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Thermal Restructuring of Fractal Clusters: The Case of a Strawberry-like Core-Shell Polymer Colloid Zichen Jia, Hua Wu, and Massimo Morbidelli* Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, 8093 Zurich, Switzerland ReceiVed NoVember 7, 2006. In Final Form: January 31, 2007 Thermal restructuring of fractal styrene-acrylate copolymer clusters dispersed in water has been investigated experimentally in the temperature range between 313 and 363 K. The particles constituting the clusters are of strawberrylike core-shell structure with a soft core and a rigid shell grafted on the core polymer chains. Due to the incomplete coverage of the core, the rather soft core may “flow out” through the open areas of the shell, leading to coalescence with the neighboring particles. The clusters were generated under diffusion-limited cluster aggregation conditions, and the restructuring kinetics was monitored by small-angle light scattering. Two sets of thermal restructuring experiments have been performed at various temperatures: (1) restructuring of growing clusters during aggregation and (2) restructuring of preformed clusters in the absence of aggregation. It is found that restructuring occurs only at temperature values above 323 K. In the absence of aggregation, restructuring leads to an increase of the fractal dimension and a decrease of the radius of gyration of the clusters. At sufficiently long times, both quantities reach a plateau value due to the presence of the grafted rigid shell, which constrains the coalescence of the soft core. A simple model, based on coalescence theory of liquid droplets and accounting for the incomplete coalescence and its dependence on temperature, has been developed to interpret the restructuring kinetics in the absence of aggregation. It is found that the proposed model can represent the measured experimental data well.
1. Introduction Emulsion polymerization is one of the dominant industrial processes to produce polymers, in the form of latexes, which are typical colloidal systems with submicrometer polymer particles. In many cases, the polymer particles need to be separated from the disperse medium through a proper coagulation process. The resulting aggregates or clusters may undergo a sintering process at a temperature higher than the polymer glass transition temperature Tg (for amorphous) or melt temperature Tm (for semicrystalline), where the particles become soft or rubbery or molten, leading to coalescence or fusion among neighboring particles.1 This eventually leads to more compact clusters. Restructuring during sintering is of essential importance for the size, size distribution, and morphology of clusters, which significantly affect the properties of the final polymer products. For instance, polymer aggregates with more compact structure contain less residual of the dispersion medium (including electrolytes or other chemical species present in it), which clearly affects the quality and properties of the final products. Particle coalescence and restructuring has been studied in the case of aerosol clusters, which are made in the gas phase at high temperature,2 where the clusters grow and restructure with simultaneous coagulation and coalescence of particles. One of the commonly used theoretical frames for particle coalescence was developed by Frenkel,3 who studied the early stage of viscous coalescence of two equal-sized droplets. He derived the characteristic coalescence time as a function of viscosity, surface tension, and particle size, and modeled the reduction of particle surface area during coalescence assuming a constant rate. Koch and Friedlander4 proposed that the particle surface area decays * Corresponding author. E-mail: 0041-44-6321082.
[email protected]. Fax:
(1) Narkis, M., Rosenzweig, N., Eds. Polymer Powder Technology; John Wiley & Sons Ltd.: Chichester a.o., 1995. (2) Pratsinis, S. E.; Vemury, S. Powder Technol. 1996, 88, 267. (3) Frenkel, J. J. Phys. (Moscow) 1945, 9, 385.
exponentially with time during coalescence and incorporated this sintering law into the population balance equation to describe the dynamics of the aerosol particle size distribution. Their model was further improved5 by accounting for the restructuring of clusters. Besides the reduction rate of particle surface area, the growth rate of the neck radius of two coalescing particles has been used to describe particle coalescence kinetics.6-8 More recently, a model for particle coalescence has been proposed,9,10 which is based on the rate of reduction of the interparticle distance, derived through geometrical transformations from the reduction rate of the surface area. This model is able to describe the evolution of aggregate size and structure in terms of radius of gyration and fractal dimension when coagulation and sintering are simultaneously taking place. On the other hand, the restructuring of polymer clusters dispersed in aqueous solutions has received little attention in the literature. Some studies have been reported on the coalescence of polymer particles due to sintering6,11-17 with reference to either the postprocessing of polymer materials in the absence of (4) Koch, W.; Friedlander, S. K. J. Colloid Interface Sci. 1990, 140, 419. (5) Xiong, Y.; Pratsinis, S. E. J. Aerosol Sci. 1993, 24, 283. (6) Bellehumeur, C. T.; Bisaria, M. K.; Vlachopoulos, J. Polym. Eng. Sci. 1996, 36, 2198. (7) Bellehumeur, C. T.; Kontopoulou, M.; Vlachopoulos, J. Rheol. Acta 1998, 37, 270. (8) Pokluda, O.; Bellehumeur, C. T.; Vlachopoulos, J. AIChE J. 1997, 43, 3253. (9) Schmid, H. J.; Tejwani, S.; Artelt, C.; Peukert, W. J. Nanopart. Res. 2004, 6, 613. (10) Schmid, H. J.; Al-Zaitone, B.; Artelt, C.; Peukert, W. Chem. Eng. Sci. 2006, 61, 293. (11) Mahr, T. G. J. Phys. Chem. 1970, 74, 2160. (12) Bertha, S. L.; Ikeda, R. M. J. Appl. Polym. Sci. 1971, 15, 105. (13) Narkis, M. Polym. Eng. Sci. 1979, 19, 889. (14) Rosenzweig, N.; Narkis, M. Polymer 1980, 21, 988. (15) Rosenzweig, N.; Narkis, M. J. Appl. Polym. Sci. 1981, 26, 2787. (16) Mazur, S. Coalescence of polymer particles. In Polymer powder technology; German, R. M., Messing, G. L., Cornwall, R. G., Eds.; John Wiley & Sons: New York, 1995. (17) Dobler, F.; Pith, T.; Lambla, M.; Holl, Y. J. Colloid Interface Sci. 1992, 152, 1.
10.1021/la063254s CCC: $37.00 © 2007 American Chemical Society Published on Web 04/04/2007
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Figure 1. (a) SEM picture of primary particles; (b) scheme of the core-shell structure of a primary particle.
dispersion medium or the film formation process. Moreover, the particles undergoing restructuring studied in the literature usually have regular morphology, e.g., smooth surface, homogeneous melting, or glass transition temperature. This is not the case for particles having, for example, a core-shell structure with different properties for the core and the shell. In this work, the thermal restructuring process of fractal styreneacrylate copolymer clusters dispersed in water has been investigated experimentally in the temperature range between 313 and 363 K. The particles constituting the clusters exhibit a strawberry-like core-shell structure, with significantly different glass transition temperatures: the core is very soft, while the shell is rigid in the given temperature range. Due to these peculiar structures, the soft core is not fully covered by the rigid shell, and it may “flow out” through the open areas of the shell, leading to coalescence with the neighboring particles. This peculiar coalescence mechanism has been monitored by small-angle light scattering, during two sets of thermal restructuring experiments starting with polymer clusters generated under diffusion-limited cluster aggregation conditions. These refer to the restructuring of the growing clusters during aggregation and to the restructuring of preformed clusters in the absence of aggregation, respectively. A simple model, based on the coalescence theory available in the literature for soft particles (droplets), is then developed to interpret the experimental data in the absence of aggregation. 2. Experimental Section 2.1. Materials and Methods. The Colloidal System. The colloidal system used in this work is a styrene-acrylate copolymer latex, produced through emulsion polymerization. The particles exhibit a core-shell structure with the core of acrylic ester elastomer and the shell of polystyrene. Figure 1a shows the SEM picture of primary particles. It can be seen that the shell does not cover completely the core of the particle, leading to a somewhat strawberry-like structure with some regions of the core surface not covered by the shell, as schematically represented in Figure 1b. Since the glass transition temperatures Tg of the core and the shell materials are different, being 373 K, respectively, the core is very soft and behaves like a viscous droplet at sufficiently high temperatures. Then, when clusters are formed from such particles, the incompletely
Jia et al. covered core of the particles can easily be deformed and lead to restructuring of the clusters. It is this restructuring process that we investigate in this study. Although the particles are not perfectly spherical, both static and dynamic light scattering measurements show that they are rather monodisperse with average radius a0 ) 52 nm. The original latex was first diluted with demineralized water to the particle volume fraction φ ) 10-2, which was then used as the reference latex for all subsequent experiments. Two sets of experiments have been carried out in order to investigate the cluster thermal restructuring. In the first set, at a fixed particle volume fraction, the thermal restructuring of clusters during the growth of the clusters under diffusion-limited cluster aggregation (DLCA) conditions has been monitored at various temperatures. In the second set, the restructuring of DLCA clusters preformed at room temperature has been investigated as a function of time for various temperatures. Characterization of Cluster Structure Using SALS. A small-angle light scattering (SALS) instrument, Mastersizer 2000 (Malvern, U.K.), has been used to characterize the structure of the clusters in all experiments. The scattering angle ranges from θ ) 0.02° to 40°, and the wavelength of the laser beam is λ0 ) 633 nm. More details about the instrument may be found elsewhere.18 In particular, for each sample, we determine the normalized average structure factor of the clusters, 〈S(q)〉, from the measured angle-dependent scattered intensity, I(q)19,20 〈S(q)〉 )
I(q) I(0)P(q)
(1)
where I(0) is the scattered intensity at zero angle, P(q) is the form factor of the primary particles, which is measured through the SALS of the original latex, and q is magnitude of the wavevector defined as q)
4πn0 θ sin λ0 2
()
(2)
with n0 being the refractive index of the dispersion medium. From the so-determined 〈S(q)〉, the average radius of gyration of the clusters, 〈Rg〉, can be estimated using the Guinier plot21,22 〈S(q)〉 ) exp(-q2〈Rg〉2/3)
for q < 1/〈Rg〉
(3)
Moreover, for well-developed mass fractal clusters, the slope of the power-law regime of 〈S(q)〉 gives an estimate of the mass fractal dimension, Df 〈S(q)〉 ∝ q-Df
for 1/〈Rg〉 , q , 1/a0
(4)
The average radius of gyration 〈Rg〉 and the fractal dimension Df are the main quantities used in this work for quantifying the evolution of the structure of the cluster population during aggregation and restructuring. 2.2. Restructuring of Growing Clusters during Aggregation. In this set of experiments, particle aggregation has been conducted under DLCA conditions at a fixed particle volume fraction and salt concentration but at various temperature values. In order to have sufficient time to observe the restructuring process, we have slowed down substantially the aggregation rate by initiating the DLCA experiments at a rather low particle volume fraction (φ ) 4 × 10-5). The coagulant used for destabilizing the system is MgSO4. The ccc (critical coagulant concentration) of MgSO4 for the (18) Wu, H.; Lattuada, M.; Sandkuhler, P.; Sefcik, J.; Morbidelli, M. Langmuir 2003, 19, 10710. (19) Bohren, C. F.; Huffman, D. R. Absorption and scattering of light by small particles; Wiley: New York, 1983. (20) Kerker, M. The Scattering of Light; Academic Press: New York, 1969. (21) Sorensen, C. M. Aerosol Sci. Technol. 2001, 35, 648. (22) Bushell, G. C.; Yan, Y. D.; Woodfield, D.; Raper, J.; Amal, R. AdV. Colloid Interface Sci. 2002, 95, 1.
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Figure 2. Scheme of the sampling procedure for the measurements in the SALS unit. given colloidal system, determined experimentally, is 0.025 mol/L. Thus, the MgSO4 concentration for the experiments has been chosen at Cs ) 0.08 mol/L, much larger than the ccc, so as to ensure DLCA conditions. In order to initiate a DLCA experiment, we added a MgSO4 solution, preheated at the desired temperature, to a non-preheated colloidal dispersion to reach the desired final salt concentration and particle volume fraction (i.e., Cs ) 0.08 mol/L and φ ) 4 × 10-5). The obtained mixture was kept in a closed bottle and immediately introduced in a thermostat preset at the desired temperature. Note that the volume of the MgSO4 solution is always four times larger than that of the colloidal dispersion, and the mixing induced by pouring the salt solution into the colloidal dispersion is enough to get quick homogeneity of the mixture without using any agitator. For the same reason, the effect of mixing of the preheated salt solution with the non-preheated colloidal dispersion on the initial temperature reduction of the system is negligible. During aggregation, samples were taken using a pipet at different times, which were immediately diluted 20 times (to φ ) 2 × 10-6) in demineralized water in order to stop aggregation and then characterized by SALS. It should be noted that, since the clusters generated under stagnant conditions can be rather weak, in order not to damage them during sampling, pipettes with a tip opening of at least 2 mm were used and the samples were sucked with the pipet very slowly. Moreover, in order to reduce the shear effect on the cluster structure during the injection of the sample into the SALS cell, a procedure schematically shown in Figure 2 was used. In particular, instead of injecting, we sucked the sample into the SALS unit cell through a short rubber tube with an inner diameter of 6 mm. The sucking rate was controlled by a syringe pump at 8.5 mL/min, which corresponds to a mean shear rate of 4 l/s in the tube, which should not significantly affect the cluster size and structure. This was checked by changing the speed of the syringe pump to lower values, and no effect on the final aggregates was observed. Figure 3a shows the time evolution of the original average structure factor I(q)/P(q) of the clusters, measured by SALS, as a function of the wavevector q, for the DLCA experiment at T ) 313 K. It is seen that, as the aggregation time t increases, the I(q)/P(q) curve bends down progressively and shifts toward the small q range, and the plateau intensity increases, representing the typical growth of clusters. When the Guinier plot, eq 3, is applied to all the curves in Figure 3a, we obtain the time evolution of the average radius of gyration 〈Rg〉. Note that, for the Guinier plot, it is necessary to transfer the original average structure factor to the normalized one, 〈S(q)〉, as defined by eq 1, such that 〈S(q)〉 ) 1 as q f 0, where I(0) is the intensity at the plateau, representing the zero-angle intensity. It is known18,23 that the I(0) value is proportional to the average mass of the clusters, which scales with 〈Rg〉 as I(0) ∝ 〈Rg〉Df. Then, if the growth of the clusters follows certain self-similarity, i.e., the fractal scaling, a plot of I(0) vs 〈Rg〉 in the log-log plane should be a straight line and the slope leads to an estimate of the fractal dimension, Df. Figure 3b shows such a plot for the data obtained from Figure 3a. Indeed, it is a straight line, and the obtained value of the fractal dimension is Df ) 1.87, which is very close to the typical Df value (23) Brown, W. Light ScatteringssPrinciples and DeVelopment; Clarendon Press: Oxford, 1996.
Figure 3. (a) The original average structure factor I(q)/P(q) as a function of the wavevector q at different aggregation times; (b) the zero-angle intensity I(0) at each aggregation time as a function of the corresponding average radius of gyration 〈Rg〉; and (c) the normalized average structure factor 〈S(q)〉 as a function of normalized wavevector q〈Rg〉 obtained from data in (a). T ) 313 K. of DLCA clusters. Another manifestation of the cluster growth following the self-similarity is that, after the concentration of the primary particles in the system becomes negligible, all the normalized average structure factors 〈S(q)〉 at different times overlap to form a single master curve when they are plotted as a function of 〈Rg〉,24-27 which is clearly shown in Figure 3c for the given system. Moreover, its power-law region gives the fractal dimension Df ) 1.87, which is identical to that estimated above from the I(0) vs 〈Rg〉 plot. The same DLCA process has been carried out at five higher temperatures, between 323 and 363 K. It turns out from the analysis of the light scattering data that the same features as in Figure 3 have been found at all the observed temperatures. However, there are distinct differences in the aggregation kinetics and cluster structures. In particular, Figure 4a compares the time evolution of the average radius of gyration 〈Rg〉 at six different temperatures, and Figure 4b shows the normalized structure factor 〈S(q)〉 as a function of q〈Rg〉 (24) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Nature (London) 1989, 339, 360. (25) Gimel, J. C.; Durand, D.; Nicolai, T. Macromolecules 1994, 27, 583. (26) Sandkuhler, P.; Sefcik, J.; Morbidelli, M. Langmuir 2005, 21, 2062. (27) Wu, H.; Xie, J.; Morbidelli, M. Biomacromolecules 2005, 6, 3189.
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Figure 5. A Cryo-SEM (cryogenic scanning electron microscopy) picture of the cluster, generated by the strawberry-like core-shell particles shown in Figure 1, after restructuring at T ) 358 K.
Figure 4. (a) Average radius of gyration 〈Rg〉 as a function of time measured by SALS at different temperatures; (b) the 〈S(q)〉 vs q〈Rg〉 master curves at three selected temperatures; and (c) the fractal dimension Df as a function of the temperature. at three selected temperatures. In Figure 4c are reported the values of the fractal dimension Df as a function of temperature, obtained from the 〈S(q)〉 vs q〈Rg〉 plots. It is seen that, when the temperature increases from T ) 313 to 323 K, the growth rate of 〈Rg〉 becomes faster. On the other hand, the Df values shown in Figure 4c at the two temperatures are practically the same. This means that the faster growth rate of 〈Rg〉 at T ) 323 K with respect to T ) 313 K is only due to the effect of temperature on the system physicochemical properties (e.g., viscosity, diffusivity, etc.). This can be clearly seen by comparing the values of the characteristic (or Smoluchowski) time of the Brownian aggregation, tB ) 3µ/(8kBTN0) (where kB is the Boltzmann constant, µ the kinematic viscosity, and N0 the initial number concentration of the primary particles) at the two temperatures, which are equal to 0.559 s and 0.453 s, respectively. In fact, if the 〈Rg〉 values at the two temperatures are plotted as a function
of the reduced time, t/tB, they collapse on a single master curve (data not shown). Starting from T ) 333 K, however, the growth rate of 〈Rg〉 in Figure 4a does not follow the master curve anymore and progressively slows down as the aggregation temperature increases. Moreover, the Df value in Figure 4c does not keep constant and increases as the temperature increases. This is clearly evidenced in Figure 4b, where the slope of the power-law regime increases continuously as the temperature increases. The above results clearly indicate the presence of an additional process, i.e., restructuring, during the aggregation for T g 333 K, while for T < 333 K, no restructuring occurs. Since for a given number of primary particles the cluster size decreases as Df increases, this explains why the growth rate of 〈Rg〉 in Figure 4a decreases as T increases for T > 333 K. The mechanism leading to such cluster restructuring may be understood by recalling that the primary particles are of strawberrylike core-shell morphology, as shown in Figure 1. Since the Tg values of the core and shell are 373 K, respectively, the shell material remains rather rigid in the entire investigated temperature range. However, the incompletely covered, very soft core may “flow out” through the open areas. This may lead to partial fusion between the neighboring particles, similar to the coalescence process between two droplets. This is supported by the cryo-SEM picture of the cluster restructured at T ) 358 K, as shown in Figure 5. It is seen that the original primary particles of a0 ) 52 nm can be hardly identified after the restructuring. On the other hand, since the polymer chains of the particle shell are grafted on those of the particle core, such core fusion is strongly constrained by the presence of the shell. This makes this process different from the case of droplet coalescence. In particular, droplet coalescence is typically driven by viscous flow,3 eventually leading to molecular interdiffusion, but in the present case, macromolecular interdiffusion hardly occurs. Most likely in this case, the viscous flow results only in local rubbery chain interpenetration or even just in local deformation, and all the chains remain within the original primary particles. Thus, the observed restructuring phenomenon occurs basically only in local areas, and the overall shape of the clusters may remain unchanged. In fact, the structure of the cluster shown in Figure 5 remains open after the restructuring. At T ) 313 and 323 K, although they are substantially above the Tg of the core, no restructuring is observed, confirming the strong restriction imposed by the shell to the “flowing-out” of the core. Since the softness of the core increases as the temperature increases, increasing the temperature accelerates the restructuring process, and as a result, the compactness (Df) of the clusters increases with temperature. It is clear that the data shown in Figure 4 at each temperature value result from the coexistence of the two processes, the DLCA and restructuring. The former generates clusters of open structure, and the latter compacts them. On the other hand, at a given temperature, the normalized structure factor 〈S(q)〉 obtained at different times collapses on a single master curve, as, for example,
Thermal Restructuring of Fractal Clusters shown in Figure 4b, indicating that the clusters generated at different times are self-similar. This may imply that the restructuring process is faster than the aggregation such that, although the restructuring leads to compactness, the aggregation proceeds following the fractal scaling. 2.3. Restructuring of Preformed Clusters. Although the above experiments provide clear evidence of cluster restructuring at high temperature, the simultaneous presence of aggregation does not allow us to quantify its kinetics. For this reason, we have performed the second set of experiments that is composed of two steps: (1) preforming the DLCA clusters to a certain average size at room temperature, and (2) diluting substantially the preformed clusters into demineralized water preheated at the desired temperature to avoid aggregation, and observing their restructuring kinetics. We choose DLCA, instead of RLCA, to generate the preformed clusters, because the cluster size distribution is much narrower in a DLCA process than in an RLCA one.28 The particle volume fraction (φ ) 4 × 10-5), the MgSO4 concentration (Cs ) 0.08 mol/L), and the procedure for initiating the aggregation are identical to those adopted in the first set of experiments discussed above. The size of the preformed clusters used for all the restructuring experiments at different temperatures corresponds to the situation when aggregation has proceeded for 1 h at ambient temperature. Moreover, in order to better characterize the initial conditions of the restructuring process in terms of 〈Rg〉0 and Df, the aggregation process for producing the preformed aggregates was monitored in situ in the SALS sample cell. The dilution ratio for the restructuring experiments was 20 times, i.e., the clusters preformed at φ ) 4 × 10-5 were diluted to φ ) 2 × 10-6. Such a dilution reduces the salt concentration from Cs ) 0.08 mol/L to Cs ) 0.004 mol/L, which is smaller than the ccc ()0.025 mol/L). As a consequence of both dilution and salt concentration reduction, we can safely assume that aggregation is negligible during the restructuring process. In addition, in order to further confirm the absence of aggregation, a small amount (2 mmol/ L) of sodium dodecyl sulfate (SDS) was added to the preheated water prior to adding the preformed clusters, to further stabilize the clusters. After the dilution and the restructuring process started, samples were taken at different times and characterized by SALS. Note that, since the restructuring experiments were already performed at very dilute conditions, no further dilution was done for the characterization. Instead, the sampling procedure described in Figure 2 was applied to directly suck the sample from the vessel where restructuring occurs to the SALS instrument sample cell. Since the previous results from the first set of experiments indicate that no restructuring occurs for T < 333 K, the restructuring experiments have been carried out at four higher temperatures, i.e., at T ) 333, 343, 353, and 363 K, respectively. The obtained time evolutions of the average radius of gyration 〈Rg〉 and the fractal dimension Df are shown in Figure 6a,b, respectively. From the data in Figure 6a, we see that the 〈Rg〉 value in all cases decreases in time and eventually reaches a plateau or steady-state value. Starting from practically the same initial 〈Rg〉 value, the plateau decreases as the temperature increases. The corresponding Df shown in Figure 6b instead increases as the temperature increases and also reaches a plateau at sufficiently long times. At T ) 363 K, the plateau Df value is as large as 2.9, indicating very compact clusters. These results clearly confirm the presence of thermal restructuring in the polymer clusters under examination. When the plateau Df value in Figure 6b is compared with the Df value in Figure 4c, it appears that, although both increase as temperature increases, the former at a given temperature is substantially larger than the latter, i.e., the clusters are more compact when the aggregation process is absent. Thus, the plateau Df value in Figure 6b can be considered as the upper compactness of the clusters that one can obtain at each given temperature, while the one in Figure 4c represents an equilibrium structure when both the aggregation and restructuring processes are simultaneously present. (28) Sandkuhler, P.; Lattuada, M.; Wu, H.; Sefcik, J.; Morbidelli, M. AdV. Colloid Interface Sci. 2005, 113, 65.
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Figure 6. (a) Average radius of gyration 〈Rg〉 and (b) fractal dimension Df measured by SALS as a function of time for various temperature values: 333 K (O), 343 K (0), 353 K (4), and 363 K (]). Accordingly, we would expect that the former does not depend on the particle volume fraction φ, while the latter does, since the φ value alters the aggregation but not the restructuring rate. In order to further support the conclusion drawn above, it is worth checking whether restructuring also occurs in the case of clusters formed by rigid primary particles. For this, we have selected a standard polystyrene colloid with a0 ) 250 nm and performed the same restructuring experiments as those shown in Figure 6, at T ) 353 K. Since the restructuring temperature is smaller than the Tg of polystyrene, the polystyrene particles can be considered rigid particles in this case. It is found (data not shown) that both the 〈Rg〉 and Df values are practically constant over time. This means that, in the case of clusters made of rigid polystyrene particles, no thermal restructuring occurs, thus confirming that the restructuring behavior observed in Figures 4 and 6 is due to the specific rubbery core and primary particle morphology of the latex considered in this work.
3. Modeling the Kinetics of Cluster Restructuring 3.1. The Restructuring Model. As concluded in the previous section, the observed thermal restructuring of the fractal clusters is strongly related to the strawberry-like core-shell morphology of the primary particles with a very soft rubbery core, as shown in Figure 1. When the temperature is high enough, the soft core of a particle may partially coalesce with those of its neighboring particles within the cluster. However, the presence of the grafted rigid shell imposes some constraints to their coalescence process, which therefore most likely occurs only at the local level. Figure 5 supports this mechanism. Thus, in order to describe the kinetics of such a restructuring process, we propose a model that accounts for partial coalescence of the particles and assumes that the shape of the backbone of the cluster remains unchanged during restructuring. Restructuring between Two Particles at Contact. Let us first consider two neighboring particles in a cluster. On the basis of the general theoretical framework of coalescence between two droplets,3,4 in order to minimize the surface free energy, two
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(droplets) rubbery particles tend to coalesce upon contact. Considering two equal-size spherical particles with radius a0, such a coalescence process is illustrated schematically in Figure 7. At t ) 0, the center-to-center distance of the two particles is l0 ) 2a0. As coalescence proceeds, the particles partially overlap, and their center-to-center distance l decreases. Accordingly, the particles have to deform in order to preserve mass. To simplify the mathematical treatment, let us assume that the two particles keep their spherical shape during coalescence, with a radius a that increases with time, in order to account for the mass in the overlapping region which has to be redistributed between the two spheres so as to conserve the total mass. Thus, at sufficiently long times, the two particles completely coalesce (overlap), resulting in l ) 0 and a single spherical particle with radius a 3 ) x2a0. Let us now apply such a coalescence model to describe the restructuring between two neighboring particles in our clusters. The commonly used approach for describing the coalescence process is monitoring the time evolution of the total surface area of the coalescing particles S, which follows the linear decay law as derived by Friedlander and co-workers4,29
S - Sm dS )dt τc
(5)
where Sm is the minimum surface area of the coalesced particles, which for a completely coalescing system is equal to the surface area of the final sphere. For our core-shell particles, however, due to the presence of the constraint given by the rigid shell, the coalescence is incomplete and its extent depends on temperature. Therefore, the Sm value can be significantly larger than that corresponding to complete coalescence and is a function of temperature. In eq 5, τc is the characteristic coalescence time of the particles, which, according to Frenkel’s model,3 may be expressed as
τc ∝
ηa0 σ
Figure 7. Schematic illustration of the coalescence of two equalsize particles with a > a0.
(6)
where η is the material viscosity of the coalescing particles and σ the surface tension. Since the viscosity and surface tension are strong functions of temperature, τc, together with Sm, describes the temperature dependence of the S decay rate in eq 5. Moreover, it has been shown4 that τc can be described as a function of temperature by an Arrhenius-type law
Figure 8. An aggregate undergoing coalescence of the constituent particles during restructuring: (a) at time t ) 0; (b) restructuring time t > 0.
A τc ∝ exp T
EVolution of the Cluster Radius of Gyration. Let us consider a cluster, as schematically shown in Figure 8, which is composed of i equal-sized spherical primary particles. At t ) 0 (Figure 8a), no overlapping occurs among the constituent primary particles, and its radius of gyration, Rg,i, is given by32
()
(7)
where A is a constant, which in the case of oxide aerosols is on the order of 104 K.30,31 Once the time evolution of S is obtained from eq 5, the time evolutions of a and l can be easily computed purely on the basis of geometrical considerations about continuously overlapping spheres as shown in Figure 7
(
S -1 l ) 2a 4πa2
)
(8)
and
4a03 a ) l 2 l 1+ 22a 2a 3
(
)(
)
(9)
Rg,i2 ) Rg,p2 +
1 i
2
i
i
rjm2 ∑ ∑ j)1 m)1
(10)
where Rg,p ) x3/5a0 is the radius of gyration of the primary particle, and rjm is the distance between the centers of the jth and mth particles. Let us assume that coalescence between particles occurs only along their central line, and then, the “backbone” of the cluster scales down progressively, following the decay of the center-to-center distance l, but keeping the shape similarity, (29) Friedlander, S. K.; Wu, M. K. Phys. ReV. B 1994, 49, 3622. (30) Kobata, A.; Kusakabe, K.; Morooka, S. AIChE J. 1991, 37, 347. (31) Xiong, Y.; Akhtar, M. K.; Pratsinis, S. E. J. Aerosol Sci. 1993, 24, 301. (32) Jullien, R. Croat. Chem. Acta 1992, 65, 215.
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as illustrated by the thick lines in Figure 8. With such an assumption, it is straightforward to have
rjm l ) (j, m ) 1 to i) rjm,0 l0
(11)
where the subscript 0 refers to the initial time t ) 0. Equation 11 states that the scaling in distance between any two particles in the cluster during restructuring is equal to that between two partially coalescing particles as described by eqs 5-9. When eq 11 is substituted into eq 10, it is readily seen that the following scaling of the radius of gyration of the cluster during restructuring
Rg,i2 - Rg,p2 Rg,i,0 2
Rg,p,02
)
() l l0
2
(12)
where Rg,p and Rg,p,0 are the gyration radii of the deformed and initial primary particles, respectively. Since both the Rg,p and Rg,p,0 values are much smaller than those of the clusters, eq 12 can be simplified to
Rg,i2 2
Rg,i,0
)
()
Rg,i l 2 l w ) l0 Rg,i,0 l0
(13)
In the case of a populated cluster system, the average radius of gyration of the clusters, 〈Rg〉, is given by21,33 imax
〈Rg〉2 )
Nii2Rg,i2 ∑ i)1 (14)
imax
Nii2 ∑ i)1 where Ni is the number of clusters containing i primary particles, and imax is the largest possible number of primary particles in a cluster. Let us assume that, during restructuring the population of the clusters, Ni does not change in time. This corresponds to the case of restructuring of preformed clusters, investigated experimentally in the previous section. Then, from eqs 13 and 14, we have
〈Rg〉 〈Rg〉0
)
l l0
(15)
i.e., the scaling of 〈Rg〉 with time is also equal to that of l. It should be emphasized that eqs 13 and 15 are valid only for partially coalescing systems, such as the core-shell particles under examination here. For completely coalescing systems, the center-to-center distance between particles l approaches zero, while the Rg value does not, since it approaches the radius of the finally formed sphere. In this case, the simplification from eq 12 to 13 cannot be applied. EVolution of the Cluster Fractal Dimension. The experiments shown above indicate that the structure of the clusters during restructuring can still be approximated as that of fractal objects, with fractal dimension Df (average radius gyration 〈Rg〉) increasing (decreasing) with time. It is known32,34-38 that, for a fractal cluster, (33) Lattuada, M.; Sandkuhler, P.; Wu, H.; Sefcik, J.; Morbidelli, M. AdV. Colloid Interface Sci. 2003, 103, 33. (34) Mandelbrot, B. The Fractal Geometry of Nature; W. H. Freeman: San Francisco, 1982. (35) Meakin, P. Phys. ReV. Lett. 1983, 51, 1119. (36) Kolb, M.; Herrmann, H. J. J. Phys. A: Math. Gen. 1985, 18, L435. (37) Meakin, P. AdV. Colloid Interface Sci. 1988, 28, 249.
Figure 9. Experimental values (O) and the power-law fitting curve of the prefactor k as a function of the fractal dimension Df corresponding to the data shown in Figure 6 in the temperature range from 333 to 363 K. The open circles (O) and open triangles (3) represent the numerically simulated data reported by Sorensen and Roberts39 and Lattuada et al.40
the mass i and the radius of gyration Rg follow the fractal scaling rule
i)k
() Rg a0
Df
(16)
where k is the prefactor. Equation 16 shows that, for a fixed cluster mass i, the Df value increases as the Rg value decreases. The relation between 〈Rg〉 and Df observed in our restructuring experiments is qualitatively consistent with eq 16. However, to quantitatively describe the experimental results, one has to consider the fact that the prefactor k in eq 16, though constant for a given Df value, depends on the Df value. This is clearly demonstrated by the results of numerical simulations reported in the literature.39,40 In particular, Sorensen and Roberts39 performed various simulations of diffusion-limited aggregations and found that the k value decreases as Df increases. Lattuada et al.40 showed that k ) 1.117 and 0.94 for average clusters with Df ) 1.85 and 2.05, respectively. Then, to model the observed time evolution of the Df value, we need to first establish a relation between k and Df. Unfortunately, little information is available in the literature about this relation, and therefore it has been determined by fitting the 〈Rg〉 and Df data for the restructuring experiments shown in Figure 6. In particular, we first estimate the average mass of the clusters 〈i〉 from 〈Rg〉0 and Df,0 at t ) 0, using the k value of 1.1, typical of DLCA clusters. Due to the absence of aggregation, such an 〈i〉 value can be considered to be constant during restructuring. Then, on the basis of eq 16, we can estimate the k value for each pair of 〈Rg〉 and Df values in Figure 6. The so-obtained k values are shown in Figure 9 as a function of Df, together with the numerical results from the literature.39,40 It is seen that the obtained k values exhibit a clear tendency to decrease with Df, and they follow the same trend exhibited by the values from the numerical simulations. All the k values in Figure 9 can be well-represented by a power-law function as follows:
k ) 7.32Df-3.0
(17)
Therefore, eq 17 will be used in eq 16 to simulate the time evolution of Df from the time evolution of the average radius of gyration 〈Rg〉 modeled in the previous subsection. (38) Meakin, P. Croat. Chem. Acta 1992, 65, 237. (39) Sorensen, C. M.; Roberts, G. C. J. Colloid Interface Sci. 1997, 186, 447. (40) Lattuada, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 106.
5720 Langmuir, Vol. 23, No. 10, 2007
Jia et al.
Figure 10. Comparison of model results (s) and experimental data of the dimensionless average radius of gyration 〈Rg〉/〈Rg〉0 as a function of time at various restructuring temperature values: 333 K (O), 343 K (0), 353 K (4), and 363 K (]).
Figure 11. Characteristic coalescence time τc as a function of the inverse of temperature 1/T. The straight line represents the best fitting by an exponential function.
Table 1. Model Parameters Obtained by Fitting the Experimental Values temperature (K) τc (s) Sm/S0
333 7660 0.977
343 2560 0.926
353 893 0.85
363 229 0.827
3.2. The Model Results. Let us now apply the above proposed model to simulate the experimentally observed restructuring kinetics reported in the previous section. Note that, in the first set of experiments, since restructuring and aggregation occur simultaneously, numerical simulations are rather complex, requiring two internal coordinates. In this work, we perform the simulation only for the second set of restructuring experiments, where aggregation is absent. The first step is to simulate the time evolution of 〈Rg〉 at different temperatures in Figure 6a, through eq 5 together with eqs 8, 9, and 15. These equations involve two unknown model parameters, τc and Sm. As given by eq 6, the characteristic coalescence time τc is a function of viscosity, surface tension, and initial size of particles, and similarly, the minimum surface area Sm depends on the properties of the particle material, and particularly on the core-shell structure of the particles. Obviously, no information about the τc and Sm can be found in the literature for the specific particles under consideration here, and they are also very difficult to estimate on the basis of some a priori principle. Thus, they have been used as fitting parameters during the simulations. A nonlinear regression procedure, based on the least-squares method, has been used to find the values for τc and Sm that lead to the best fit of the experimental 〈Rg〉/〈Rg〉0 data shown in Figure 6a. The comparison between model results and experimental data is shown in Figure 10. It is seen that the agreement between model results and experimental data is very satisfactory for all the investigated temperatures. The obtained values for τc and Sm at different temperatures are reported in Table 1, where S0 ) 8πa20 is the total surface area of two particles at t ) 0. As expected, both quantities decrease as T increases, indicating that both the coalescence extent and the coalescence rate increase as the restructuring temperature increases. Moreover, when the τc values are plotted against 1/T in a semilog plane, as shown in Figure 11 (symbols), they can be represented well by a straight line. This confirms that the dependence of τc on T follows the typical Arrhenius relationship, as given by eq 7, and is best given by the following expression:
τc ) 4.75 × 10-15 exp(1.4 × 104/T)
(18)
The obtained value of the constant A ()1.4 × 104) in eq 7 is on the same order of magnitude as reported in the literature for the
Figure 12. Surface area ratio Sm/S0 as a function of temperature T. The open circles (O) are experimental data; the filled circle (B) is the extrapolation to T > Tg,shell. The curve represents the best fitting of the data.
Figure 13. Comparison of the model results (s) and experimental data of the fractal dimension Df as a function of time for various restructuring temperature values: 333 K (O), 343 K (0), 353 K (4), and 363 K (]).
coalescence of oxide aerosols.30,31 Figure 12 shows the Sm/S0 values reported in Table 1 as a function of the restructuring temperature. In the same figure, the two Sm/S0 values at T ) 313 and 323 K, which are equal to 1 because no restructuring is observed, are also shown. Moreover, since the Tg value of the shell material (PS) is around 380 K, we may assume that complete coalescence of the present core-shell particles occurs at T g 400 K, and the corresponding Sm/S0 value reaches a minimum equal to 0.794, which is given by the filled circle in Figure 12. In conclusion, it appears that the data in Figure 12 indicate that restructuring tends to a limiting Sm/S0 value in time that decreases continuously with temperature until the lower boundary provided by complete coalescence is reached.
Thermal Restructuring of Fractal Clusters
We can now apply the time evolution of 〈Rg〉, calculated through the model, to predict the time evolution of the fractal dimension Df, using eq 16, where k is given by eq 17. The initial values of 〈Rg〉0 ) 2100 nm and Df,0 ) 1.87 are taken directly from the experimental results. The predicted results (curves) are compared to the experimental ones (symbols) in Figure 13, and the agreement is rather satisfactory.
4. Concluding Remarks The thermal restructuring process of fractal styrene-acrylate copolymer clusters dispersed in water has been investigated experimentally in the temperature range between T ) 313 and 363 K. The clusters were generated under diffusion-limited cluster aggregation (DLCA) conditions. The particles constituting the clusters are of strawberry-like core-shell structure. The core is made of polyacrylate with glass transition temperature Tg < 253 K, and the shell of polystyrene with Tg > 373 K. Thus, in the given temperature range, the core is very soft, while the shell is rigid. The latter was grafted on the core polymer chains. Due to the incomplete coverage of the core, the rather soft core may “flow out” through the open areas of the shell, leading to coalescence with the neighboring particles. Two sets of thermal restructuring experiments have been carried out: (1) restructuring of growing clusters during aggregation and (2) restructuring of preformed clusters in the absence of aggregation. In the first set of experiments, several DLCA runs have been conducted at different temperatures. It is found that, for T e 323 K, the aggregation rate increases as T increases, but the fractal dimension Df is independent of T and equal to 1.87, typical of DLCA clusters. This indicates that no restructuring occurs in this temperature range, and the increase of the aggregation rate with T is only due to changes in the system physicochemical properties (e.g., viscosity), which affect the aggregation rate constant under DLCA conditions. For T > 323 K, however, as T increases, the aggregation rate decreases, while the Df value increases. This clearly indicates the occurrence of restructuring of the clusters. Since in all cases the average radius of gyration
Langmuir, Vol. 23, No. 10, 2007 5721
〈Rg〉 still grows with time, it is clear that aggregation and restructuring occur simultaneously. Thus, the cluster structure results from the balancing between aggregation and restructuring. The former gives rather open while the latter leads to compact structures. For the second set of restructuring experiments, i.e., in the absence of aggregation, the 〈Rg〉 value decreases monotonically with time, while the Df value increases. At sufficiently long times, both the 〈Rg〉 and Df values reach a plateau value. However, at a given temperature, the plateau value of Df is significantly larger in the absence of aggregation than in the presence of aggregation. This confirms that, in the first set of experiments, the steady-state structure of the clusters results from a balance between aggregation and restructuring. Instead, for the second set of experiments, the steady-state structure represents the limiting state of the restructuring process. For the given temperature range, although the measured limiting Df value increases with T, all the Df values are smaller than 3. This indicates that the extent of the restructuring (coalescence), although increasing with T, is never complete. This arises because of the presence of the grafted rigid shell, which constrains the coalescence of the soft core. A simple model, based on coalescence theory, has been developed to interpret the restructuring kinetics of the clusters in the absence of aggregation. It accounts for incomplete coalescence and its dependence on temperature, by introducing a minimum surface area that depends on the temperature. It is found that the proposed model can represent well the observed restructuring kinetics, and the computed time evolutions of the average radius of gyration and the fractal dimension are in good agreement with the ones measured experimentally. Acknowledgment. Financial support of BASF AG (Ludwigshafen, Germany) and the Swiss National Science Foundation (grant no. 200020-113805/1) are gratefully acknowledged. LA063254S
