- Email: [email protected]
stickiness development in amorphous sugar particles to the classical Arrhenius law
Journal of Food Engineering 79 (2007) 675–680 www.elsevier.com/locate/jfoodeng
Conformability of the kinetics of cohesion/stickiness development in amorphous sugar particles to the classical Arrhenius law Xiao Dong Chen
*
Department of Chemical Engineering, Faculty of Engineering, Monash University, Clayton Campus, Vic. 3800, Australia Received 11 November 2005; accepted 20 February 2006 Available online 18 April 2006
Abstract Glass-transition related stickiness or cohesiveness has been a topic of significant scientific and practical interest in the last decade. The concept is very important in both food and pharmaceutical product developments. It is more appropriate to consider its development as a rate process but the time-response has been very difficult to measure accurately until lately. Here, the classical rate law for chemical reaction has been tested for its fitness in correlating the time-response of cohesion or stickiness developed in packed amorphous particles during temperature/humidity treatments. The rate law has been benchmarked against the T Tg dependent rate function established in the previous studies [Foster, K. D., Bronlund, J. E., & Paterson, A. H. J. Glass transition related cohesion of amorphous sugar particles. Journal of Food Engineering, in press.], using the original data sets for amorphous sucrose particles measured by Foster. It has been shown that the classical rate law is an excellent approach to accurately account for the process. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Stickiness; Rate of stickiness development; Amorphous sugar particles
1. Introduction The stickiness or cohesion of packed amorphous sugar powders is recognized as a function of temperature and water content (or relative humidity). Once the material, being subjected to certain temperature and humidity conditions that exceed the corresponding Tg, the material is known to undergo glass-transition, transforming from amorphous glass to a rubbery/viscous state (Aguilera, de Valle, & Karel, 1995; Aguilera, Levi, & Karel, 1993; Roos & Karel, 1990, 1991a, 1991b, 1992; Slade & Levine, 1991). This transformation is known to have a significant impact on product stickiness and caking (Lloyd, Chen, & Hargreaves, 1996; Wallack & King, 1988) and dried product quality (Bhandari & Howes, 1999). Tg can be plotted as a function of water content on dry basis (X), as shown qual*
Tel.: +64 9 373 7599; fax: +64 9 373 7463. E-mail addresses: [email protected], [email protected]. edu.au. 0260-8774/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2006.02.027
itatively in Fig. 1. For some years in recent time, the sticking point temperatures (SPT) have been measured using a variety of devices as summarized by Boonyai, Bhandari, and Howes (2004). Most methods only record the transition points for each combination of temperature and humidity. If the equilibrium isotherm of the particular material of concern is not sensitive to temperature change in the range considered, X can be replaced by water activity (aw) or relative humidity (RH). The sticking phenomena treated as a finite rate process, rather than an instantaneous process, has been contem¨ zkan, Waliplated in recent years (Levine & Slade, 1986; O ¨ zkan, Withy, & Chen, 2003; singhe, & Chen, 2002; O Paterson, Brooks, Bronlund, & Foster, 2005). Quantitatively, the approach is only possible when the development of stickiness or cohesion over time can be reliably and accurately recorded. Paterson et al. (2005), based on their experimental data, have proposed that the rate of glass-transition related cohesion in powdered material is only a function of (T Tg) and
676
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680
Foster, Bronlund, and Paterson (2005) carried out similar experimental tests on several other powders (pure substance and mixtures) and they generalized rate law where the rate of change in strength was assumed to be a function of T Tg only.
Tg or SPT
Sticking Point Temperature
dS 10aðT T g Þþb dt Glass-transition Temperature 0
X or a w
Fig. 1. Glass-transition temperature (Tg) and sticking point temperature (SPT) curves.
linking the rate of stickiness development in amorphous powders through the Frenkel’s equation for inter-particle bridge radius with time for coalescence of particles due to surface energy driven viscous flow (Frenkel, 1945), 3rp rt ð1Þ r2b ¼ 2l They then used the viscosity equation described by Williams, Landel, and Ferry (1955), i.e., the WLF equation, which is ! 2 rb 3rp r C 1 ðT T g Þ log ð2Þ þ ¼ log 2lg C 2 þ ðT T g Þ t to correlate the rate against (T Tg). It was suggested that the strength of a liquid bridge is proportional to the cross-sectional area ðpr2b Þ. They devised the ‘blow test’ to establish the time course of the stickiness development, which allows one sample to be subjected to a constant set of air conditions (T and RH) and the forces holding the particles together and the stickiness to be measured incrementally with time (Brooks, 2000). The rate of change in the strength of the packed particle bed was then correlated using Eq. (2). The LHS of Eq. (2) is replaced by the blow test end point (the strength with the unit of L min1) (S) to the RHS of the WLF equation, i.e., logðSÞ ¼ a þ b
C 1 ðT T g Þ C 2 þ ðT T g Þ
ð3Þ
The plot following Eq. (3) has produced an approximate linear relationship between the LHS and the 2nd term on the RHS. The value obtained in these sticking tests for C1 was 1.7 which is very different from the universal constant of 17.44 proposed by Williams et al. (1955). The constant C2 of 3.5 °C was also found to be quite different from the original WLF model constant of 51.6. This may indicate the mechanism proposed based on the Frenkel and WLF theories is not totally correct at least from a quantitative view point. It is noted that the symbol S can generically represent a sticking force or an indirect measurement such as the air flow rate required to move a deposited mass etc.
ð4Þ
where a and b are correlation coefficients. All these have pointed to the same insight intended to illustrate by these authors, i.e., the rate of change in strength (stickiness or cohesiveness) is directly (and only) related to T Tg. It is noted that the scatters appeared in the log rate plots are sometimes fairly large. These data sets (Foster, 2002) are highly valuable and must have been very time-consuming to obtain them in laboratory. An attempt was made to correlate the sticking/caking ¨ zkan et al. (2003) for the skim milk powders time by O using essentially the following first order equation: dS E ¼ f ðX Þrm exp ð5Þ ðS 1 SÞ dt RT where r is the pressure, m is the pressure index, E is the activation energy, R is the universal gas constant (8.31 J mol1 K1), T is the absolute temperature in Kelvin (K), and f(X) is a parameter whose value depends on the moisture content of the powders. The influence of the moisture content of the powders on the sticking/caking time was however not investigated. It is expected that the influence of the moisture content of the powders can be included in Eq. (5). The stickiness in skim milk powder mainly arises from the glass-transition of the amorphous lactose content which takes up about half of the solids amount in the powder (Richard Lloyd & Xiao Dong Chen, 1993; unpublished data). ¨ zkan et al. (2003), S is the penetration In the study by O force measured for 6 mm penetration depth into a skim milk powder compact. In these tests, the milk powder compacts were made first and then placed in the Instron machine (Model 5567, UK). The force to penetrate into the sample was then measured to a depth of 6 mm. The highest force recorded would represent the strength of the ¨ zkan et al., 2003). Assuming that the temperacompacts (O ture dependence on the caking time follows the Arrhenius behaviour, i.e., the classical approach, the activation energy was estimated from the slope of the ln (1/t) against (1/T) plot (the solution of Eq. (5)), where t is the sticking time. The Arrhenius plots were made for the skim milk powder compacts consolidated at 14.1 and 23.4 kPa and were evaluated with the average activation energy found to be 270 kJ mol1. Eq. (5) is the ‘classical approach’ to modeling chemical reaction kinetics. In the following section, a detailed development of the classical model incorporating the effect of water content (therefore the effect of Tg) will be presented. The work is based on the laboratory results obtained (and kindly provided to this author) by Foster (2002).
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680
The question will be answered is whether or not Eq. (5) is appropriate for describing the kinetics of the sticking/cohesion problem. 2. The classical approach The temperature dependence is one of the most important characteristics of chemical reactions, especially for the elementary reactions. The rate of reaction is usually written as the product of a temperature dependent term and a composition dependent term: r_ ¼ k f ðcompositionÞ
ð6Þ
The reaction rate constant k is a function of temperature: k ¼ k o eE=RT
ð7Þ
Eq. (7) is the very well known Arrhenius’ law. ko is called the frequency factor (s1) and E is called the activation energy of the reaction (J mol1). This expression has been found to fit the experimental results on chemical reaction rates well over wide temperature ranges. This expression has been proven to be good approximation to temperature dependency of both collision and transition-state theories (Levenspiel, 1972). For species A undergoing reaction, the rate of conversion of species A can be expressed as follows: dC A ¼ k C nA dt
ð8Þ
C denotes concentration (kg m3) and n is the reaction order. The glass-transition is an endothermic reaction process which can be facilitated by the existence of the plasticizer. For amorphous sugar systems, water is a good plasticizer. The process is marked by the change in the phase (glassy material to rubbery state—a kind of liquid). The development of the cohesiveness in amorphous particulate material is therefore related to glass-transition which signifies the on-set of the viscous nature of the surface of the particles. The most related property to this is perhaps the viscosity of the surface material, which can be correlated to the WLF equation incorporating the glass-transition temperature Tg as mentioned earlier. The Tg has been taken as either a function of equilibrium relative humidity RH (or water activity aw) or a function of water content on dry basis X. These two concepts are inter-exchangeable when the temperature dependence of the equilibrium isotherms in the temperature range of concern is insignificant. This is approximately true for amorphous sugar particles in the relatively narrow range concerned in industry (such as 10–50 °C). According to the Arrhenius’ law, a single E would mean there is a single mechanism that dominates the reaction process. As such, when applying the classical approach, E is expected to be the same for all Tg conditions. This represents that the mechanism of the transition is taken to be the same irrespective of what the water content or temperature
677
is. This restricts the classical approach, as far as the fitting capacity is concerned, having to have an adjustable reaction frequency factor k as a function of water content (thus a function of Tg). As suggested earlier, Eq. (5) can be seen as a derivative of the classical chemical reaction rate equation of the 1st order. The reaction species A is now (S1 S): dS E ¼ k o exp ð5Þ ðS 1 SÞ dt RT where ko should ‘account for’ all the effects of water content and compaction pressure. Since there was no deliberate compaction and the packing condition was the same for all the tests by Foster (2002), this effect of compaction pressure is not investigated in the current work. Only the water content dependence function for ko is of the interest. Re-arranging Eq. (5), gives: dS E dt ko ¼ exp ð9Þ RT ðS 1 SÞ Practically we are more interested when the stickiness starts to develop significantly so that the initial (high) rate is very important. Considering S ! So, the above Eq. (9) can be re-written as: dS E dt o ko exp ð10Þ RT ðS 1 S o Þ Since E is expected to be quite large following the result gi¨ zkan et al. (2003), a preliminary analysis can be ven by O based on the well-known simplification (Bowes, 1984) using the transition temperature Tg as the reference temperature, i.e., dS E dt o ko exp RT ðS 1 S o Þ ! dS E E dt o exp exp 2 ðT T g Þ ð11aÞ RT g ðS 1 S o Þ RT g or ! dS E E k o ðS 1 S o Þ exp ðT T g Þ exp dt o RT g RT 2g ð11bÞ Tg in the above two equations has the unit of Kelvin (K). The effect of T 2g in the last exponential term is not significant in determining the accuracy of the above equation as shown in previous works on self-heating (or spontaneous heating) research at low temperatures (e.g., Wake, 1982) such as the range investigated here. For instance, an average temperature between the Tg and the maximum temperature of the range may be used instead. The last exponential term in Eq. (11b) effectively coincides with the RHS of Eq. (4) by Foster et al. (2005). As such, in order for the reaction rate Eq. (5) to conform to the new (T Tg) dependence model
678
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680
to a large extent, i.e., Eq. (4), k o ðS 1 S o Þ exp RTE g
7.0E+49
Thereshould be somewhat constant at different T g values. E fore, ko should have a function such as exp to be able RT g E to ‘absorb’ the effect of exp RT g .
ðS 1 S o Þ
1
E RT g
was plotted against exp
, a linear relation-
ship was obtained, as shown in Fig. 3. It was found that the best fit activation energy E was 304 kJ mol1, which com¨ zkan et al. (2003) for skim milk pares well with the result of O powder, which was 270 kJ mol1. As such, one rate law, based on the classical approach, for sticking and cohesion development due to glass-transition, can be further derived:
(dS/dt)o /(S_inf -So)/ exp(-E/RT)
Since the initial ‘linear’ rate of stickiness development is of particular interest as it sets off the fastest increase in stickiness or cohesiveness, Foster’s data (2002) on amorphous sucrose particles, which had the most complete set of conditions evaluated previously (Foster et al., 2005), have been employed. These data were kindly provided by Foster which have been re-analyzed to focus more on the estimation of the initial rate (the number of data used was therefore in general fewer than the results used by Foster et al., 2005). Any significant sign of ‘trend leveling’ has been avoided. Most importantly, the values for So and S1 are necessary for the current ‘classical’ approach. As such only the sets that showed the apparent asymptotic values have been collected. The So values can be estimated accurately to be within ±0.3 L s1 but S1 was more difficult to estimate accurately (within approximately ±1 L s1 error range). It must be noted that there was no obvious trend found in the present work for S1 to be a single function of Tg. In order to show that the current analysis does not really affect the general conclusion of Foster et al. (2005), Fig. 2 has been produced which shows that the rates obtained by re-calculations do not alter the perception in the previous investigation that the rate is a function of T Tg. As a significant indication of the conformability dS of Eq. (5) E to Eq. (4), is that when the experimental exp RT dt o
y = 0.000820x
6.0E+49
R2 = 0.9666
5.0E+49 4.0E+49 3.0E+49 2.0E+49 1.0E+49 0.0E+00 0.E+00
2.E+52
4.E+52
6.E+52
8.E+52
exp (E/RTg ) Fig. 3. The approximate relationship established from Eq. (11b).
dS E E ¼ Ao exp ðS 1 SÞ dt RT RT g
ð12Þ
which seems to be able to accurately simulate the laboratory data. Note again here both T and Tg in the above equation are in unit of Kelvin (K). Based on the results shown in Fig. 3, the new rate constant Ao 0.00082. Eq. (12) predicts a non-linear development of S over time. Integration of Eq. (12) at constant T and Tg, yields the following: S ln SS11S o t¼ ð13Þ E Ao exp RT RTE g When the stickiness or cohesiveness reaches 90% of its maximum, the time required (t90) is given below: t90 ¼
lnð0:1Þ E RTE g Ao exp RT
ð14Þ
The model such as Eq. (4) does not have this feature. Fig. 4 shows the predicted results for the amorphous sucrose particles investigated by Foster (2002). One would notice that at the more instantaneous sticking region, the spread
100 1.E+03 Tg = 20 °C
Current analysis
10
1 0
5
10
15
20
25
0.1
0.01
T-Tg ( °C)
Fig. 2. The rate of the stickiness development in amorphous sucrose particles versus T Tg.
Time to reach 90% maximum stickiness (hr)
(dS/dt) o (L.min-1 .hr -1)
Foster et al (2005)
Tg = 30 °C
1.E+02
Tg = 40 °C Tg = 50 °C
1.E+01
Tg = 60 °C
1.E+00 0
10
20
30
40
50
60
1.E-01 1.E-02 1.E-03
T-T g ( °C)
Fig. 4. Time-to-reach 90% maximum stickiness/cohesiveness predicted using Eq. (13).
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680 T = 58.7 °C, RH = 12.9% 25
S (L.min-1)
20
crystallization stage
15 10
5 glass-transition stage 0 0.0
0.5
1.0
1.5
Time (hr) Fig. 5. Two possible stages of the sticking/cohesion/caking process.
of times predicted is significant for the same T Tg. This shows that there is a possibility of the independent acts of T and Tg (or water content) making the rate of sticking not solely a function of the temperature difference T Tg. 1 1 Instead the numeric ‘driving force’ is now T T g , which is an interesting outcome of the current analysis. When T ! Tg, on the other hand, the two driving forces become very similar. 3. Further remarks on the sticking or cohesion development process Based on the data provided by Foster (personal communications, 2005) and also on the previous understanding established in literature, once the glass-transition has been triggered off, provided also that the temperature is relatively far away from Tg, the development of the crystallization is expected. As time goes, a more aggressive increase of the bonding strength between the contacting particle surfaces is resulted, as crystallization reinforces (or consolidates) the already formed amorphous liquid bridges. Some of the experimental results are in favor of this argument. Fig. 5 shows one of those, the solid line indicates the glass-transition induced sticking and then a more significant rise in bonding strength happens, which may be directly linked to the crystallization process. When T Tg is very large, the apparent asymptotic trend due to amorphous material’s transition from glass would disappear due to the fast development of the crystallization process taking over from the glass-transition process. If one is interested in the whole process from sticking through to caking, a two-stage reaction model may be more appropriate. However, this is beyond the scope of the current study. 4. Conclusions In this study, it has been clearly demonstrated that the classical rate law of chemical reaction is a good alternative
679
to correlate the rates of the development of stickiness or cohesiveness resulted from the glass-transition phenomena. A simple formula (Eq. (12)) has been derived to accommodate the experimental data sets provided by Foster (2002), which enables the time to a finite level of stickiness or cohesiveness to be calculated. It can be shown that in this approach, the ‘driving force’ for the stickiness or cohesion development as a result of glass-transition of the amorphous material is T1 T1g rather than (T Tg). The two different approaches converge as T gets very close to Tg. Furthermore, it has been suggested through a brief discussion, the whole process of sticking/cohesion/caking process may be modeled by a two stage reaction model. In any case, the rate model established in this work can be used for predicting the transient stickiness/cohesiveness profiles inside a packed particle bed when the bed is subjected to temperature gradient or humidity gradient. It may also be useful in formulating the rate equations for other glass-transition related phenomena such as degradation or leakage of valuable compounds which are encapsulated in an amorphous sugar matrix. Acknowledgements The author is very grateful to the original spreadsheet data sets kindly provided by Dr. Kylie D. Foster at Institute of Food, Nutrition and Human Health, Albany Campus, Massey University, North Shore, Auckland. References Aguilera, J. M., de Valle, J. M., & Karel, M. (1995). Caking phenomenon in amorphous food powders. Trends in Food Science and Technology, 6, 149–155. Aguilera, J. M., Levi, G., & Karel, M. (1993). Effect of water content on the glass transition and caking of fish protein hydrolyzates. Biotechnology Progress, 9, 651–654. Bhandari, B. R., & Howes, T. (1999). Implication of glass transition for the drying and stability of dried foods. Journal of Food Engineering, 40, 71–79. Boonyai, P., Bhandari, B., & Howes, T. (2004). Stickiness measurement techniques for food powders: a review. Powder Technology, 145(1), 34–46. Bowes, P. C. (1984). Self-heating: evaluating and controlling the hazards. Amsterdam: Elsevier, pp. 26–27. Brooks, G. F. (2000). The sticking and crystallization of amorphous lactose. Masters of Technology Thesis. Massey University, Palmesrton North, New Zealand. Foster, K. D. (2002). The prediction of sticking in dairy powders. PhD Thesis. Massey University, Palmerston North, New Zealand. Foster, K. D., Bronlund, J. E., & Paterson, A. H. J. Glass transition related cohesion of amorphous sugar particles. Journal of Food Engineering, in press, doi:10.1016/j.jfoodeng.2005.08.028. Frenkel, J. (1945). Viscous flow of crystalline bodies under action of surface tension. Journal of Physics (USSR), 9(5), 385–391. Levenspiel, O. (1972). Chemical reaction engineering (2nd ed.). New York: John Wiley and Sons. Levine, H., & Slade, L. (1986). A polymer physico-chemical approach to the study of commercial starch hydrolysis products (SHPs). Carbohydrate Polymers, 6, 213–244.
680
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680
Lloyd, R. J., Chen, X. D., & Hargreaves, J. B. (1996). Glass transition and caking of spray dried lactose. International Journal of Food Science and Technology, 31, 305–311. ¨ zkan, N., Walisinghe, N., & Chen, X. D. (2002). Characterization of O stickiness and cake formation in milk powders. Journal of Food Engineering, 55, 293–303. ¨ zkan, N., Withy, B., & Chen, X. D. (2003). Effects of time, temperature, O and pressure on the cake formation of milk powders. Journal of Food Engineering, 58, 355–361. Paterson, A. H. J., Brooks, G. F., Bronlund, J. E., & Foster, K. D. (2005). Development of stickiness in amorphous lactose at constant T Tg levels. International Dairy Journal, 15, 513–519. Roos, Y., & Karel, M. (1990). Differential scanning calorimetry study of phase transitions affecting the quality of dehydrated materials. Biotechnology Progress, 6, 159–163. Roos, Y., & Karel, M. (1991a). Plasticizing effect of water on thermal behaviour and crystallisation of amorphous food models. Journal of Food Science, 56, 38–43.
Roos, Y., & Karel, M. (1991b). Water and molecular weight effects on glass transitions in amorphous carbohydrates and carbohydrate solutions. Journal of Food Science, 56, 1676–1681. Roos, Y., & Karel, M. (1992). Crystallisation of amorphous lactose. Journal of Food Science, 57(3), 775–777. Slade, L., & Levine, H. (1991). Beyond water activity: recent advances based on an alternative approach to the assessment of food quality and safety. Critical Reviews in Food Science and Nutrition, 30, 115. Wake, G. C. (1982). An analytic criterion for the disappearance of criticality. In Mathematics and models in engineering science (pp. 111–119). Wellington: DSIR. Wallack, D. A., & King, C. J. (1988). Sticking and agglomeration of hygroscopic, amorphous carbohydrate and food powders. Biotechnology Progress, 4(1), 31–35. Williams, M. L., Landel, R. F., & Ferry, R. D. (1955). The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. Journal of the American Chemical Society, 77(14), 3701–3707.
Conformability of the kinetics of cohesion/stickiness development in amorphous sugar particles to the classical Arrhenius law Xiao Dong Chen
*
Department of Chemical Engineering, Faculty of Engineering, Monash University, Clayton Campus, Vic. 3800, Australia Received 11 November 2005; accepted 20 February 2006 Available online 18 April 2006
Abstract Glass-transition related stickiness or cohesiveness has been a topic of significant scientific and practical interest in the last decade. The concept is very important in both food and pharmaceutical product developments. It is more appropriate to consider its development as a rate process but the time-response has been very difficult to measure accurately until lately. Here, the classical rate law for chemical reaction has been tested for its fitness in correlating the time-response of cohesion or stickiness developed in packed amorphous particles during temperature/humidity treatments. The rate law has been benchmarked against the T Tg dependent rate function established in the previous studies [Foster, K. D., Bronlund, J. E., & Paterson, A. H. J. Glass transition related cohesion of amorphous sugar particles. Journal of Food Engineering, in press.], using the original data sets for amorphous sucrose particles measured by Foster. It has been shown that the classical rate law is an excellent approach to accurately account for the process. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Stickiness; Rate of stickiness development; Amorphous sugar particles
1. Introduction The stickiness or cohesion of packed amorphous sugar powders is recognized as a function of temperature and water content (or relative humidity). Once the material, being subjected to certain temperature and humidity conditions that exceed the corresponding Tg, the material is known to undergo glass-transition, transforming from amorphous glass to a rubbery/viscous state (Aguilera, de Valle, & Karel, 1995; Aguilera, Levi, & Karel, 1993; Roos & Karel, 1990, 1991a, 1991b, 1992; Slade & Levine, 1991). This transformation is known to have a significant impact on product stickiness and caking (Lloyd, Chen, & Hargreaves, 1996; Wallack & King, 1988) and dried product quality (Bhandari & Howes, 1999). Tg can be plotted as a function of water content on dry basis (X), as shown qual*
Tel.: +64 9 373 7599; fax: +64 9 373 7463. E-mail addresses: [email protected], [email protected]. edu.au. 0260-8774/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2006.02.027
itatively in Fig. 1. For some years in recent time, the sticking point temperatures (SPT) have been measured using a variety of devices as summarized by Boonyai, Bhandari, and Howes (2004). Most methods only record the transition points for each combination of temperature and humidity. If the equilibrium isotherm of the particular material of concern is not sensitive to temperature change in the range considered, X can be replaced by water activity (aw) or relative humidity (RH). The sticking phenomena treated as a finite rate process, rather than an instantaneous process, has been contem¨ zkan, Waliplated in recent years (Levine & Slade, 1986; O ¨ zkan, Withy, & Chen, 2003; singhe, & Chen, 2002; O Paterson, Brooks, Bronlund, & Foster, 2005). Quantitatively, the approach is only possible when the development of stickiness or cohesion over time can be reliably and accurately recorded. Paterson et al. (2005), based on their experimental data, have proposed that the rate of glass-transition related cohesion in powdered material is only a function of (T Tg) and
676
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680
Foster, Bronlund, and Paterson (2005) carried out similar experimental tests on several other powders (pure substance and mixtures) and they generalized rate law where the rate of change in strength was assumed to be a function of T Tg only.
Tg or SPT
Sticking Point Temperature
dS 10aðT T g Þþb dt Glass-transition Temperature 0
X or a w
Fig. 1. Glass-transition temperature (Tg) and sticking point temperature (SPT) curves.
linking the rate of stickiness development in amorphous powders through the Frenkel’s equation for inter-particle bridge radius with time for coalescence of particles due to surface energy driven viscous flow (Frenkel, 1945), 3rp rt ð1Þ r2b ¼ 2l They then used the viscosity equation described by Williams, Landel, and Ferry (1955), i.e., the WLF equation, which is ! 2 rb 3rp r C 1 ðT T g Þ log ð2Þ þ ¼ log 2lg C 2 þ ðT T g Þ t to correlate the rate against (T Tg). It was suggested that the strength of a liquid bridge is proportional to the cross-sectional area ðpr2b Þ. They devised the ‘blow test’ to establish the time course of the stickiness development, which allows one sample to be subjected to a constant set of air conditions (T and RH) and the forces holding the particles together and the stickiness to be measured incrementally with time (Brooks, 2000). The rate of change in the strength of the packed particle bed was then correlated using Eq. (2). The LHS of Eq. (2) is replaced by the blow test end point (the strength with the unit of L min1) (S) to the RHS of the WLF equation, i.e., logðSÞ ¼ a þ b
C 1 ðT T g Þ C 2 þ ðT T g Þ
ð3Þ
The plot following Eq. (3) has produced an approximate linear relationship between the LHS and the 2nd term on the RHS. The value obtained in these sticking tests for C1 was 1.7 which is very different from the universal constant of 17.44 proposed by Williams et al. (1955). The constant C2 of 3.5 °C was also found to be quite different from the original WLF model constant of 51.6. This may indicate the mechanism proposed based on the Frenkel and WLF theories is not totally correct at least from a quantitative view point. It is noted that the symbol S can generically represent a sticking force or an indirect measurement such as the air flow rate required to move a deposited mass etc.
ð4Þ
where a and b are correlation coefficients. All these have pointed to the same insight intended to illustrate by these authors, i.e., the rate of change in strength (stickiness or cohesiveness) is directly (and only) related to T Tg. It is noted that the scatters appeared in the log rate plots are sometimes fairly large. These data sets (Foster, 2002) are highly valuable and must have been very time-consuming to obtain them in laboratory. An attempt was made to correlate the sticking/caking ¨ zkan et al. (2003) for the skim milk powders time by O using essentially the following first order equation: dS E ¼ f ðX Þrm exp ð5Þ ðS 1 SÞ dt RT where r is the pressure, m is the pressure index, E is the activation energy, R is the universal gas constant (8.31 J mol1 K1), T is the absolute temperature in Kelvin (K), and f(X) is a parameter whose value depends on the moisture content of the powders. The influence of the moisture content of the powders on the sticking/caking time was however not investigated. It is expected that the influence of the moisture content of the powders can be included in Eq. (5). The stickiness in skim milk powder mainly arises from the glass-transition of the amorphous lactose content which takes up about half of the solids amount in the powder (Richard Lloyd & Xiao Dong Chen, 1993; unpublished data). ¨ zkan et al. (2003), S is the penetration In the study by O force measured for 6 mm penetration depth into a skim milk powder compact. In these tests, the milk powder compacts were made first and then placed in the Instron machine (Model 5567, UK). The force to penetrate into the sample was then measured to a depth of 6 mm. The highest force recorded would represent the strength of the ¨ zkan et al., 2003). Assuming that the temperacompacts (O ture dependence on the caking time follows the Arrhenius behaviour, i.e., the classical approach, the activation energy was estimated from the slope of the ln (1/t) against (1/T) plot (the solution of Eq. (5)), where t is the sticking time. The Arrhenius plots were made for the skim milk powder compacts consolidated at 14.1 and 23.4 kPa and were evaluated with the average activation energy found to be 270 kJ mol1. Eq. (5) is the ‘classical approach’ to modeling chemical reaction kinetics. In the following section, a detailed development of the classical model incorporating the effect of water content (therefore the effect of Tg) will be presented. The work is based on the laboratory results obtained (and kindly provided to this author) by Foster (2002).
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680
The question will be answered is whether or not Eq. (5) is appropriate for describing the kinetics of the sticking/cohesion problem. 2. The classical approach The temperature dependence is one of the most important characteristics of chemical reactions, especially for the elementary reactions. The rate of reaction is usually written as the product of a temperature dependent term and a composition dependent term: r_ ¼ k f ðcompositionÞ
ð6Þ
The reaction rate constant k is a function of temperature: k ¼ k o eE=RT
ð7Þ
Eq. (7) is the very well known Arrhenius’ law. ko is called the frequency factor (s1) and E is called the activation energy of the reaction (J mol1). This expression has been found to fit the experimental results on chemical reaction rates well over wide temperature ranges. This expression has been proven to be good approximation to temperature dependency of both collision and transition-state theories (Levenspiel, 1972). For species A undergoing reaction, the rate of conversion of species A can be expressed as follows: dC A ¼ k C nA dt
ð8Þ
C denotes concentration (kg m3) and n is the reaction order. The glass-transition is an endothermic reaction process which can be facilitated by the existence of the plasticizer. For amorphous sugar systems, water is a good plasticizer. The process is marked by the change in the phase (glassy material to rubbery state—a kind of liquid). The development of the cohesiveness in amorphous particulate material is therefore related to glass-transition which signifies the on-set of the viscous nature of the surface of the particles. The most related property to this is perhaps the viscosity of the surface material, which can be correlated to the WLF equation incorporating the glass-transition temperature Tg as mentioned earlier. The Tg has been taken as either a function of equilibrium relative humidity RH (or water activity aw) or a function of water content on dry basis X. These two concepts are inter-exchangeable when the temperature dependence of the equilibrium isotherms in the temperature range of concern is insignificant. This is approximately true for amorphous sugar particles in the relatively narrow range concerned in industry (such as 10–50 °C). According to the Arrhenius’ law, a single E would mean there is a single mechanism that dominates the reaction process. As such, when applying the classical approach, E is expected to be the same for all Tg conditions. This represents that the mechanism of the transition is taken to be the same irrespective of what the water content or temperature
677
is. This restricts the classical approach, as far as the fitting capacity is concerned, having to have an adjustable reaction frequency factor k as a function of water content (thus a function of Tg). As suggested earlier, Eq. (5) can be seen as a derivative of the classical chemical reaction rate equation of the 1st order. The reaction species A is now (S1 S): dS E ¼ k o exp ð5Þ ðS 1 SÞ dt RT where ko should ‘account for’ all the effects of water content and compaction pressure. Since there was no deliberate compaction and the packing condition was the same for all the tests by Foster (2002), this effect of compaction pressure is not investigated in the current work. Only the water content dependence function for ko is of the interest. Re-arranging Eq. (5), gives: dS E dt ko ¼ exp ð9Þ RT ðS 1 SÞ Practically we are more interested when the stickiness starts to develop significantly so that the initial (high) rate is very important. Considering S ! So, the above Eq. (9) can be re-written as: dS E dt o ko exp ð10Þ RT ðS 1 S o Þ Since E is expected to be quite large following the result gi¨ zkan et al. (2003), a preliminary analysis can be ven by O based on the well-known simplification (Bowes, 1984) using the transition temperature Tg as the reference temperature, i.e., dS E dt o ko exp RT ðS 1 S o Þ ! dS E E dt o exp exp 2 ðT T g Þ ð11aÞ RT g ðS 1 S o Þ RT g or ! dS E E k o ðS 1 S o Þ exp ðT T g Þ exp dt o RT g RT 2g ð11bÞ Tg in the above two equations has the unit of Kelvin (K). The effect of T 2g in the last exponential term is not significant in determining the accuracy of the above equation as shown in previous works on self-heating (or spontaneous heating) research at low temperatures (e.g., Wake, 1982) such as the range investigated here. For instance, an average temperature between the Tg and the maximum temperature of the range may be used instead. The last exponential term in Eq. (11b) effectively coincides with the RHS of Eq. (4) by Foster et al. (2005). As such, in order for the reaction rate Eq. (5) to conform to the new (T Tg) dependence model
678
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680
to a large extent, i.e., Eq. (4), k o ðS 1 S o Þ exp RTE g
7.0E+49
Thereshould be somewhat constant at different T g values. E fore, ko should have a function such as exp to be able RT g E to ‘absorb’ the effect of exp RT g .
ðS 1 S o Þ
1
E RT g
was plotted against exp
, a linear relation-
ship was obtained, as shown in Fig. 3. It was found that the best fit activation energy E was 304 kJ mol1, which com¨ zkan et al. (2003) for skim milk pares well with the result of O powder, which was 270 kJ mol1. As such, one rate law, based on the classical approach, for sticking and cohesion development due to glass-transition, can be further derived:
(dS/dt)o /(S_inf -So)/ exp(-E/RT)
Since the initial ‘linear’ rate of stickiness development is of particular interest as it sets off the fastest increase in stickiness or cohesiveness, Foster’s data (2002) on amorphous sucrose particles, which had the most complete set of conditions evaluated previously (Foster et al., 2005), have been employed. These data were kindly provided by Foster which have been re-analyzed to focus more on the estimation of the initial rate (the number of data used was therefore in general fewer than the results used by Foster et al., 2005). Any significant sign of ‘trend leveling’ has been avoided. Most importantly, the values for So and S1 are necessary for the current ‘classical’ approach. As such only the sets that showed the apparent asymptotic values have been collected. The So values can be estimated accurately to be within ±0.3 L s1 but S1 was more difficult to estimate accurately (within approximately ±1 L s1 error range). It must be noted that there was no obvious trend found in the present work for S1 to be a single function of Tg. In order to show that the current analysis does not really affect the general conclusion of Foster et al. (2005), Fig. 2 has been produced which shows that the rates obtained by re-calculations do not alter the perception in the previous investigation that the rate is a function of T Tg. As a significant indication of the conformability dS of Eq. (5) E to Eq. (4), is that when the experimental exp RT dt o
y = 0.000820x
6.0E+49
R2 = 0.9666
5.0E+49 4.0E+49 3.0E+49 2.0E+49 1.0E+49 0.0E+00 0.E+00
2.E+52
4.E+52
6.E+52
8.E+52
exp (E/RTg ) Fig. 3. The approximate relationship established from Eq. (11b).
dS E E ¼ Ao exp ðS 1 SÞ dt RT RT g
ð12Þ
which seems to be able to accurately simulate the laboratory data. Note again here both T and Tg in the above equation are in unit of Kelvin (K). Based on the results shown in Fig. 3, the new rate constant Ao 0.00082. Eq. (12) predicts a non-linear development of S over time. Integration of Eq. (12) at constant T and Tg, yields the following: S ln SS11S o t¼ ð13Þ E Ao exp RT RTE g When the stickiness or cohesiveness reaches 90% of its maximum, the time required (t90) is given below: t90 ¼
lnð0:1Þ E RTE g Ao exp RT
ð14Þ
The model such as Eq. (4) does not have this feature. Fig. 4 shows the predicted results for the amorphous sucrose particles investigated by Foster (2002). One would notice that at the more instantaneous sticking region, the spread
100 1.E+03 Tg = 20 °C
Current analysis
10
1 0
5
10
15
20
25
0.1
0.01
T-Tg ( °C)
Fig. 2. The rate of the stickiness development in amorphous sucrose particles versus T Tg.
Time to reach 90% maximum stickiness (hr)
(dS/dt) o (L.min-1 .hr -1)
Foster et al (2005)
Tg = 30 °C
1.E+02
Tg = 40 °C Tg = 50 °C
1.E+01
Tg = 60 °C
1.E+00 0
10
20
30
40
50
60
1.E-01 1.E-02 1.E-03
T-T g ( °C)
Fig. 4. Time-to-reach 90% maximum stickiness/cohesiveness predicted using Eq. (13).
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680 T = 58.7 °C, RH = 12.9% 25
S (L.min-1)
20
crystallization stage
15 10
5 glass-transition stage 0 0.0
0.5
1.0
1.5
Time (hr) Fig. 5. Two possible stages of the sticking/cohesion/caking process.
of times predicted is significant for the same T Tg. This shows that there is a possibility of the independent acts of T and Tg (or water content) making the rate of sticking not solely a function of the temperature difference T Tg. 1 1 Instead the numeric ‘driving force’ is now T T g , which is an interesting outcome of the current analysis. When T ! Tg, on the other hand, the two driving forces become very similar. 3. Further remarks on the sticking or cohesion development process Based on the data provided by Foster (personal communications, 2005) and also on the previous understanding established in literature, once the glass-transition has been triggered off, provided also that the temperature is relatively far away from Tg, the development of the crystallization is expected. As time goes, a more aggressive increase of the bonding strength between the contacting particle surfaces is resulted, as crystallization reinforces (or consolidates) the already formed amorphous liquid bridges. Some of the experimental results are in favor of this argument. Fig. 5 shows one of those, the solid line indicates the glass-transition induced sticking and then a more significant rise in bonding strength happens, which may be directly linked to the crystallization process. When T Tg is very large, the apparent asymptotic trend due to amorphous material’s transition from glass would disappear due to the fast development of the crystallization process taking over from the glass-transition process. If one is interested in the whole process from sticking through to caking, a two-stage reaction model may be more appropriate. However, this is beyond the scope of the current study. 4. Conclusions In this study, it has been clearly demonstrated that the classical rate law of chemical reaction is a good alternative
679
to correlate the rates of the development of stickiness or cohesiveness resulted from the glass-transition phenomena. A simple formula (Eq. (12)) has been derived to accommodate the experimental data sets provided by Foster (2002), which enables the time to a finite level of stickiness or cohesiveness to be calculated. It can be shown that in this approach, the ‘driving force’ for the stickiness or cohesion development as a result of glass-transition of the amorphous material is T1 T1g rather than (T Tg). The two different approaches converge as T gets very close to Tg. Furthermore, it has been suggested through a brief discussion, the whole process of sticking/cohesion/caking process may be modeled by a two stage reaction model. In any case, the rate model established in this work can be used for predicting the transient stickiness/cohesiveness profiles inside a packed particle bed when the bed is subjected to temperature gradient or humidity gradient. It may also be useful in formulating the rate equations for other glass-transition related phenomena such as degradation or leakage of valuable compounds which are encapsulated in an amorphous sugar matrix. Acknowledgements The author is very grateful to the original spreadsheet data sets kindly provided by Dr. Kylie D. Foster at Institute of Food, Nutrition and Human Health, Albany Campus, Massey University, North Shore, Auckland. References Aguilera, J. M., de Valle, J. M., & Karel, M. (1995). Caking phenomenon in amorphous food powders. Trends in Food Science and Technology, 6, 149–155. Aguilera, J. M., Levi, G., & Karel, M. (1993). Effect of water content on the glass transition and caking of fish protein hydrolyzates. Biotechnology Progress, 9, 651–654. Bhandari, B. R., & Howes, T. (1999). Implication of glass transition for the drying and stability of dried foods. Journal of Food Engineering, 40, 71–79. Boonyai, P., Bhandari, B., & Howes, T. (2004). Stickiness measurement techniques for food powders: a review. Powder Technology, 145(1), 34–46. Bowes, P. C. (1984). Self-heating: evaluating and controlling the hazards. Amsterdam: Elsevier, pp. 26–27. Brooks, G. F. (2000). The sticking and crystallization of amorphous lactose. Masters of Technology Thesis. Massey University, Palmesrton North, New Zealand. Foster, K. D. (2002). The prediction of sticking in dairy powders. PhD Thesis. Massey University, Palmerston North, New Zealand. Foster, K. D., Bronlund, J. E., & Paterson, A. H. J. Glass transition related cohesion of amorphous sugar particles. Journal of Food Engineering, in press, doi:10.1016/j.jfoodeng.2005.08.028. Frenkel, J. (1945). Viscous flow of crystalline bodies under action of surface tension. Journal of Physics (USSR), 9(5), 385–391. Levenspiel, O. (1972). Chemical reaction engineering (2nd ed.). New York: John Wiley and Sons. Levine, H., & Slade, L. (1986). A polymer physico-chemical approach to the study of commercial starch hydrolysis products (SHPs). Carbohydrate Polymers, 6, 213–244.
680
X.D. Chen / Journal of Food Engineering 79 (2007) 675–680
Lloyd, R. J., Chen, X. D., & Hargreaves, J. B. (1996). Glass transition and caking of spray dried lactose. International Journal of Food Science and Technology, 31, 305–311. ¨ zkan, N., Walisinghe, N., & Chen, X. D. (2002). Characterization of O stickiness and cake formation in milk powders. Journal of Food Engineering, 55, 293–303. ¨ zkan, N., Withy, B., & Chen, X. D. (2003). Effects of time, temperature, O and pressure on the cake formation of milk powders. Journal of Food Engineering, 58, 355–361. Paterson, A. H. J., Brooks, G. F., Bronlund, J. E., & Foster, K. D. (2005). Development of stickiness in amorphous lactose at constant T Tg levels. International Dairy Journal, 15, 513–519. Roos, Y., & Karel, M. (1990). Differential scanning calorimetry study of phase transitions affecting the quality of dehydrated materials. Biotechnology Progress, 6, 159–163. Roos, Y., & Karel, M. (1991a). Plasticizing effect of water on thermal behaviour and crystallisation of amorphous food models. Journal of Food Science, 56, 38–43.
Roos, Y., & Karel, M. (1991b). Water and molecular weight effects on glass transitions in amorphous carbohydrates and carbohydrate solutions. Journal of Food Science, 56, 1676–1681. Roos, Y., & Karel, M. (1992). Crystallisation of amorphous lactose. Journal of Food Science, 57(3), 775–777. Slade, L., & Levine, H. (1991). Beyond water activity: recent advances based on an alternative approach to the assessment of food quality and safety. Critical Reviews in Food Science and Nutrition, 30, 115. Wake, G. C. (1982). An analytic criterion for the disappearance of criticality. In Mathematics and models in engineering science (pp. 111–119). Wellington: DSIR. Wallack, D. A., & King, C. J. (1988). Sticking and agglomeration of hygroscopic, amorphous carbohydrate and food powders. Biotechnology Progress, 4(1), 31–35. Williams, M. L., Landel, R. F., & Ferry, R. D. (1955). The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. Journal of the American Chemical Society, 77(14), 3701–3707.