.

The earlier work of van den Hout (1997) demonstrated that the inactivation of TIs when heated in shearless environments showed a 2-phase inactivation behavior:

  • image(16)

where TIAt and TIA0 are TI activity after heating time t and without heating, respectively. k1 and k2 are inactivation constants in the 1st and 2nd phase, and expressed as the Arrhenius relation with their corresponding preexponent factors k1,0= 0.142 s−1 and k2,0= 0.0148 s−1, and activation energies E1= 200 kJ/mol and E2= 158 kJ/mol at reference temperature Tr= 110 °C. α is a fitting parameter = 0.583.

When shear effect on TI inactivation is considered, E1 and E2 are reduced by ΔE calculated using Eq. (15). TIA was calculated numerically from the feed port of the extruder to die exit using Eq. (16) with the reduced E1 and E2. The predicted trypsin inhibitors activity (TIA) levels were comparable with the measured values of the extrudates.

While modeling shear-induced inactivation of α-amylase in a plain shear field, van der Veen and others (2004) assumed that the shear-induced reduction of the activation energy for bond breakage is equal to the work done by the external applied force. Although the exact relation between the force applied on the bond and the external applied shear force is unknown, a positive relation between the external applied shear stress and the bond energy reduction may exist. They proposed a simple power law relation to describe the reduction of activation energy due to the external shear stress:

  • image(17)

where ζ and m are constants to be determined experimentally.

Substituting Eq. (17) into Eq. (1) and integrating gives:

  • image(18)

where C denotes the enzyme activity at time t, C0 the initial enzyme activity. Equation (18) was used to fit all experimental data at once, using ζ, k, and m as adjustable parameters and was numerically integrated over the complete experimental τ-t profile of each shearing treatment. The estimated parameters were k= 6.05 × 10−5 s−1, ζ= 25.1 m2 mol−1 Pa1−n, and m = 0.57. The range of τ was not presented in the article. Taking τ= 18375 Pa (half of the τmax average), we calculated ΔE using Eq. (17)= 6.77 kJ/mol. The activation energy for thermal inactivation E0 is on the order of 240 kJ/mol (De Cordt and others 1994).

The E0 for starch gelatinization (45.64 J/mol, Cai and Diosady 1993) is higher than the strength of hydrogen bindings (10 to 40 kJ/mol, Semenova and Dickinson 2010) slightly. This suggests that other interactions stronger than hydrogen bindings may be involved in starch gelatinization. The E0s for the loss of 3 amino acids (67.60 to 124.71 kJ/mol, Ilo and Berghofer 2003), thiamin destruction (99.41 kJ/mol, Ilo and Berghofer 1998), inactivation of TIs (158 to 220 kJ/mol, van den Hout and others 1998), and α-amylase (240 kJ/mol, De Cordt and others 1994) are all lower than the strengths of C-C bonds (360 kJ/mol in C2H6, Israelachvili 1992) and C-O bonds (340 kJ/mol in CH3OH, Israelachvili 1992) that are very common in food molecules. This indicates that these processes may involve interactions weaker than the covalent bonds.

The activation energy reduction and the rate constant increase induced by thermomechanical processes are summarized in Table 1. Comparing the estimated ΔE values showed that the reduction of the activation energy for starch gelatinization is about one order of magnitude smaller than that for amino acid loss, thiamin destruction, and inactivation of TI and α-amylase. This suggests that shear stress has lower efficiency in catalyzing starch gelatinization than in catalyzing other reactions. In other words, thermal effects are the major contributor to the starch gelatinization in the experimental range investigated. The size of starch granules is much larger than that of amino acid, thiamin, and TI. Theoretically speaking, the larger starch granules should be more easily deformed by shear force and demonstrate more activation energy reduction than the smaller. A possibility is that shear leads to viscous dissipation and results in an additional thermal effect on the loss of smaller molecules.

Table 1–.  The activation energy reduction and rate constant increase induced by thermomechanical processes.
Thermomechanical processKinetic topicActivation energy and its reductionk/kT*Source
  1. aIn the original article E = 45.6369 J/mol, ΔE = 0.063 to 0.289 J/mol. The authors may have made a mistake on the unit of general gas constant. Please refer to text for detail. b Calculated as stated in text. cFor run 1 in Lei and others (2005) only. dVarying with the location in extruder. eFrom De Cordt and others (1994). fCalculated using Eq. (17) as taking τ= 18375 Pa (half of the τ max average).

  2. *Calculated using k/kT= exp (Δ E/RT). inline image and inline image for Cai and Diosady (1993). inline image and inline image for Ilo and Berghofer (1998, 2003). The central levels of N, T, and M were used to calculate inline image in Ilo and Berghofer (1998). For van den Hout and others (1998), the temperature in the die, about 100 °C, was used.

ExtrusionStarch gelatinizationE = 45.6369 kJ/mol, ΔE = 0.063 to 0.289 kJ/mol a1.02 to 1.10Cai and Diosady (1993)
ExtrusionThiamine destructionE = 99.41 kJ/mol, ΔE = 16.09 kJ/mol b76.88Ilo and Berghofer (1998)
ExtrusionAmino acid lossLys: E = 124.71 kJ/mol, ΔE = 1.89 to 3.18 kJ/mol blys: 1.67 to 2.42Ilo and Berghofer (2003)
 Arg: E = 67.60 kJ/mol, ΔE = 0.81 to 1.36 kJ/mol bArg: 1.25 to 1.46 
 Cys: E = 74.33 kJ/mol, ΔE = 1.75 to 2.95 kJ/mol bCys: 1.61 to 2.27 
ExtrusionWater solubility index increaseE0= 94.99 kJ/mol ΔE = 77.96 kJ/mol c8.13 × 108 to 3.14 × 1010Lei and others (2005)
ExtrusionInactivation of trypsin inhibitorE1= 200 kJ/mol, E2= 158 kJ/mol1.03 to 1.25van den Hout and others (1998)
 ΔE = 0.1 to 0.7 kJ/mol d  
Shear treatmentInactivation of α-amylaseE = 240 kJ/mol e, ΔE = 6.77 kJ/mol fT = 90 °C: 9.41van der Veen and others (2004)
  T = 110 °C: 8.37 

Different from the commonly used linear relationship ΔE = , Eq. (17) shows that the reduction of the activation energy for α-amylase inactivation was less than linear with increasing shear stress (m < 1). A possible explanation given by van der Veen and others (2004) is that the α-amylase is somewhat flexible and allows a certain alignment to the shear field. Alignment to the shear field reduces the hydrodynamic diameter of the enzyme, as a result of which the enzyme becomes less sensitive to the shear forces.

Application of the Basedow model

Shear-induced rate constants can be modeled by the Basedow model, while thermal induced by the Arrhenius relation. The problem is how to combine these 2 effects when both shear and thermal effects contribute to rate constants? One approach is that of Zhurkov and Korsukov (1974) by modifying the Arrhenius equation. Wang and others (1992) proposed another approach. They assumed that the contributions of thermal and shear energy to starch conversions are additive based on laws of thermodynamics. The overall rate constant k is equal to the summation of shear-induced rate constant kS, thermally induced rate constant kT, and shear and thermal interaction term.

  • image(19)

When kT is sufficiently small to be neglected, Eq. (19) is reduced to Eq. (4) and was used by Wang and others to model shear-induced starch conversion in extruder and capillary rheometer (Wang and others 1992; Zheng and Wang 1994; Gogoi and others 1995; Wang and Zheng 1995). In their reports, Wang and others took the molar volume of anhydroglucose units in the starch–water mixture as the activation volume V in Basedow model, V = 16200/(Psρs+ (100 − Psw), where Ps and Pw are the weight percentage of starch and water, respectively; ρs and ρw are the density of starch and water, respectively.

The activation energy for starch conversion by shear in extruder (5.10 to 19.19 J/mol, Zheng and Wang 1994; 167.4 J/mol, Gogoi and others 1995) and capillary rheometer (272.1 to 1023.8 J/mol, Zheng and Wang 1994) is about 2 or 3 orders of magnitude lower than that by heating only at extrusion temperatures and moisture contents (141.2 to 185.0 kJ/mol, Wang and others 1989). This suggests that shear energy is orders of magnitude more efficient in initiating the conversion of starch to cooked starch when compared with thermal energy. An interesting finding is that the activation energy for starch gelatinization in absence of shear from Cai and Diosady (1993) (45.6369 kJ/mol) is much lower than that for starch conversion also in absence of shear from Wang and others (1989) (141.2 to 185.0 kJ/mol). The starch gelatinization degree was determined based on formation of a blue iodine complex by released during gelatinization, while starch conversion degree was determined based on the enthalpy change of starch using a differential scanning calorimeter. Maybe, the difference in determining methods leads to the discrepancy in activation energy.

The results of Zheng and Wang (1994) indicated that shear activation energy for starch conversion decreases with temperature increase, and their relation can be expressed byES= A exp (BT). For starch extruded in a rheometer at 30% moisture content, coefficients A and B are 1.740 × 106 kJ/mol and −0.05068 k−1, at 35% moisture content, 4.443 × 106 kJ/mol and −0.0555 k−1. The exponential dependence of ES on temperature indicates that thermal effect still plays an important role in the shear conversion of starch. In addition, multiple-pass extrusion in a capillary rheometer showed that shear activation energy increases with shear rate. An unexpected finding of Zheng and Wang (1994) is that the shear activation energy from extruder treatment (8.0 to 19.2 J/mol at 80 to 110 °C and 30% moisture content) is much lower than that from rheometer treatment (464.0 to 1116.1 J/mol at 21 to 50 °C and same moisture content). Although the temperature for the rheometer treatment is lower, it is difficult to attribute the much higher activation energy from rheometer treatment to the lower temperature only.

Equation (19) was applied later by Qu and Wang (2002) to predict starch conversions in a single-screw extruder. In this study, kS was calculated using Basedow model with parameters obtained by Zheng and Wang (1994), kT was calculated using the empirical model developed by Wang and others (1989).

Application of a simplified Basebow model

If the range of shear stress in the Basedow model is small, the exponential function for rate constants is simplified to a linear function of shear stress:

  • image(20)

where k′ is a constant. Davidson and others (1984) used this linear relation to model starch degradation during extrusion. Assuming n = 1, they derived the following equation from Eq. (1)

  • image(21)

where C =[η]/[η]0, d is a constant. [η], [η]0 are the intrinsic viscosity of starch after and before extrusion, respectively.

Diosady and others (1985) assumed that only fully cooked starch was susceptible to shear degradation and modified Eq. (21) by introducing gelatinization degree of the extruded starch to give

  • image(22)

While determining shear stress, Diosady and others (1984, 1985) calculated apparent viscosity using the model of Harper and others (1971) and shear rate using NπD/H (D: extruder screw diameter, H: screw channel depth). Results showed that, for starch with 20% to 25% moisture content, k′= 1.7 × 10−6 m2/N-s and d =−0.05 (Davidson and others 1984), with 25% to 30% moisture content, k′= 0.39 × 10−6 m2/N-s, and d =−0.08 (Diosady and others 1985). This indicated that lower moisture content leads to a pronounced response of degradation rate constant to shear stress.

Cai and others (1995) also used Eq. (21) to model starch degradation in extrusion. Different from Davison and others (1984) and Diosady and others (1985), they calculated the shear rate using Eq. (9). Like Cai and others (1993), they rewrote Eq. (22) to give

  • image(23)

It is known from Eq. (11) that the term τG−0.51 in Eq. (23) is equal to inline image. Its value can be calculated from the material temperature, moisture content, and screw speed for a given extrusion run. The values of kG0.51 and d were obtained by fitting starch degradation data to Eq. (23), being 6.01 × 10−7 m2/N-s and −0.559, respectively.

Assuming n = 1, the following starch gelatinization kinetic model is obtained by substituting Eq. (6) in Eq. (1):

  • image(24)

In this case, C in Eq. (1) donates 1−Dg. Substituting Eq. (24) in Eq. (22) gives (Cai and others 1995)

  • image(25)

This model can predict the degradation of extruded starch based on material temperature, feed moisture, extruder screw speed, and starch gelatinization degree.

Equation (20) was used in an alterative way by Cha and others (2003) to describe shear effects on thiamin retention in extruded wheat flour. They obtained the following empirical model by regressing the shear history on thiamin retention:

  • image(26)

where b1 and b2 are constants, RΦ is the thiamin retention when only shear history Φ (inline image) is considered, RΦ = CΩ,Φ/CΩ,Φ=0, CΩ,Φ is the thiamin concentration after extrusion, CΩ,Φ=0 is the thiamin concentration after a treatment where the thermal history Ω is equal to that in extrusion and Φ= 0. Then

  • image(27)

Differentiating both sides with respect to time, we get

  • image(28)

At relatively low reaction conversions, CΩ,Φ≈ CΩ,Φ=0, then Eq. (28) can be written as

  • image(29)

Clearly, Eq. (29) suggests that thiamin retention as a function of shear history follows a 1st-order kinetic model, and its rate constant is inline image. Equation (20) shows that the rate constant is k′τ. When η is constant, also noticing inline image, Eq. (29) is in line with Eq. (20).

Using the empirical shear rate model obtained by Cha and others (2003), inline image, we calculated shear rates from screw speed (N = 1.67 to 5 round per second). The resulting inline image is 44.1 to 119.8 s−1. With b1=−3 × 10−5 given by Cha and others (2003), the rate constants inline image is calculated, 1.31 × 10−3 to 3.59 × 10−3 s−1.

Bueche- and Zhurkov-like models

In the Basedow and Zhurkov models, the shear energy input is expressed as the product of shear stress and activation volume. Another way to quantify the shear effects in a thermomechanical treatment, especially in extrusion cooking, is specific mechanical energy (SME). SME is defined as shear energy per unit weight of treated material. It reflects the thermomechanical history of the material during the entire mechanical treatment. To overcome the constraint by using only product temperature, Lei and others (2005) constructed an SME model (Eq. (30)) by adapting the Arrhenius equation, and SME-Arrhenius models (Eqs. (31) and (32)) by incorporating SME into the Arrhenius equation to describe the increase in water solubility index of rice flour after extrusion.

  • image(30)
  • image(31)
  • image(32)

where ESME is SME constant, ET is temperature constant, ESME-T is SME and temperature constant. Regression analysis showed that, compared with the Arrhenius relationship, all these 3 models gave improved correlation coefficients.

Equation (30) has a similar form of the Basedow model, except that τV is replaced by SME in this case. So, Eq. (30) can be called Basedow-like model. Equation (32) can be rewritten as:

  • image(33)

where E0= ETR. When ESME is negative, term (−RT-ESME)/SME can be regarded as the ΔE in the Zhurkov model. So, Eq. (32) can be called as Zhurkov-like model. In this case, ΔE is a function of temperature, SME and ESME. After copying the Figure15 of Lei and others (2005) into Photoshop 5.0, we estimated the extrusion temperatures of the run 1 and its corresponding rate constants with the aid of the ruler in the operating window. The obtained data were regressed using Eq. (33) with taking (RT·ESME)/SME as one parameter. The resulting E0 and ΔE are 94.99 and 77.96 kJ/mol, respectively. The calculation of k/kT (see Table 1) showed that mechanical effect dominates the water solubility index increase. However, Lei and others (2005) demonstrated that the correlation coefficients with the Arrhenius equation, even lower than those of Eqs. (30) to (32), were still considerable (0.87). This discrepancy may suggest that the Zhurkov theory is not valid for the very complicated event such as water solubility increase, which involves starch degradation and gelatinization, and even protein degradation.

Maximum stress-based degradation model

As for shear-induced starch degradation, Davison and others (1984) and Diosady and others (1985) related the extent of degradation to the product of shear stress and time. This means that starch degradation is time dependent. Contrary to these studies, the fundamental investigations of Grandbois and others (1999) and Beyer (2000) suggested that breakdown of a covalent bond in an amylose molecule takes place almost instantaneously in the case that the force applied exceeds the bond strength. Inspired by this finding, van den Einde and others (2004a) proposed that shear degradation of starch during a thermomechanical treatment will occur only when the shear stress exceeds a critical value τcri. The degradation is maximum stress dependent, other than time dependent, and follows an exponential relation with the maximum stress.

  • image(34)

where XM is relative intrinsic viscosity of the molten fraction of the treated starch due to mechanical breakdown (=[η]/[η]0), c1 and c2 are constants, τmax is the maximal shear stress occurring during a treatment. van den Einde and others (2004b) indicated that, at 43% moisture content, c1= 1.46, c2=−1.63 × 10−5, τcri= 2.32 × 104 N/m2; at 30% moisture content, c1= 1.15, c2=−2.18 × 10−5, τcri= 7.74 × 103 N/m2.

The above model gave a good prediction of the relative intrinsic viscosity of extruded waxy corn starch with assuming that shear and elongation in an extruder have the same effect on the molecular breakdown and that the combined effect of simultaneous shear and elongation be equal to the effect of the larger one (van den Einde and others 2005). In another work, van den Einde and others (2004c) proposed a molecular scale model that relates the molecular breakdown of waxy corn starch with the maximum shear stress on the polymer and molecular structural characteristics. The application of this model to their experimental data yielded a fractal dimension of 1.9 of amylopectin and a C-O bond strength of 272 kJ/mol, which were in good agreement with literature data.

van den Einde and others (2004b) indicated that the critical stress necessary for the degradation of waxy corn starch with 30% moisture content is 2.32 × 104 N/m2. This value is very close to the shear stress threshold (2.5 × 104 N/m2) above which α-amylase is rapidly deactivated in a plain shear field (van der Veen and others 2004). The minimum stress necessary for the conversion of wheat starch with 30% moisture content at 100 °C is a little lower (1.0 × 104 N/m2, Zheng and Wang 1994). With temperature increase (Zheng and Wang 1994) or moisture decrease (van den Einde and others 2004b), the minimum stress increases. The critical shear stress beyond which the viability of Chaetoceros muelleri in solution drop sharply is 4 orders of magnitude lower (1 to 1.3 N/m2, Michels and others 2010). Probably, having a larger size than starch and α-amylase makes C. muelleri prone to be deformed and lose viability.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Kinetic Models for Mechanochemical Reactions
  5. Mechanochemical Kinetics in Food Processing
  6. Discussion
  7. Conclusion
  8. References

Limitations of the mechanochemical approach to food kinetics

Originally, both the Basedow and Zhurkov models were developed to describe the scission of covalent bonds in polymers. In application to food, only starch degradation involves covalent breakage. Many events occurring in food processing involve other molecular interactions. For example, starch conversion or gelatinization involves mainly hydrogen bonding. The inactivation of enzymes and their inhibitors may involve hydrogen bonding, electrostatic interactions, hydrophobic interactions, and even covalent bonds. The loss of small molecules (such as amino acids, thiamin) may result from degradation and/or complexion with other food components. The water solubility increase of cereal flour after extrusion may be a result of starch degradation and gelatinization, and even protein degradation. These very complicated events usually involve various interactions at the atomic, molecular, and supramolecular levels. Different interactions will respond distinctively to the applied mechanical force. So, the detected change is actually the combination of various responses at various levels. In addition, both off-reaction and on-reaction can occur during the time course of experiment. So, strictly speaking, the mechanochemical kinetic models are only borrowed to describe the events phenomenologically, rather than mechanistically.

When both shear and extensional stresses take effects

In most kinetics literature reported up to now, only shear stress was accounted for while calculating mechanical effects. Actually, forces are not only put onto materials due to shear flow, but also due to elongation flow. Estimation from the Trouton ratio of extruded corn meal (Bhattacharya and others 1994) and the average elongation rates in a twin-screw extruder (van der Wal and others 1996) showed that the extensional stress in an extruder can be high compared to the shear stress. The effect of elongation flow on starch breakdown during extrusion was not negligible (van den Einde and others 2005). Hence, elongation stress must be taken into account while calculating mechanical effects on reaction rates. Extensional and shear stresses may have different efficiency in activating a reaction. When both stresses occur in a thermomechanical treatment, van den Einde and others (2005) thought that the higher one dominated the reaction rates. According to the theory of Zhurkov, it is the energy input that modifies the rate constants. Both shear and extensional forces can reduce the activation energy. So, we suppose that their effects would be synergistic.

When shearing has opposite effects to heating

The Zhurkov model assumes that the shear action catalyzes thermal degradation by lowering its reaction energy barrier. In some cases, shearing and heating may have opposite effects on the direction of a reaction. For example, during flour dough mixing, heating causes polymerization of gluten and glutenin, while shear force leads to their degradation. Assuming that the polymerization results from the cross-linking of thiol groups, and the degradation results from the breaking of disulfide linkage, Strecker and others (1995) derived the following mechanistic-based reaction rate model:

  • image(35)

where S and SH are the concentration of disulfide and thiol groups, respectively; kp and kd are rate constants of polymerization and degradation, respectively, and expressed as the Arrhenius relation; np and nd are their corresponding reaction orders.

As an alternative way, the shear force can be considered to prevent the thermo-induced reaction through increasing its activation energy. Then, the following modified version of the Zhurkov model may be used to model the rate constants of disulfide group disappearance and other similar reaction processes.

  • image(36)

The step accelerated by mechanical force for a multistep reaction

A reaction process may include several steps. The step with lowest speed will control the final reaction rate and is called rate-limited step. For a low-moisture reaction system, the diffusivity of reactants is small. The diffusion of 2 reactant molecules before they colloid is the rate-limited step. Taking starch gelatinization for example, at least 3 steps are involved: the diffusion of water into starch granules, hydration of the starch granules accompanied with other phenomena such as radial swelling, loss of optical birefringence and crystalline order, and amylose leaching (Hoover 2001). When there is sufficient moisture, diffusion is fast, then the hydration of starch through hydrogen bonding will be the rate-limited step. In this case, starch gelatinization will be accelerated by the applied force acted on the hydrogen bonding. When there is insufficient moisture, the moisture availability will be the rate-limited factor (Donovan 1979). In this case, only when the diffusivity of water is speeded up, can the starch gelatinization be accelerated. So, the shear force can accelerate starch gelatinization through lowering the activation energy for starch hydration or through disrupting the starch granules and making moisture be available by their inner parts.

Time or stress dependent?

van den Einde and others (2004a, 2004b) found that the degradation of starch by mechanical treatment is an instantaneous process. Actually, similar findings have been documented. Van Lengerich (1990) found that the molecular weight of extruded wheat starch decreased sharply when SME reaches to about 130 Wh/kg. Van der Veen and others (2004) found that peak stresses play an important role in determining the shear-induced inactivation of α-enzyme despite both time and shear stress being involved. According the theory of Zhurkov, these instantaneous processes can be explained. The reduction in activation energy is so large that the time scales become small compared to those of the experiment.

Theoretically speaking, once the force acted on a chemical bond exceeds its strength, the bond will be broken. However, Eq. (34) indicated that further breakdown of starch needs higher shear stress. Actually, when a polymer molecule is stretched, the macroscopically applied force is not evenly distributed along the chain. The tension on the center link of the chain is maximal (Bueche 1960). When the maximal tension exceeds bond strength, the polymer molecule will spilt into 2 segments with almost half the size. The nub of the problem is that the bulk stress necessary to make the tension on the center link up to bond strength increases with molecular size decrease (Bueche 1960). So, only when the applied stress is increased further, can it lead to additional breakdown.

The mechanical breakdown of starch is time independent; however, its thermal breakdown is time dependent, at least within the first 300 s of the thermomechanical treatment (van den Einde and others 2004b). Then, thermomechanical breakdown at higher temperatures (≥140 °C), which is the sum of thermal and mechanical breakdown, will be time dependent in the first 300 s. The mean residence time in an extruder is usually not more than 2 min. This can explain partially the time-dependent observation of starch degradation at 121 to 177 °C (Davidson and others 1984; Diosady and others 1985).

Molecular- and bulk-based kinetics

SMFS, an atomic force microscopy-based technique, has been widely used to interpret the noncovalent and covalent interactions of both biomacromolecules and synthetic polymers (Grandbois and others 1999; Janshoff and others 2000; Zhang and others 2003; Linke and Grützner 2008). In this technique, molecules of interest are pulled apart under an external loading force. The distribution of rupture forces at a constant loading rate enables the prediction of the most probable rupture force, which is a quantitative measure of the interaction at that loading rate. A force spectrum of the single-molecule interaction is generated by mapping the most probable rupture force as a function of the loading rate. Fitting the force spectrum data to different theoretical models will yield kinetic and thermodynamic information for the single-molecule interaction. The Bell-Evans equation is the most commonly used model (Porter-Peden and others 2008).

The SMFS technique would shed fresh light on the reaction kinetics induced by mechanical forces. While applying SMFS to the kinetic analysis of a bulk food material, 3 problems need to be solved. First, what is the relationship of the effects of the macroscopically applied force and the force exerted locally to molecule bonds on reaction rates? Only tensional force is applied in SMFS study, while bulk food materials are usually treated by shear force. This makes the molecule-level force different from the bulk force in direction. In addition, the bulk food materials are usually characterized as viscoelastic, which make the 2 forces different in magnitude also. Second, just as stated above, an often-seen event in food may involve several kinds of molecular interactions that may occur simultaneously or successionally. So, which interaction dominates the rate of the reaction needs to be answered. Third, the reaction environments (such as pH, moisture content) while taking SMFS experiments is usually different the bulk reaction conditions. This makes it very difficult to apply the kinetic results from SMFS analysis to bulk reaction prediction directly.

For a reaction occurring in a practical food processing operation, such as extrusion cooking, the situation will be further complicated. The rate-influencing factors, such as temperature, shear rate, and shear stress, may change with location. In addition, for the reactions such as starch gelatinization and degradation, their development will in turn modify the material (for example, viscosity) and make these factors change also with time. The ideal approach to reveal the kinetic process of a reaction in an extruder is establishing the flow model of the material in the extruder, the heat transfer models within the melt and between the melt and extruder barrel, and the kinetic model of the reaction. Then, the 3 models are solved coupledly. Assuming the molecule in SMFS analysis has the same reaction environments as in bulk, the kinetic results from SMFS analysis can be used in the above overall models. Van den Hout and others (1998) took this approach to predict TI inactivation during extrusion except that they determined the parameters of the single-molecule kinetic model from theoretical analysis of molecule deformation other than SMFS analysis. Usually, the assumption cannot be met. Then, we can assume the bulk kinetic model takes the form of the Zhurkov or Basedow models with parameters to be determined. After measuring the content of the reactant, the model parameters can be estimated reversely. This approach was used by van der Veen and others (2004) to determine the relationship between activation energy reduction and bulk shear force.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Kinetic Models for Mechanochemical Reactions
  5. Mechanochemical Kinetics in Food Processing
  6. Discussion
  7. Conclusion
  8. References

In some food processing operations, such as extrusion, ball milling, and high-pressure homogenization, raw materials, are sheared and heated simultaneously. Mechanochemical kinetics, which has been deeply studied in polymer science, has been applied to food reactions induced by the thermomechanical treatments. Up to now, the mechanochemical approach has been used to model starch degradation, gelatinization or conversion, the loss of amino acids and thiamin, and the inactivation of α-amylase and TIs, occurring mainly during extrusion cooking.

The Basedow and Zhurkov models are often used to describe the mechanically activated reactions. Their simplified and variational versions were also used in food research. When thermal effects are negligible, the Basedow model is suitable. When thermal effects are also considered, the Zhurkov model is applicable. Wang and others (1992) proposed another way to combine the thermal and shear effects on reaction rates. The linear relationship ΔE = τV is often used to calculate activation energy reduction. Other power law relations have been proposed theoretically or empirically.

Since the changes occurring in food systems often involve interactions at the atomic, molecular, and supramolecular levels, the mechanochemical models can describe them only phenomenologically rather than mechanistically. Single-molecule technique may open a new way to elucidate the relationship between the reaction rates and the applied forces at molecular scale. However, considering the complication of the changes in food and the difference between macroscopically applied and the molecule-level forces, there is still a long distance between the single-molecular kinetics and the bulk food kinetics.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Kinetic Models for Mechanochemical Reactions
  5. Mechanochemical Kinetics in Food Processing
  6. Discussion
  7. Conclusion
  8. References