Kinetics of Cold-Set Diffusion-Limited Aggregations of Denatured

Kinetics of Cold-Set Diffusion-Limited Aggregations of Denatured Whey ... kinetic model have also been carried out in order to support the experimenta...
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Biomacromolecules 2005, 6, 3189-3197

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Kinetics of Cold-Set Diffusion-Limited Aggregations of Denatured Whey Protein Isolate Colloids Hua Wu, Jianjun Xie, and Massimo Morbidelli* Swiss Federal Institute of Technology Zurich, ETHZ, Institut fu¨r Chemie und Bioingenieurwissenschaften, ETH-Ho¨nggerberg/HCI, CH-8093 Zu¨rich, Switzerland Received July 28, 2005; Revised Manuscript Received September 10, 2005

The CaCl2-induced cold-set aggregation kinetics of the denatured whey protein isolate (WPI) colloids has been investigated under dilute diffusion-limited cluster aggregation (DLCA) conditions, using small-angle light scattering. In particular, the structure factor, the scattered intensity at zero angle and the average radius of gyration have been measured for the aggregating system as a function of time. It is found that the fractal dimension of the clusters is df ) 1.85, in the range typical of clusters aggregated under DLCA conditions. The aggregation kinetics in this transition region can be described by a power law relation in the initial stage of the aggregation, but the exponent of the power law is equal to 0.7, i.e., significantly larger than 1/df ) 0.54, which is the typical value of the DLCA kinetics. Since it is found that the average gyration radius of the clusters has reached a value of 80 µm, leading to a cumulative volume fraction of clusters equal to 0.25, it is legitimate to expect that the process is in the region of transition from aggregation to gelation. This confirmed by the fact that, at the later stage of the aggregation, the growth of the average cluster size further accelerates with time and eventually becomes explosive, leading to gelation. The observed aggregation kinetics has been compared with that reported in the literature from DLCA Monte Carlo simulations, and a good agreement has been found with the data corresponding to the transition region from aggregation to gelation. Numerical simulations using the Smoluchowski kinetic model have also been carried out in order to support the experimental findings. Introduction Whey proteins (WPs), due to their functional characteristics, high nutritive value, and generally recognized as safe (GRAS) status, are widely used in foods as gelling agents, emulsifiers, texture modifiers, thickening agents, and foaming agents.1,2 Thus, a great number of studies have been carried out in the literature to investigate the WP aggregation and particularly gelation behavior and the structures of the corresponding clusters and gels.1-20 Traditionally, when WPs are added to foods to form gels or thicken solutions, the obtained mixture has to be heated to a temperature of at least 65 °C. This often limits the application of WPs, since this heating procedure may alter the quality of the food products. Thus, a technique usually referred to as cold-set aggregation and gelation has been developed.3-5 This technique divides the addition of WPs in foods into two steps: (1) preparation of a heat-denatured WP colloidal dispersion (often called soluble aggregates); and (2) salt- (often CaCl2 or NaCl) or pH-induced gelation and aggregation of the WP colloidal dispersion with the food components at room or even lower temperatures. A large number of studies have been carried out to investigate the aggregation mechanism and aggregates properties in the first step.5-12 However, for the second step, most of the studies in the literature have focused mainly on the structure * Corresponding author. E-mail: 0041-44-6323034.

[email protected]. Tel:

properties of the final gels.13-16 Little information can be found about the kinetic behavior of the cold-set aggregation of the WP colloidal dispersions and the time evolution of the cluster structures during the gelation, which is instead essential to understand the final gel structures. In this work, we investigate the cold-set aggregation kinetics of the denatured whey protein isolate (WPI) colloidal dispersion and monitor the time evolution of the average size, the scattering structure factor and the other structure information, using small-angle light scattering technique. In particular, since the WPI solution after denaturation becomes a WPI colloidal dispersion, we apply the methodologies typically used in colloidal aggregation studies.21-31 We investigate the aggregation process of the WPI dispersions in the diffusion-limited cluster aggregation (DLCA) regime, where the colloidal particles are fully destabilized. This implies that, in order to be able to measure the process evolution, we have to slow the aggregation process by operating at very high dilution conditions. Since the aggregation kinetics and the aggregate structure under DLCA conditions are well understood, it is interesting to find out if these systems agree with or deviate from such classical behaviors. In this respect, since the density difference between WPI particles and water is negligible, we have the opportunity of monitoring the aggregation kinetics till reaching substantially large cluster sizes or even gelation without significant sedimentation.

10.1021/bm050532d CCC: $30.25 © 2005 American Chemical Society Published on Web 10/13/2005

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Table 1. SLS and DLS Characterization of the Denatured WPI Colloidal Dispersion at Various WPI Concentrations

C, g/L Rg,o, nm Rh,o, nm

0.1 68.1 32.3

0.3 67.9 32.5

1.0 67.6 32.4

3.0 61.8 31.3

Experiments Preparation and Characterization of WPI Dispersions. Whey protein isolate (WPI) powder (BiPro, JE345-3-420) was supplied by Davisco Foods International Inc. (Eden Prairie, MN). The WPI solution was prepared by dissolving the WPI powder in distilled water, and adjusting the pH of the solutions to 7 using either a NaOH or a HCl solution. To make sure that all of the proteins in the WPI solution form aggregates,5,12 we have prepared the WPI solution at the concentration of C )100 g/L, and performed the WPI denaturation by heating the WPI solution at 90 °C for 30 min. Then, the solution was quickly filtered under hot conditions using 0.45 µm GD/X syringe filters (Whatman, Clifton, NJ) to eliminate possible big particles that may disturb the light scattering measurements. The denatured WPI colloidal dispersion was further diluted to C ) 50 g/L in sodium azide solution (at a final NaN3 concentration of 200 ppm) to avoid bacterial growth, and the obtained WPI dispersion was then kept in a fridge at 5∼8 °C, ready for the experiments. The obtained WPI colloidal dispersion was characterized at various WPI concentrations in dilute regime by both dynamic and static light scattering (DLS and SLS), using the instrument BI-200SM (Brookhaven) with an argon-ion laser (M95-2, Lexel) operating at a wavelength of 488 nm and a goniometer in the angle range between 15° and 150°. The values of the hydrodynamic radius, Rh,o at four WPI concentrations, determined by DLS based on the method of cumulants of a third-order fit,32 are shown in Table 1. It is seen that for a WPI concentration value below 1.0 g/L, the Rh,o values are practically constant and equal to ∼32.5 nm, which is also very close to the value reported in the literature under similar conditions.5,12 However, in the case of C ) 3 g/L, the Rh,o value is slightly smaller, indicating the presence of some limited correlations (interactions) among the WPI particles. Similar behavior was also observed in the intensity curves of the SLS measurements, as shown in Figure 1, where q is the wave vector, defined as q)

4πn θ sin λ0 2

()

(1)

λ0 is the wavelength of the incident light, n is the refractive index of water, and θ is the scattering angle. It is seen that the I(q) curve at C ) 3 g/L is significantly different from the other ones shown in the same figure. The radius of gyration, Rg,o obtained from the Zimm plot of the I(q) curve,33 is in fact significantly smaller for C ) 3 g/L than in all the other cases, as shown in Table 1. It is worth noting that all of the Rg,o values in Table 1 are larger than the corresponding Rh,o values. This reflects the nonspherical shape of the WPI aggregates.9 Polydispersity of the aggregate sizes can also contribute to the difference between Rg,o and Rh,o. In general,

Figure 1. Normalized static light scattering intensity curves for the denatured WPI colloidal system at various WPI concentrations, C. The solid curve is the Zimm plot using Rg,o ) 68 nm.

Figure 2. Experimental curve for determining the critical coagulant (CaCl2) concentration (CCC), where tc is the coagulation time. WPI concentration, C ) 1 g/L.

for narrowly distributed samples, the polydispersity, which is defined in the method of cumulants as 2µ2/Γ h 2 (where Γ h and µ2 are the first and the second cumulant, respectively32), is smaller than 0.025, whereas a polydispersity value of 0.2 has been determined by DLS for all of the four cases in Table 1. Cold WPI Aggregation under Dilute DLCA Conditions. In this work, we use CaCl2 to induce the cold aggregation of WPI dispersions. To make sure that WPI aggregation is under diffusion-limited cluster aggregation (DLCA) conditions, we need first to determine the critical coagulant (CaCl2) concentration (CCC) for fast aggregation of the system. Aggregation experiments carried out at CaCl2 concentrations larger than CCC are certainly going to be in the DLCA regime. To estimate the CCC value, we add 0.4 mL of the prepared WPI dispersion at C ) 50 g/L through a pipet into a series of beakers containing 20 mL of the CaCl2 solution but at different CaCl2 concentrations. Then, we observe visually the time when large clusters appear. This time is referred to in the following as the coagulation time, tc. Figure 2 reports such measured coagulation times as a function of the CaCl2 concentration. It is seen that, when the CaCl2 concentration is larger than 0.025 mol/L, the coagulation time is practically zero, indicating fast or diffusion-limited aggregation (DLCA) conditions. In these cases, when a WPI droplet falls in the CaCl2 solution, big WPI clusters are observed instantaneously. However, when the CaCl2 concentration decreases below 0.025 mol/L, the coagulation time

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starts to increase. Note that in these cases with the CaCl2 concentrations smaller than 0.025 mol/L, the reported tc values are only qualitative, since if the large clusters cannot form instantaneously the WPI particles disperse into the CaCl2 solution and the visual observation of the time for the appearance of large clusters becomes subjective. It is thus concluded from Figure 2 that the CCC value for our WPI dispersion is about 0.025 mol/L. When the CaCl2 concentration is larger than 0.025 mol/L, aggregation proceeds under DLCA conditions. When the CaCl2 concentration is smaller than 0.025 mol/L, aggregation proceeds under RLCA conditions, where the aggregation rate is a function of the partial electrostatic repulsive interactions. The CCC value of 0.025 mol/L is in good agreement with the value reported by Hongsprabhas and Barbut,13 who observed (their Figure 1) that the surface lightness of the cold-set WPI gels increases substantially when the CaCl2 concentration increases from 0.01 to 0.03 mol/L. All our DLCA aggregations are carried out at the CaCl2 concentration of 0.1 mol/L, which is much larger than the critical coagulant concentration, CCC ) 0.025 mol/L. To initiate a DLCA experiment, we added a concentrated CaCl2 solution to the WPI dispersion to reach a 0.1 mol/L final CaCl2 concentration under gentle agitation with a magnetic bar, which took about 5 s. The dilute DLCA experiments were carried out at two WPI concentrations, C ) 0.007 and 0.02 g/L. Monitoring WPI Aggregation Using in Situ SALS. The kinetics of the cold WPI aggregation and the corresponding cluster structures were investigated by monitoring the time evolution of the average structure factor of the clusters, I(q)/ P(q) measured with a small-angle light scattering (SALS) instrument (Malvern, U.K.) with scattering angle ranging between θ ) 0.02 and 40° and wavelength λ0 ) 633 nm. In particular, after initiating the aggregation by adding the CaCl2 solution to the WPI dispersion as described above, the sample was immediately and gently injected into the sample cell of the SALS instrument using a syringe. This procedure was repeated twice: the first time it was done to wash the cell, whereas the sample injected for the second time was kept in the cell for the in situ measurement of the entire aggregation process. The light intensity scattered by the sample was measured by the SALS instrument as a function of time. The instrument measures the intensity values at all of the detecting angles at the same time, and it has an intensity acquisition frequency of 1000 s-1. It was found for the colloidal systems considered in this work that a measurement duration of 20 s was sufficient to obtain each intensity curve, I(q) with good statistics. This is essentially an instantaneous measurement with respect of the characteristic time of the aggregation process. From the angle-dependent intensity curve measured at each time, the normalized average structure factor of the WPI clusters in the system, 〈S(q)〉, can be estimated as follows:33-35 〈S(q)〉 )

I(q) I(0)‚P(q)

form factor of the primary particles, which was measured directly from the original denatured WPI dispersion at C ) 0.3 g/L. The average radius of gyration of the clusters, 〈Rg〉*, is then determined from 〈S(q)〉, using the Guinier plot:36,37 〈S(q)〉 ) exp[-q2(〈Rg〉*)2/3]

(3)

Note that this procedure is correct only when the clusters are substantially large. For small clusters, such as those generated at the initial stage of the aggregation process, the contribution of the primary particles to the radius of gyration has to be taken into account 2 + (〈Rg〉*)2 〈Rg〉2 ) R g,o

(4)

where Rg,o is the average radius of gyration of the primary particles (equal to 68 nm in this case, as given in Table 1) and 〈Rg〉* is the value determined from 〈S(q)〉 using eq 3. In this way, since at the very beginning of the aggregation 〈S(q)〉 ≈ 1, we have from eq 3 that 〈Rg〉* ≈ 0 and then 〈Rg〉 ≈ Rg,o, while for substantially larger clusters, the contribution of Rg,o can be neglected, so that 〈Rg〉* ≈ 〈Rg〉. Further information about the cluster structure may be obtained under the assumption that this follows the mass fractal scaling. In particular, it is well-known that, for sufficiently large mass fractal clusters, the normalized average structure factor, 〈S(q)〉, scales with q as follows:35,36 〈S(q)〉 ∝ q-df

for 1/〈Rg〉 , q , 1/Rg,o

(5)

where df is the mass fractal dimension. Thus, from a loglog plot of 〈S(q)〉 vs q, the slope in the given q range gives an estimate of df. This same quantity has also been estimated from other two independent sets of data as discussed in the following. (a) Small Angle Light Scattered Intensity at q ) 0, I(0). This quantity, when no correlations (interactions) are present among clusters, scales as follows:35,38 I(0) ∝ 〈i〉 ∝

( ) 〈Rg〉 Rh,o

df

(6)

where 〈i〉 is average number of primary particles per cluster. Thus, the log-log plot of I(0) vs 〈Rg〉/Rh,o must be a straight line, and the corresponding slope gives the estimate of df. (b) Aggregation Kinetics. If the cold WPI aggregation follows the universal DLCA kinetics under very dilute conditions,22,23 the number concentration of clusters during aggregation can be described by the second-order kinetics originally proposed by Smoluchowski39 KBN0 N0 t ) 〈i〉 ) 1 + t)1+ N 2W tB

(7)

where N0 and N are the initial number of primary particles and the number of clusters at a given time, t, respectively. The rate constant KB and the characteristic time tB of the Brownian aggregation process are given by

(2)

where I(0) is the scattered intensity at q ) 0, and P(q) is the

for q〈Rg〉* < 1

KB )

8kBT 2W and tB ) 3µ KBN0

(8)

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Figure 3. (a) Time evolution of the average structure factor for the cold aggregation of the denatured WPI dispersion under DLCA conditions measured by SALS. (b) Normalized average structure factor, 〈S(q)〉, as a function of the normalized wave vector, q×〈Rg〉. Rh,o ) 32.5 nm; C ) 0.02 g/L; CCaCl2 ) 0.1 mol/L.

where µ is dynamic viscosity of the disperse medium and W is the Fuchs stability ratio, whose value in DLCA conditions is typically in the range between 1 and 5.40-43 Since N0/N ) 〈i〉, we obtain from eqs 6 and 7

( )

〈Rg〉 t ∝ 1+ Rh,o tB

1/df

(9)

Thus, for sufficiently long aggregation times (t.tB), the average radius of gyration, 〈Rg〉, increases with time following the power law (〈Rg〉 ∝ t1/df), and therefore, the slope of the double logarithmic plot of 〈Rg〉 vs t gives an estimate of df. Note that the above eq 9 is valid only when the cumulative volume fraction occupied by clusters, φ*, is substantially smaller than unity. When the φ* value approaches unity, transition from aggregation to gelation occurs. In the latter case, as clearly shown through Monte Carlo (MC) DLCA simulations,44 the growth of 〈Rg〉 with time becomes faster than that given by eq 9. Results and Discussion Aggregation Kinetics and Cluster Structure. Figure 3a shows the time evolution of the original average structure factor, I(q)/P(q) for the cold WPI aggregation under DLCA conditions measured by the SALS instrument at the WPI concentration, C ) 0.02 g/L. It is seen that a power law region is present in all of the curves of the structure factors and enlarges as time increases, whereas the bending region of the curve moves toward lower q values. This indicates

Figure 4. Time evolution of the average radius of gyration 〈Rg〉, (a), and intensity at q ) 0, I(0), as a function of the corresponding 〈Rg〉/ Rh,o, (b), for the WPI aggregation system shown in Figure 3a.

that the average radius of gyration of the clusters, 〈Rg〉, increases with time. The normalized average structure factor, obtained by dividing each structure factor in Figure 3a by its value at q ) 0, is shown in Figure 3b as a function of the normalized wave vector, q×〈Rg〉. It is seen that all of the data collapse in a single curve, which exhibits a power-law region covering 2 orders of magnitudes. This clearly indicates that the growth of the clusters follows the mass fractal scaling, and the slope of the power-law region gives the estimate for the value of the mass fractal dimension, df ) 1.85. Using the data in Figure 3 in the Guinier plot, the time evolution of the average radius of gyration 〈Rg〉 is obtained as shown by the symbols in Figure 4a. As discussed above, in a typical DLCA process under very dilute conditions, the growth of 〈Rg〉 with time follows the power-law given by eq 9. Indeed, the 〈Rg〉 values in Figure 4a follow a power-law with time, at least up to time values of the order of 300 min, above which the growth of 〈Rg〉 accelerates. However, the slope of the power-law region in Figure 4a is about 0.7, which is significantly larger than the expected 1/df ) 1/1.85 ) 0.54. This would actually correspond to a mass fractal dimension of df ) 1/0.7 ) 1.43. To verify whether the df value obtained from 〈S(q)〉 is correct, we have estimated it from the plots of I(0) vs 〈Rg〉/ Rh,o, i.e., using the technique described at point (a) in the previous section, and the obtained results are shown in Figure 4b. It is seen that except for the very initial and the very final stages, the log-log plot of I(0) vs 〈Rg〉/Rh,o can be well represented by a straight line with slope equal to df ) 1.85. It is therefore confirmed that the df value of the WPI clusters

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formed under DLCA conditions as estimated from measurements related to the cluster structure is indeed 1.85. Let us now go back to consider the aggregation kinetics discussed in the context of Figure 4a. Recently, Rottereau et al.44 have systematically investigated the DLCA kinetics in a large range of particle volume fractions, using the MC off-lattice and on-lattice simulations. In particular, they have simulated the time evolution of the average mass and size of the clusters during aggregation till gelation occurs. It was found that the aggregation kinetics follows eq 9 only when the average mass, 〈i〉 of the clusters, is substantially smaller than a characteristic mass, ic. When 〈i〉 becomes larger than ic, the aggregating system switches progressively to gelation, and the time evolution of the average mass, 〈i〉, accelerates and eventually follows that of a static percolation process. This could explain the acceleration of the DLCA kinetics observed in Figure 4a. This can be verified by estimating the cumulative volume fraction occupied by clusters, φ* 4 φ* ) π〈Rg〉3N 3

(10)

based on the maximum 〈Rg〉 ()8.07 × 104 nm) value measured experimentally in Figure 4a, to see whether the φ* value is comparable to unity. Note that in order to estimate the number concentration of clusters in eq 10, N (1/m3)45 N)

N0 〈i〉

)

N0 1.12(〈Rg〉/Rh,o)1.85

(11)

it is necessary to know the initial number concentration of the WPI particles, N0, which may be expressed as N0 )

NAC × 103 MW

(12)

where NA is the Avogadro number, C (g/L) is the WPI concentration, and MW (g/mol) is defined as the molar mass of the WPI particles, i.e., the mass of one mole of WPI particles. Ju and Kilara5 investigated the effect of WP concentration on the MW value of the WP particles after the thermal denaturation at T ) 80 °C for 30 min. It was found that the MW is equal to about 1.03 × 106 and 1.25 × 106 g/mol at C ) 30 and 50 g/L, respectively, whereas it is larger than 1 × 107 g/mol at C ∼ 80 g/L; that is, MW increases almost exponentially with the WPI concentration, at least in this concentration range. Considering that our WPI particles were prepared at C ) 100 g/L, we can extrapolate from the data above at 80 °C a MW value of 4 × 107 g/mol. Further, considering that the WPI denaturation was carried out at T ) 90 °C, an even larger MW value is expected. Thus, in the following computations, we consider the molar mass of the WPI particles equal to MW ) 5 × 107 g/mol. Accordingly, in the case of Figures 3 and 4 where C ) 0.02 g/L, we can estimate from eq 12 the value of N0 ) 2.41 × 1017 1/m3 (the corresponding volume fraction of the WPI particles based on Rh,o ) 32.5 nm is φ0 ) 3.46 × 10-5). It follows from eqs 10 and 11 that the cumulative volume fraction occupied by clusters at the maximum measured 〈Rg〉 value in Figure 4a is φ* ) 0.248. This value is not far from unity,

Figure 5. Time evolution of the average structure factor in the later stage (t > 600 min) of the WPI aggregation process considered in Figure 3a.

indicating that the DLCA system under investigation is in the transition from aggregation to percolation mechanism and approaching gelation.44 This would suggest that the observed slope of 0.7, larger than 1/df, in the region of t < 300 min in Figure 4a represents the transition kinetics from aggregation to percolation, whereas the experimental results in the region of t > 300 min are already dominated by the static percolation process leading to gelation. Further evidence of approaching gelation may be obtained from the average structure factor curves at very long aggregation times, three of which are shown in Figure 5. It is seen that for q > 1 × 10-4 1/nm, the curve can still be fitted with a straight line with a slope corresponding to df ) 1.85, but for q < 5 × 10-4 1/nm, the average structure factor starts bending upward with time, which is typical of the transition to gelation as observed for other colloidal systems.46 Note that since the data at q < 5 × 10-4 1/nm in Figure 5 reflect the secondary structure due to the interconnections among clusters, which affects the information about the average cluster size (for example the small local bending around q ) 1 × 10-4 1/nm could already be related to the forming of gel network), no accurate 〈Rg〉 value can be estimated. This is why these data have not been used in Figure 4a. Comparison between Experiments and Monte Carlo Simulations. From the DLCA MC simulations in a large range of particle volume fractions, Rottereau et al.44 have obtained a master curve of the normalized average cluster mass 〈i〉/ic as a function of the normalized aggregation time, t/tg, given by the solid curve in Figure 6. The quantities, tg and ic, are the gelation time and the characteristic cluster mass representing the crossover from aggregation to percolation, mentioned above. The broken curve in Figure 6 represents the linear relation between 〈i〉 and t given by eq 7, i.e., corresponding to the typical DLCA conditions far from the transition region. The master curve given by MC simulations indicate that when the average mass of the clusters 〈i〉 reaches a certain value, its growth accelerates progressively and does not follow anymore the linear relation. This occurs in Figure 6 for normalized aggregation time, t/tg > 0.01, which is where the system reaches the transition region from aggregation to percolation. When the t/tg value approaches unity, the growth of 〈i〉 with t becomes explosive and dominated by the percolation process, leading to gelation.

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〈Rg〉2 )

Nii2Rg,i2 ∑ i)1 (14)

∞

Nii2 ∑ i)1 ∞

〈S(q)〉 )

Nii2Si(q) ∑ i)1 (15)

∞

Nii2 ∑ i)1 Figure 6. Normalized average cluster mass, 〈i〉/ic, as a function of the normalized aggregation time, t/tg: solid curve, calculated from DLCA MC simulations;44 symbols, from the 〈Rg〉 experimental data in Figure 4a, with tg ) 900 min and ic ) 6.0 × 104.

Let us now consider the kinetic behavior measured experimentally in Figure 4a. To do this, we need to estimate the values of ic and tg for our system. However, the ic value reported in the above-mentioned work was estimated based on the dependence of the mass i on Rg of single clusters. Since in our system we have a distribution of cluster sizes, of which only the average mass, 〈i〉 ) 1.12(〈Rg〉/Rh,o)1.85, is accessible experimentally, the previous correlation cannot be used. Thus, the values of tg and ic have been estimated in order to get the best fitting of the mastercurve, leading to the experimental data (symbols) shown in Figure 6. It is seen that the agreement between experiments and simulations is rather satisfactory, and particularly the shape of the curves is surprisingly similar, which seems to indicate that the system is in the transition region. This is also supported by the estimated values for tg and ic, which are 900 min and 6.0 × 104, respectively. The tg value is consistent with the growth rate of 〈Rg〉 in Figure 4a and the upward bending of the structure factor in Figure 5. On the other hand, from the ic value, we may estimate the characteristic radius of gyration for the crossover from aggregation to percolation based on the fractal scaling, Rg,c ) Rh,o×(ic/1.12)1/1.85, and the so obtained Rg,c value is 1.17 × 104 nm, which corresponds to the region in Figure 4a where the acceleration in the growth rate of 〈Rg〉 starts. PBE Simulations. The Smoluchowski kinetic model based on population balance equations (PBE) is commonly used to investigate Brownian aggregation processes.23-31 Let us now apply it to simulate the kinetic and structure data obtained from the SALS measurements in this transition region. The relevant PBE model is given by29-31 dNi(t) dt

∞

)-

1 i-1

Ki,jNi(t)Nj(t) + ∑Ki-j,jNi-j(t)Nj(t) ∑ 2 j)1 j)1

(13)

where Ni(t) is number of clusters with mass i at time t and Ki,j is the aggregation kernel between clusters with mass i and j. The solution of this model gives the entire cluster mass distribution (Ni, i ) 1, 2, ...) of the aggregating system, from which the average radius of gyration, 〈Rg〉, the normalized average structure factor, 〈S(q)〉, and the intensity of scattered light at q ) 0, I(0), can be computed, based on the following expressions:

∞

I(0) ∝ N0〈i〉 ) N0

Nii ∑ i)1 (16)

∞

Ni ∑ i)1 where Rg,i and Si(q) are the radius of gyration and the structure factor of the cluster of mass i, which are obtained from DLCA MC simulations.31 Details about the Rg,i and Si(q) expressions can be found elsewhere.45 The classical aggregation kernel for the DLCA processes far from the transition region is Ki,j )

(

2kBT 1/df 1 1 (i + j1/df) 1/d + 1/d f 3µW i j f

)

(17)

which has been proved to describe the typical DLCA behavior of various colloids.30,31,45,47 However, when kernel (17) is applied to simulate the time evolution of 〈Rg〉 in the transition region shown in Figure 4a, it is found that the growth rate of 〈Rg〉 with time is too slow to represent the experimental trend. This predicts in fact the classical power law dependence of 〈Rg〉 with t with the power of 1/df ) 1/1.85 ) 0.45, instead of 0.7 as observed experimentally. To account for the faster growth rate of 〈Rg〉 in the transition region, we introduce a product term, (ij)λ, into kernel (17), thus leading to Ki,j )

(

)

2kBT 1/df 1 1 (i + j1/df) 1/d + 1/d (ij)λ f 3µW i j f

(18)

This product term is often used to simulate the reactionlimited cluster aggregation (RLCA) kinetics21,24,26-31 and the exponent λ is a fitting parameter. In this work, we use this same term, regarding again λ as an adjustable parameter, to empirically reproduce the reactivity of the clusters that increases with their size in the transition region. The mass fractal dimension df has been set equal to 1.85, as determined by the experimental data described above. The value of the Fuchs stability ratio W, as mentioned in the previous section, is generally in the range between 1 and 5 for the typical DLCA kinetics far from the transition region. For the present DLCA kinetics in the transition region, however, no information is available in the literature, and thus, it becomes another fitting parameter. The algorithm proposed by Kumar and Ramkrishna48,49 has been used to solve the above PBE.

Aggregations of Denatured WPI Colloids

Figure 7. Comparison of model results with SALS data for the WPI aggregation system shown in Figures 3 and 4, at C ) 0.02 g/L. Time evolution of the normalized average structure factor, 〈S(q)〉, (a), time evolution of the average radius of gyration 〈Rg〉, (b), and intensity at q ) 0, I(0), as a function of the corresponding 〈Rg〉/Rh,o, (c). df ) 1.85; W ) 8.5; λ ) 0.1.

The simulation results corresponding to the aggregation system in Figure 3 are compared with the experimental SALS data in Figure 7a-c for 〈S(q)〉, 〈Rg〉, and I(0), respectively. It is seen that for the aggregation time, t < 300 min, the time evolutions of the normalized average structure factor 〈S(q)〉 and the average radius of gyration 〈Rg〉 are very well reproduced by the model, as well as the scaling of the zero intensity I(0) with 〈Rg〉. The fitted values for the Fuchs stability ratio W and the exponent λ in eq 18 are 8.5 and 0.1, respectively. The W value is slightly larger than that for typical DLCA processes, but this could be due to specific kinetic characteristics in this transition region. If in fact one uses kernel (17) in the PBE simulation, the obtained results, represented by the broken curve in Figure 7b, reproduce only the first experimental point at the smallest aggregation time, and the fitted W value is 4.5, well in the range between 1 and 5. A positive value of λ ()0.1) indicates that the

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aggregation between large clusters becomes favorable in the transition region. The simulation results in Figure 7 indicate that by introducing the product term in the aggregation kernel, as given in eq 18, the Smoluchowski kinetic model based on PBE can reproduce the DLCA kinetics and cluster structure in the transition region, up to a certain point. Beyond this, which in our case means t>300 min, the percolation process becomes dominant and the PBE model does no longer reproduce the system kinetics. To further confirm the observed aggregation kinetics and clusters structure, we have repeated the cold WPI aggregation at a lower WPI concentrations, C ) 0.007 g/L. Figure 8a-c (symbols) shows the measured time evolution of the normalized average structure factor, 〈S(q)〉, and of the average radius of gyration, 〈Rg〉, together with the I(0) vs 〈Rg〉/Rh,o curve, respectively. In the same figures are also shown the PBE simulation results, using the values of the parameters, df ) 1.85, W ) 7.8, and λ ) 0.1. It is seen that again the agreement between experimental data and simulation results is satisfactory, indicating that also at C ) 0.007 g/L the aggregation system is located in the transition region. It should be pointed out that a slightly smaller W ()7.8) value at C ) 0.007 g/L with respect to that ()8.5) at C ) 0.02 g/L is reasonable, because as the WPI concentration decreases, the system progressively shifts to the typical DLCA behavior, where the W value is in the range between 1 and 5. Note that in Figure 8c when the 〈Rg〉 value is small, the measured I(0) value is significantly smaller than the predicted one. This is because at the considered WPI concentration the scattered intensity at the initial stage of aggregation is very low, and the obtained I(0) values are accordingly inaccurate. Figure 9 compares the 〈i〉/ic vs t/tg curve at C ) 0.007 g/L, together with that at C ) 0.02 g/L, with the mastercurve obtained from DLCA MC simulations as given in Figure 6. In this case of the lower concentration experiment, only the value of tg has been used as a fitting parameter, whereas for ic we used the value estimated at C ) 0.02 g/L and the scaling ic ∼ C-1.4.44 This leads to ic ) 2.61 × 105 and tg ) 7.0 × 103 min. It is seen that the experimental mastercurve obtained from the aggregation experiments at two different WPI concentrations is in good agreement with the mastercurve obtained from DLCA MC simulations. Concluding Remarks In this work, we have investigated the CaCl2-induced coldset aggregation kinetics of the denatured whey proteins isolate (WPI) colloidal dispersion under dilute diffusionlimited cluster aggregation (DLCA) conditions. The average size and the structure of the resulting clusters have been monitored as a function of time using the small-angle light scattering (SALS) technique. The initial diameter of the WPI particles is about 65 nm, whereas the maximum measured average gyration radius of the clusters, 〈Rg〉, is about 80 µm, leading to a cumulative volume fraction of the clusters equal to about 0.25. It is found that all of the average structure factors of the clusters, measured at different times and different WPI

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Figure 9. Normalized average mass of clusters, 〈i〉/ic, as a function of the normalized aggregation time, t/tg, estimated from the measured 〈Rg〉 values at two WPI concentrations, compared with data from DLCA MC simulations reported in the literature.44

Figure 8. Comparison of model results with SALS data for the WPI aggregation system at C ) 0.007 g/L. Time evolution of the normalized average structure factor, 〈S(q)〉, (a), time evolution of the average radius of gyration 〈Rg〉, (b), and intensity at q ) 0, I(0), as a function of the corresponding 〈Rg〉/Rh,o, (c). df ) 1.85; W ) 7.8; λ ) 0.1.

concentrations, collapse in a single curve representing the normalized average structure factor 〈S(q)〉 as a function of the normalized wave vector q×〈Rg〉, with a slope in the power-law region equal to -1.85 ( 0.05. This indicates that the growth of the WPI clusters follows the fractal scaling with the fractal dimension, df ) 1.85 ( 0.05, which is well in the range of the common DLCA colloidal clusters. Independent data of the scattered intensity at q ) 0, I(0), confirm this value of the fractal dimension. Since the cumulative volume fraction of the clusters at the final stage of the aggregation process has been found to be not far from one, i.e., about 0.25, it is legitimate to expect that the aggregation process has reached the region of transition from aggregation to gelation. This is supported by the observation that the observed kinetics of the aggregation process does not follow the typical DLCA power law behavior with the value of the exponent equal to 1/df ) 0.54. In particular, when the aggregation time is smaller than a

certain value, although the aggregation kinetics can still be described by the power law relation, the value of the exponent is equal to about 0.70, which is significantly larger than 1/df. Beyond such time value, the growth of the cluster size with time further accelerates and eventually becomes explosive. In this later stage, the average structure factor measured by SALS tends to bend upward with time in the small q range, indicating the presence of secondary structure originating from the interconnections among clusters to form a gel. The above observed kinetics has been shown to be in good agreement with DLCA MC simulations,44 which describe the evolution of the classical DLCA process from aggregation to gelation. Numerical simulations using the Smoluchowski kinetic model, based on the population balance equations, have also been carried out. As expected, with the classical DLCA kernel, the predicted growth rate of the average cluster size is too slow to represent the aggregation kinetics in this transition region, because it reproduces the classical power law behavior with the exponent of 1/df. It turns out that the introduction of an empirical product term, as often used for the reaction-limited cluster aggregation processes, leads to a good reproduction of the time evolution of all the quantities measured by SALS, i.e., the structure factor, the scattered intensity at zero angle and the average radius of gyration. Acknowledgment. This work was financially supported by Suisse National Science Foundation (Grant No. 200020101714). The WPI powder supplied by Davisco Foods International Inc. (Eden Prairie, MN) is gratefully acknowledged. It is a pleasure to thank Taco Nicolai for carefully reading the manuscript and making fruitful suggestions. References and Notes (1) Kinsella, J. E.; Whitehead, D. M. AdV. Food Nutr. Res. 1989, 33, 343. (2) Bryant, C. M.; McClements, D. J. Trends Food Sci. Technol. 1998, 9, 143. (3) Barbut, S.; Foegeding, E. A. J. Food Sci. 1993, 58, 867. (4) McClements, D. J.; Keogh, M. K. J. Sci. Food Agric. 1995, 69, 7. (5) Ju, Z. Y.; Kilara, A. J. Agric. Food Chem. 1998, 46, 3604. (6) Roefs, S. P. F. M.; de Kruif, K. G. Eur. J. Biochem. 1994, 226, 883. (7) Aymard, P.; Nicolai, T.; Durand, D.; Clark, A. Macromolecules 1999, 32, 2542. (8) Le bon, C.; Nicolai, T.; Durand, D. Int. J. Food Sci. Technol. 1999, 34, 451. (9) Ikeda, S.; Morris, V. J. Biomacromolecules 2002, 3, 382.

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