CRYSTAL GROWTH & DESIGN
Sucrose Crystallization Kinetics in Thin Films at Elevated Temperatures and Supersaturations Terry A. Howell,
Jr.,†
Eyal
Ben-Yoseph,‡
2002 VOL. 2, NO. 1 67-72
Chetan Rao, and Richard W. Hartel*
Department of Food Science, University of Wisconsin-Madison, 1605 Linden Drive, Madison, Wisconsin 53706 Received September 15, 2001
ABSTRACT: Sucrose crystal growth from stagnant sucrose syrups in thin films was monitored by videomicroscopy at different temperatures and concentrations in order to obtain sucrose crystal growth rates as a function of sucrose total solids at four temperatures (40, 55, 70, and 82 °C). Total solids were varied from 70% to 95%, creating supersaturation values (based on concentration units in weight percent) of 1.02-1.3 at each temperature. Growth rates for individual crystals at each temperature under stagnant growth conditions in thin films were monitored by an image analysis system for up to 30 min. Additionally, theoretical growth rates based on diffusion-limited sucrose growth models were obtained at the same temperatures. A concentration gradient model was compared to a model based on chemical potential differences. At 40, 55, and 70 °C, growth rates decreased at very high supersaturation values, whereas at 82 °C growth rates continued to increase with supersaturation (within experimental limits). The diffusion-limited growth rates calculated using either the chemical potential driving force or the concentrationbased driving force approximated the actual growth rates very well. This analysis confirms that concentration gradients are an acceptable approximation to chemical potential gradients under these conditions. Introduction Sucrose growth rates for a variety of processing conditions and applications have been widely studied and reported.1,2 Shastry and Hartel3 studied the growth of sucrose crystals from sucrose syrups spread into thin films near room temperature (25 and 30 °C) as a function of sucrose concentration (total solids). Their findings relate well to crystallization processes associated with sugar coatings on foods that have temperature-sensitive centers (e.g., sugar-coated chocolate candies). Other applications (e.g., sugar-frosted cereals) typically require much higher temperatures and initial sucrose concentrations. Controlling sugar crystallization in these applications, where concentrations reach very high levels due to drying, requires a broader understanding of kinetics of crystal growth under these conditions. The growth of crystals from solution occurs by a series of consecutive steps.4 However, for sucrose crystallization, only two steps may be considered rate limiting: diffusion of sucrose to the crystal surface and incorporation of the molecules into the crystal lattice.5 When mass transfer is slow due to a highly viscous solution, stagnant condition, or other inhibition mechanisms, diffusion becomes the rate-limiting step in sucrose crystal growth. This is the case when studying crystal growth in thin, stagnant films. The true driving force for diffusion is the difference in chemical potential between the bulk solution and the crystal surface, but this driving force is traditionally assumed to be equivalent to the concentration gradient between the bulk solution and crystal-boundary layer interface.6 * To whom correspondence should be addressed. Phone: 608-2631965. FAX: 608-262-6872. E-mail:
[email protected]. † Current address: Department of Food Science, University of Arkansas, 2650 N. Young Ave., Fayetteville, AR 72704. ‡ Current address: M&M/Mars, 810 High Street, Hackettstown, NJ 07840.
Chemical-Potential-Based Growth Model. Albon and Dunning7 demonstrated that molecular diffusion is the rate-limiting step in sucrose crystal growth (as opposed to surface integration) at temperatures above 40 °C. The growth expression in terms of chemical potential can be written as
Rµg )
(
)(
)
4C*D 1 1 µb - µ* + F Lc 2δ RT
(1)
The growth rate is highly dependent on the diffusivity of sucrose within the solution and the chemical potential gradient. Each of the terms in this equation are known or calculable in a sucrose-water system. The equilibrium concentration (C*) and the solution density (F) are known (as summarized from BenYoseph8). The diffusivity (D) and boundary layer thickness (δ), however, are not as easily found. (a) Diffusivity. Ben-Yoseph8 used the data of Zhmyria9 to develop an equation for the mutual diffusion coefficient for a water-sucrose system as a function of temperature and sucrose concentration (in units of (g of sucrose)/(g of solution)). For constant temperatures above 60 °C, a linear decrease in diffusivity was observed as the concentration increased. Also, diffusivity increased with increased temperature. Equation 2 was used to fit the diffusivity at any temperature and sucrose concentration, with temperatures above 60 °C:
D ) k1e-E1/T + (C)k2e-E2/T
(2)
Here, D is in units of 109 m2/s, k1 ) 1618 m2/s, k2 ) -1.8742 m2/s, E1 ) 2492.4 K-1, and E2 ) 1755 K-1. Equation 2 linearly extrapolates the diffusivity as a function of solids concentration (total solids, TS) from the equilibrium concentration to 100% TS. Below 60 °C, the data of Zhmyria9 are taken at the highest reported concentration and linearly extrapolated to zero at the
10.1021/cg015551v CCC: $22.00 © 2002 American Chemical Society Published on Web 11/27/2001
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Howell et al.
concentration associated with the glass transition. Values for the glass transition concentration were calculated from the Gordon and Taylor equation, as applied by Roos and Karel10 for sugar solutions. (b) Boundary Layer Thickness. Kuijvenhoven and Risseeuw11 measured the diffusion layer thickness to be between 50 and 150 µm for crystals that had lengths between 1840 and 2030 µm in stagnant, concentrated sucrose films (no forced flow but some neutral flow occurred). The authors cited values for δ from other sources: δ ) 37.0 µm for low-concentration solutions (experiment done by Zagrozki) and δ ) 100-1000 µm for high-concentration solutions (experiment done by Van Hook). The thickness of a concentration boundary layer around a growing crystal can be found theoretically. The mass transfer coefficient around growing crystals can be calculated from the Sherwood number:12
( ) ()
µb2 - µ/2 γb2 Xb2 ab2 ) ln / / ) ln / RT γ2 X2 a2
∑i Xid ln γi ) 0
D δ
X
γb2
However, as molecules are built into the crystal lattice, the solution becomes less dense and natural convection around the crystals occurs. Moreover, the liberated heat of crystallization intensifies this phenomenon.11 In this case, another correlation for the Sherwood number, for free convection around a solid sphere, is suggested:13
Sh )
(
)()
kdL L3[∆F]g ) 2.0 + 0.6 D ν2F
1/4
ν D
1/3
(6)
Combining eqs 4 and 6 gives the following correlation for δ:
δ) 2.0 + 0.6
L L [∆F]g
(
3
ν2F
)() 1/4
ν D
1/3
(7)
Equation 7 shows that the boundary layer thickness is smaller than half the crystal size in situations where stagnant conditions do not exist (i.e., free convection). (c) Driving Force for Crystallization. The thermodynamic definition of the chemical potential is6,14
µ ) µ0 + RT ln(γX)
(8)
where γ is the activity coefficient and X the mole fraction of the species under consideration (i.e., sucrose). For species 2, eq 8 may be written for the chemical potential of the bulk solution and at equilibrium conditions. The
ln
ab2 a/2
) ln
γb2 γ/2
+ ln
Xb2 X/2
(12)
a
X
∫/bX21d ln X11
ln / ) γ2 (5)
(11)
∫/b d ln γ2 ) -∫/bX21 d ln γ1
(4)
L L ) Sh 2
X1 d ln γ1 X2
When eq 11 is integrated with limits being the solubility and bulk conditions
Combining eqs 3 and 4 gives
δ)
(10)
For a binary solution (i ) 2)
(3)
For crystals that grow in stagnant films with no forced or natural mass convection, Re equals zero, and as a consequence the Sherwood number equals 2. The mass transfer coefficient can also be written as11
(9)
The Gibbs-Duhem equation14,15 can be applied to the partial molar excess Gibbs energy to get
d ln γ2 ) -
kdL Sh ) ) 2.0 + 1.1Re1/2 Sc1/3 D
kd )
chemical potential driving force used in eq 1 may be derived by subtracting the chemical potential at equilibrium from the chemical potential of the bulk solution, rearranging, and simplifying:
) ln
Xb2 X/2
-
(13)
a
X
∫/b X21d ln X11
(14)
Combining eqs 9 and 14 gives
Xb2 µb2 - µ/2 ) ln / RT X 2
X
a
∫/bX21d ln X11
(15)
All the quantities on the right-hand side of the equation are estimable, and hence, the true driving force can be quantified. Usually, the term in the integral is estimated by numerical methods. Evaluation of eq 15 needs experimental data on the activity of water corresponding to sucrose concentration. Methods for experimental determination of aw are described elsewhere.16-20 However, experimental data for water activity are scarce. The water activity data reported by Scatchard21 was used to develop an empirical representation for aqueous sucrose solution that was extrapolated to high supersaturations. Figure 1 shows the original water activity data for the relevant concentration range and the extrapolation to 95% total solids. Temperatures in the range of 0-30 °C had no effect on the water activity, and this was assumed to apply up to a temperature of 80 °C. Thus, the chemical potential driving force is calculated with eq 15, the boundary layer thickness with eq 7, and the diffusivity with eq 2, allowing the crystal growth rate to be modeled as a function of the chemical potential associated with supersaturation.
Sucrose Crystallization Kinetics in Thin Films
Crystal Growth & Design, Vol. 2, No. 1, 2002 69
Figure 1. Water activity as a function of the percentage of total solids.21
Concentration-Based Growth Model. Combining eqs 1 and 9 yields
Rµg )
( )( ( ))
Figure 2. Sucrose crystal growth rate, diffusivity, and supersaturation as a function of sucrose concentration at 80 °C.8
ab2
4DC* 1 1 + ln / Fc Lc 2δ a2
(16) ing force and the other using a concentration-based driving force) to the experimental data.
For solutions with low supersaturation, the ratio of the activities may be approximated by a concentration difference, based on the assumption of a near-ideal solution, with activity coefficients approaching unity:
ln
Rcg )
() ab2
a/2
(
≈
Cb2 - C/2
(17)
C/2
)(
)
4DC* 1 1 Cb - C* + Fc Lc 2δ C*
(18)
where a concentration-based driving force is used to estimate crystal growth. Assuming that Lc . δ, the growth model is reduced to
Rcg )
2D (C - C*) δFc b
(19)
On the basis of this model, diffusivity, supersaturation, and crystal growth rates vary with concentration at 80 °C, as shown in Figure 2. Diffusivity decreases linearly as sucrose concentration increases, with supersaturation increasing from zero at the saturation concentration. Therefore, the concentration-based growth rate equation predicts a parabolic profile with a maximum growth rate at 92.5% sucrose content (at 80 °C). As the concentration surpasses this level, the limited diffusivity reduces the growth rate, whereas at low supersaturation values, the supersaturation driving force limits crystal growth. The objectives of this work were to experimentally measure the growth rates of sucrose crystals in thin films at 40, 55, 70, and 82 °C as a function of sucrose concentration and apply the diffusion-limited growth rate models (one using a chemical-potential-based driv-
Materials and Methods Sucrose syrups were prepared using a pilot plant vacuum evaporator (Groen Mfg., Chicago, IL). Approximately 100 mL of 30% (by weight) sucrose syrup was added to the evaporator. A vacuum of 24-26 in. Hg coupled with heating by a steam jacket on the vessel allowed for the syrup to be concentrated above 75% within 2 min. To obtain higher concentrations, the syrup was boiled for a longer period, up to 5 min. This rapid preparation step allowed a new syrup to be prepared for each experimental run (rather than letting an existing syrup stand unused for a period of time). However, exact replication of a specific concentration was not possible due to the uncontrolled nature of the vacuum evaporation process used here, and so numerous data points were taken over a range of concentrations. Solutions with sucrose concentrations between 80 and 90% were very difficult to generate without spontaneous nucleation. Even with the most minimal handling, many batches of syrup would crystallize in their container in a matter of minutes. These would be discarded. To study crystal growth, the concentrated syrups were heated to the desired experimental temperature in a water bath. When the syrup was at the desired temperature, a small sample was spread onto a heated microscope slide with a small depression (150 µm deep). Seed crystals were applied through a size 44 mesh sieve to lightly seed the surface of the film. A heated cover slide was then used to cover the film and ensure that drying did not occur. The slides were then placed into the growth chamber, mounted on the stage of a Nikon Optiphot microscope (Garden City, NJ) with polarizer lens. The microscope was equipped with a Cohu video camera (Model 48152000-0000). The microscope images were magnified 40×. At each condition, at least 15 crystals were analyzed, with the exception of the extremely high concentrations, where only 4-10 were available. Due to growth rate dispersion (GRD),22 a spread of growth rates was found, as given by the standard deviation of the GRD. Crystals were grown over 30 min periods, and images were collected automatically every 30 s by OPTIMAS (Bothell, WA) image analysis software. Individual crystals were digitized at each time interval, and the area and equivalent circular
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Howell et al.
Figure 3. Sucrose crystal growth in sucrose solution (86% total solids) at 82 °C in thin film (150 µm thick) at time 0 (A), 2 min (B), 4 min (C), and 10 min (D). The frame size is 1670 by 2294 µm. Table 1. Mean Sucrose Crystal Growth Rates, Rg (( Standard Deviation), Measured Experimentally in Stagnant Films 40 °C
55 °C
C (% w/w)
s
Rg (mm/min)
70.0a 71.7 74.9 78.0 84.6 85.2 90.5 93.7
0 0.04 0.09 0.15 0.20 0.29 0.39 0.46
1.4 ( 0.5 6.1 ( 1.2 9.1 ( 1.9 11.1 ( 1.5 14.5 ( 2.8 14.0 ( 1.6 12.7 ( 3.8
a
70 °C
C (% w/w)
s
Rg (mm/min)
76.1a 76.1 81.0 83.5 84.5 85.5 89.0 89.3 90.2 93.3
0.00 0.05 0.13 0.19 0.21 0.24 0.30 0.31 0.33 0.40
7.5 ( 2.6 15.9 ( 4.7 19.8 ( 8.4 18.2 ( 5.5 17.0 ( 5.4 36.6 ( 1.4 36.9 ( 2.4 32.6 ( 1.8 19.4 ( 7.8
C (% w/w) 76.4a 79.7 82.0 85.0 86.6 93.9 95.3
82 °C
s
Rg (mm/min)
C (% w/w)
s
Rg (mm/min)
0.00 0.06 0.10 0.16 0.20 0.35 0.38
11.6 ( 2.4 19.6 ( 10.2 37.0 ( 14.2 67.4 ( 7.3 77.8 ( 2.9 60.7 ( 4.0
79.9a 80.4 81.8 85.7 87.0 94.8 95.0
0.00 0.02 0.05 0.13 0.15 0.31 0.32
6.8 ( 2.3 15.1 ( 8.4 52.8 ( 23.0 65.8 ( 17.8 135.1 ( 12.0 124.5 ( 28.1
Saturation concentration at each temperature.
diameter values were generated. Initial growth rates for individual crystals were calculated as the slope of the change in size (equivalent circular diameter) with time for those crystals that were able to grow uninhibited (did not impinge on a neighboring seed crystal) for at least 10 min. At very high supersaturations at 70 and 82 °C, crystal growth was so rapid that images could only be tracked for 4 min before their boundary layer was encroached by other crystals. When crystals grew into one another or grew out of the field of vision before the experiment was completed, growth rates were determined for the time period in which the crystals were able to grow in the syrup with at least one diameter’s distance separating crystals. Most of the crystals analyzed grew unabated for the 30 min experimental time.
Results and Discussion The crystals grew rapidly at each of the temperatures and supersaturation values that were studied. Figure 3 shows the progression of growth for crystals at 82 °C
and s ) 0.13 for the first 10 min. Growth rates for each crystal were determined by linear regression of the crystal size at each time. In almost every case, the growth rate curves showed very high linearity (R2 > 0.95). Table 1 shows the mean growth rates at each temperature and supersaturation. The standard deviation in general increased with temperature and supersaturation. An increase in GRD with increasing growth rate was also observed by Fabian et al.23 and Liang et al.22 Due to spontaneous nucleation at high concentrations and temperatures, the range of relative supersaturation ratios shrinks with increasing temperature from 0-0.34 at 40 °C to 0.0-0.20 at 82 °C, even though the actual concentrations are higher. As expected, the crystal growth rates increased with temperature at similar supersaturation values. Additionally, growth rates gen-
Sucrose Crystallization Kinetics in Thin Films
Crystal Growth & Design, Vol. 2, No. 1, 2002 71
Table 2. Boundary Layer Values (δ in µM) at Each Temperature T (°C)
µb - µ*/µ*
Cb - C*/C*
40 55 70 82
240 132 143 117
68 38 39 38
erally increased with sucrose concentration at each temperature, in agreement with previous research.3 However, at all temperatures, growth rates began to decrease as supersaturation became large due to the very high viscosity and limited molecular mobility associated with stagnant, highly concentrated solutions. Boundary Layer Thickness. Values for the chemical potential driving force were calculated with eq 15 and used with eq 1 to estimate the boundary layer thickness. For comparison, the diffusion path was also parametrically estimated from the experimentally measured Rgc using eq 19. Table 2 shows the values of δ at the four temperatures studied. The boundary layer thickness is in the range between the smallest experimental values reported (37-50 µm) and the highest theoretical value (half the size of the crystal). The experimental value determined here is reasonable for crystal sizes above a few hundred micrometers, which, in fact, is the range for crystal size after a few minutes of growth. Thus, our assumption of Lc . δ in eq 19 may not be realistic in every case. However, the values of δ obtained are well within the theoretical limit of Lc/2. An improved analysis incorporating the effect of crystal size on the growth kinetics is left as a future exercise. As shown in Table 2, the boundary layer thickness found from the chemical potential driving force is different from that found from the concentration gradient. As the temperature increased, the difference between the two models decreased. This analysis is not sufficient to determine which approach is more representative of the true situation. Kinetic Models. Figures 4 and 5 show how the activity-based and concentration-based growth models fit the experimental data, respectively. In Figure 4, Rgµ was calculated from eq 1 using eq 15 for the activitybased driving force. Rgµ was calculated from eq 19 for the comparison to the experimental data in Figure 5. Both models used extrapolated values of D based on eq 2, and each model used diffusion boundary layers found in Table 2. Both models fit the data well at low temperatures, with increasing divergence as temperature increased. In both cases, Rg is overestimated at low supersaturation, and then the models predict a sharper decrease in Rg due to mobility limitations at higher supersaturation. Numerous approximations were made for these calculations. For example, diffusivity and activity data were extrapolated from less concentrated solutions at lower temperatures. Despite these approximations, a reasonable fit to the experimental data was obtained in both cases. Note that the concentration-based model fit the data as well as the activity-based model under these conditions, indicating that the approximation of activity using concentration for the driving force is reasonable even for these high concentrations.
Figure 4. Comparison of the experimental crystal growth rates (data points) with predicted growth rates from the activity-based model (eqs 1 and 15).
Figure 5. Comparison of the experimental crystal growth rates (data points) with predicted growth rates from the concentration-based model (eq 19).
Conclusions As expected, sucrose crystal growth rates increased with increasing temperature and increased initially with increasing concentration. However, as the film approached the amorphous state (at higher concentrations and lower temperatures), crystal growth rates decreased. Models using either the activity-based or concentration-based driving force and the concentration gradient driving force for crystallization represented the effects of temperature and supersaturation on crystal growth reasonably well. Values calculated for the diffusion path (boundary layer thickness) were in accordance with published values. On the basis of these results, concentration-based units for driving force give a reasonable approximation of the true driving force.
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Crystal Growth & Design, Vol. 2, No. 1, 2002
Notation Roman Variables a aw C D g kd L R Re Rg s Sc Sh T X
activity (mole per kg water) water activity sucrose concentration (% w/w) mutual diffusion coefficients (m2/s) acceleration due to gravity (m2/s) mass transfer coefficient (kg/m2/s) characteristic size of the crystal (µm) gas-law constant Reynolds number linear crystal growth rate (µm/min) relative solution supersaturation ((Cb - C*)/C*) Schmidt number Sherwood number temperature (°C or K) mole fraction
Greek Variables δ length of the diffusion path γ activity coefficient µ chemical potential ν kinematic viscosity F density Subscripts 0 pure component c concentration-based variable µ chemical-potential-based variable Superscripts * equilibrium parameter i interfacial solution parameter b bulk solution parameter 1 species number (refers to water) 2 species number (refers to sucrose)
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Howell et al. (4) Mullin, J. W. Crystallization, 3rd ed.; Butterworths: London, 1993. (5) Smythe, B. M. Sugar Technol. Rev. 1971, 1(3), 191-231. (6) Mullin, J. W.; Sohnel, O. Chem. Eng. Sci. 1977, 32, 683689. (7) Albon, N.; Dunning, W. J. Acta Crystallogr. 1960, 13, 495498. (8) Ben-Yoseph, E. Computer Modeling of Sucrose Crystallization During Drying of Thin Sucrose Films. Ph.D. Dissertation, University of WisconsinsMadison, 1999. (9) Zhmyria, L. P. Izd Krasnodarskogo ita Pishchevo i Promyshlennosti 1972, 2, 125-128. (10) Roos, Y.; Karel, M. Food Technol. 1991, 45, 66-71. (11) Kuijvenhoven, L. J.; Risseeuw, I. J. Int. Sugar J. 1983, 1010(85), 35-38. (12) Wey, J. S. In Preparation and Properties of Solid State Materials; Wilcox, W. R., Ed.; Marcel Dekker: New York, 1981; Vol. 6. (13) Treybal, R. E. Mass Transfer Operations, 3rd ed.; McGrawHill: New York, 1980. (14) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1986. (15) Kim, S.; Myerson, A. S. Ind. Eng. Chem. Res. 1996, 35, 1078-1083. (16) Cohen, M. D.; Flagan, P. C.; Seinfeld, J. H. J. Phys. Chem. 1987, 91, 4563-4574. (17) Dunning, J. W.; Evans, H. C.; Taylor, M. J. Chem. Soc. 1951, 3, 2363-2372. (18) Norrish, R. S. J. Food Technol. 1966, 1, 25-39. (19) Robinson, R. A.; Stokes, R. H. J. Phys. Chem. 1961, 65, 1954-1958. (20) Stokes, R. H.; Robinson, R. A. J. Phys. Chem. 1966, 70, 2126-2130. (21) Scatchard, G. J. Am. Chem. Soc. 1921, 43, 2406-2418. (22) Liang, B. M.; Hartel, R. W.; Berglund, K. A. Chem. Eng. Sci. 1987, 42(11), 2723-2727. (23) Fabian, J.; Hartel, R. W.; Ulrich, J. In Proceedings of the International Workshop on Crystal Growth of Organic Materials; August 1995; American Chemical Society: Washington; DC, 1996.
CG015551V
