J. Phys. Chem. B 2008, 112, 12635–12643
12635
Melting Behavior of Shear-Induced Crystals in Dense Emulsions as Investigated by Time-Resolved Light Scattering N. Freiberger, M. Medebach, and O. Glatter* Institute of Chemistry, UniVersity of Graz, Heinrichstrasse 28, 8010 Graz, Austria ReceiVed: May 7, 2008; ReVised Manuscript ReceiVed: June 20, 2008
The crystal growth of dense and almost monodisperse colloids has been investigated during recent years, but less is known about the melting behavior. The current study thus focuses on this topic. Monodisperse hard spheres were found to crystallize for certain concentrations (49-58 vol %), after sufficiently long times. The characteristics of the crystal growth change when the colloidal particles are polydisperse. Finally, when the size distribution function of the particles is broad enough, the crystallization no longer took place. Dense oil-in-water emulsions with polydispersities of around 10% were successfully produced, and in a first approximation, these emulsions behaved like hard spheres. The polydispersity of the emulsions was sufficiently high to avoid crystallization in equilibrium but low enough to induce a disorder-to-order transition under shear. The formed crystals started to melt once the shear was discontinued. The melting behavior of these “oil droplet crystals” was investigated by means of time-resolved static light scattering experiments, and it was found that crystallization could be induced in a concentration regime between 46 and approximately 74 vol %. The melting behavior of these crystals depended strongly on the concentration. The typical melting times ranged from a few seconds to several hours or days when the concentration was increased. It was speculated that this phenomenon could be explained by the strong dependence of the mobility of the oil droplets on the volume fraction, as verified by dynamic light scattering experiments on oil-in-water emulsions in a similar concentration regime. Introduction The current study reports on dense, Triton X100-stabilized oil-in-water emulsions with a polydispersity of around 10%. The low water solubility of the used silicone oil guaranteed a stability of the emulsions for at least several weeks. Under shear, a disorder-to-order transition could be induced in the emulsions, as has also been found by others.1 As soon as the shear was discontinued, the crystals started to melt. We found that the typical melting times depended strongly on the oil concentration, and the present study describes investigations of this melting behavior by means of time-resolved static light scattering experiments. A crystalline structure could be easily identified by the presence of Bragg peaks in reciprocal space, and the degree of crystallinity could be estimated by determining the area under these peaks or alternatively the peak heights. Interpreting melting as a kind of reorganization process of the system suggests that the dynamics or mobility of the oil droplets has to be linked to typical time scales of the melting process. Therefore, we performed dynamic light scattering (DLS) experiments on the oil-in-water emulsions in a comparable concentration regime. In agreement with similar DLS studies documented in literature,2,3 we found a strong dependence of the typical decay times with respect to the oil concentration, thus supporting the mentioned hypothesis. In a first approximation, oil-in-water emulsions behave like hard sphere dispersions;4 a behavior that is well-documented in literature.3,5,6 The phase diagram of monodisperse hard spheres5,6 is one-dimensional, with the only parameter being the volume fraction. For concentrations up to 49%, a fluid isotropic phase is found, and at higher concentrations (up to * Corresponding author. E-mail:
[email protected].
56%), there is a coexistence of a fluid isotropic phase and a crystalline phase. From a thermodynamic point of view, crystals should be found between 56 and 74 vol %, since the entropy in the crystalline state is larger as opposed to in the isotropic state. The concentration value of 74 vol % cannot be exceeded, due to there simply being no space left for further spheres. As was verified by experiments and predicted by the mode coupling theory, colloidal hard spheres fail to crystallize for volume fractions larger than 58% and form a glass because of the socalled caging effect.5,7-10 The mobility of the particles is frozen for such high concentrations, and the crystalline equilibrium state can thus no longer be reached. In recent years the crystallization kinetics of dense and monodisperse hard sphere colloids has been investigated, and the growth rates of colloidal crystals, in the absence of shear, are well understood.11-17 On the other hand, it has been verified that shear can induce a disorder-to-order transition,18,19 in the case of monodisperse hard spheres, for concentrations below 49 vol % as well as above 58 vol %.20-22 Interestingly, in refs 23 and 24 the crystallization of systems with concentrations above 74% is discussed. The crystallization kinetics changes as soon as the particles become polydisperse. It was shown in ref 25 that the crystallization is delayed, while the nucleation rate density is enhanced, when the polydispersity is increased from 4.8% to 5.8%. Finally, for a sufficiently large polydispersity, crystallization no longer occurs. We never had any evidence of the formation of crystals in the oil-in-water emulsions under the present investigation, even after waiting for several weeks. Consequently, a polydispersity of 10% seems to be high enough to prevent crystallization.
10.1021/jp804027z CCC: $40.75 2008 American Chemical Society Published on Web 09/13/2008
12636 J. Phys. Chem. B, Vol. 112, No. 40, 2008 The crystal growth of monodisperse and slightly polydisperse colloidal systems both under and in the absence of shear seems to be understood, whereas less attention has been paid to the mechanism of melting.26 The current study thus focuses on the melting of crystalline oil-in-water emulsions driven by their polydispersity. Materials, Methods, Preparation, and Characterization Double-distilled water, silicone oil DC 200 from Fluka Chemie AG, Buchs, Switzerland, Triton X100 from Sigma Chemicals Co., St. Louis, MO, and glycerol from VWR, Vienna, Austria were used for the preparation of dense oil-in-water emulsions with a low polydispersity by applying the methods described in refs 1 and 27-30. All materials were used without further purification. In a first step, the surfactant Triton X100 was dissolved in double-distilled water. To ensure that any oil added to this solution would be covered by a surfactant film, a relatively high concentration of the surfactant, i.e., 33 wt %, was chosen. The silicone oil was then added dropwise to this mixture under permanent gentle stirring up to a concentration of 50 wt % relative to the surfactant solution. The so-obtained raw emulsion consisted of very large (approximately 10 µm) and polydisperse oil droplets in a continuous water matrix. In order to decrease the mean droplet diameter and the degree of polydispersity, the premixed emulsion was sheared with a laboratory-built shearing device of Couette type.4,31 In the shear gap, the droplets of the raw emulsion are stretched to cylinders, and at a certain cylinder diameter the cylinders break into droplets of similar size. In literature, this phenomenon is referred to as the so-called Rayleigh instability.27 The mentioned shearing device consists of a fixed outer hollow cylinder, and a rotating inner cylinder. The inner cylinder can be replaced by others with varying diameters. In this way, the size of the shear gap and thus also the shear rate can be tuned. As a second possibility, the shear rate can be tuned by varying the rotation speed of the inner cylinder. The raw emulsion was pressed through the shear gap via a piston.4,31 A shear rate of 9409 s-1 was chosen, which led to a mean oil droplet diameter of 1.5 µm. The sheared emulsion was constituted of oil droplets with a polydispersity of around 15% sa value still too high for the following crystallization experiments. Thus, in the next preparation steps, the polydispersity was further decreased by means of fractionated crystallization1,27-30 as described in the following. As already mentioned, there was a relatively large amount of surfactant micelles present in the emulsion at this stage of the preparation, and these micelles led to an attractive interaction potential between the oil droplets, a so-called depletion interaction,32-34 and finally to an aggregation and creaming of the oil droplets. Among other things, the attractive potential depends on the size of the oil droplets and on the concentration of the micelles. For a very high amount of micelles, nearly all of the oil droplets were found to cream, except for the smallest ones. Thus, the smallest droplets could be separated from the others. For a very low amount of micelles, only the largest droplets were found to aggregate and cream and could therefore also be easily separated. In other words, by tuning the surfactant concentration, the size distribution of the emulsion droplets could be narrowed step by step,28 and this technique permitted a polydispersity of around 10% to be reached. In a next preparation step, the amount of micelles was drastically reduced in order to change the interaction potential from attractive to hard sphere-like. These emulsions were stable
Freiberger et al. over several weeks. Finally, a highly concentrated (approximately 75 vol %) stock emulsion was obtained by centrifugation. Once the emulsions were prepared, the concentration, the type of the interaction potential, the mean value of the particle size, and the size distribution function of the droplets had to be determined. Concentration. A well-defined amount of the mentioned stock emulsion was taken and stored on a glassy platter at 40 °C for 24 h. This procedure ensured a complete evaporation of the solvent (water), whereas the oil remained on the platter for a subsequent weight measurement. The oil concentration of the stock emulsion could thus be easily calculated from the measured mass values. The less concentrated emulsions, used for the shearing experiments (see below), were created by a well-controlled dilution of the stock emulsion. Free Surfactant and Depletion Interaction. As previously described, the preparation of the emulsions was started with a surfactant concentration of 33 wt % relative to the solvent, in order to ensure that all the added oil would be covered by surfactant molecules. During the applied procedure of the fractionated crystallization, the concentration of the micelles was reduced stepwise, and consequently neither the concentration of the remaining micelles nor the corresponding attractive interaction potential (depletion interaction) was known at this stage. The best way to end up with a pure hard sphere interaction potential would be to remove all of the micelles from the system and then, if desired, add the surfactant again, but now in a controlled manner. In addition, it is known from literature35,36 that micelles can “transport” the oil from one droplet to another (via the solvent), thereby destabilizing the emulsions in the process. At a first view, the removal of the micelles can be easily carried out by a sufficient dilution of the emulsion. On the other hand, one should not go below the critical micellar concentration (cmc ) 0.24 mM ) 0.015 wt % (molar mass 625 g/mol) in the case of Triton X100), since the oil droplets could then become destabilized. The ideal value within these two limits, i.e., a low enough amount of surfactant in order for the influence of the depletion forces to be negligible and a high enough amount of surfactant to ensure that the stability of the particles is preserved, is difficult to determine. The density of the surfactant molecules in the shell of the oil droplets can possibly vary, e.g., by absorbing monomers from the solvent. We performed static light scattering experiments on the emulsions with an oil concentration of 50 vol % (droplet diameter of around 1.5 µm) and varied the surfactant concentration. To avoid multiple scattering, the sample was index-matched with glycerol and measured in a flat cell. By neglecting prefactors, the scattering intensity can be approximated as the product of the form factor P(q) (intraparticle contribution) and the structure factor S(q) (interparticle contribution).37 In the limit q f 0, the structure factor S(q) depends very sensitively on the volume fraction, the interaction potential (isothermal compressibility), and the polydispersity.38 In our case the polydispersity of the system was known from measurements of diluted emulsions (see below). In a system of hard spheres, the values of S(0) will be increased when the volume fraction is decreased. For a fixed volume fraction and polydispersity, S(0) will also be increased when the interaction potential is varied from hard sphere-like to attractive-like (see Figure 1). In other words, the interpretation of the scattering data on the basis of a hard sphere structure factor model with the volume fraction as the parameter would provide the wrong (undersized) value for the volume fraction when an attractive interaction potential is present. This
Melting Behavior of Shear-Induced Crystals
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Figure 1. Results from static light scattering on dense (50 vol %) emulsions with various surfactant concentrations. Left: scattering patterns of oil-in-water emulsions with different amounts of added surfactant. The highest concentration of surfactant was ∼0.5 wt % (dash-dotted), followed by ∼0.3 wt % (dashed), ∼0.1 wt % (dotted), and finally ∼0.075 wt % (solid). Right: scattering pattern of an emulsion with 50 vol % oil and a surfactant concentration of 0.075 wt % (open circles), in comparison with its GIFT-fit function (solid line). The GIFT evaluation of the scattering data was based on a hard sphere structure factor model resulting in the correct volume fraction of 49.8%.
fact renders it possible to identify the type of interaction via scattering methods. In the present case, the volume fraction (i.e., 50 vol %) and the degree of polydispersity were known but not the type of interaction, as a function of the surfactant concentration. The surfactant concentration was decreased stepwise down to a value of 0.075 wt % relative to the solvent, and the obtained scattering curve was evaluated with a pure hard sphere structure factor model (using the GIFT method)39-43 taking the influence of polydispersity into account. The obtained value of 49.8 vol % for the volume fraction was in good agreement with the known value of 50 vol %. We concluded that a possible attractive interaction potential (depletion interaction) is negligible for a surfactant concentration of 0.075 wt % but that this is not the case for higher concentrations. As mentioned, a too low surfactant concentration can lead to the droplets being destroyed. According to our experience, the surfactant concentration can be decreased down to at least 0.05 wt % (relative to the solvent) without jeopardizing the particles’ survival or affecting the size distribution. The droplet size distribution of the emulsion was determined via static light scattering. For this purpose, the samples were sufficiently diluted to exclude any interaction effects. Typically, the size of the emulsion droplets was around 1 µm or larger, and the refractive index of the used silicone oil was 1.405, which is significantly larger than that of water (1.33). The samples thus had to be index-matched with glycerol, and measured in a flat cell with a thickness of 0.43 mm, in order to avoid multiple scattering. The concentration of glycerol was chosen to 48 wt % to obtain a refractive index of 1.395 for the solvent. In general, for the evaluation of micrometer-sized particles with a refractive index that strongly deviates from that of the solvent, the so-called Lorenz-Mie (LM) theory has to be applied instead of the Rayleigh-Debye-Gans theory (RDG). To quantify the limit between these two approaches, the following criterion44 can be used:
2
( )
2πR nP -1 ,1 λ nS
(1)
In essence, if the circumference of the particle2πR is smaller than the wavelength λ of the used laser light, and/or if the real part of the refractive index of the particles nP is relatively close
Figure 2. Scattering patterns (symbols) for diluted oil-in-water emulsions with different polydispersity, as well as the corresponding fit functions (solid lines), for emulsion A (open circles) and emulsion B (triangles).
Figure 3. Normalized intensity size distribution functions (NISDF) for emulsion A (left) and emulsion B (right) extracted from the scattering curves from Figure 2 according to the LM theory and the RDG theory.
to that of the solvent nS, the RDG theory remains valid. In the present case (R ≈ 750 nm, λ ) 632.8 nm), the left-hand side of eq 1 yielded a typical value of 0.1 which is clearly below 1. In a first step, the scattering data (see Figure 2) were evaluated via the GIFT program which is based on the RDG theory. For comparison, the size distribution was again determined via the ORT program45 based on the LM theory (see Figure 3), to check for possible deviations of the RDG theory in this special case of nearly index-matched emulsions. Figure 2 displays the scattering patterns of two diluted emulsions, where emulsion A is much less polydisperse than emulsion B. This lower polydispersity of emulsion A is indicated by the pronounced minima of its scattering curve. For both emulsions, normalized intensity size distribution functions (NISDFs) were extracted, according to the LM theory as well as to the RDG theory (see Figure 3). The width of the NISDF was determined to 10% in the case of emulsion A and to 18% in the case of emulsion B. The deviations between the LM and RDG approaches were within the experimental error, thus confirming that the refractive index of the solvent nS had been chosen (index match) close enough to that of the particles nP. These characterized emulsions were subsequently used for time-resolved static light scattering experiments under shear, as discussed later on. Time-Resolved Light Scattering under Shear Figure 4 illustrates the experimental setup for the timeresolved light scattering experiments. The sample is sheared between two glass plates and illuminated by a laser. The scattered light is collected on a milky screen, converting the
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Figure 4. Light scattering under shear. Left: a sketch of the setup for the shearing experiment. Right: the shearing cell in more detail; (A) driven cylinder, (B) cell holder, (C) distance ring, (D) fixed part of the shearing cell screwed to the cell holder, (E) glass plates, (F) sample, and (G) rotating part of the shearing cell. To avoid a possible evaporation of the solvent, the distance ring of the shearing cell can be moistened with water, to have a humid atmosphere.
oriented scattered light into diffuse light, which can then be projected via a mirror and a lens on the chip of a CCD camera (CM9-1E/KAF-0261E, from Finger Lakes Instrumentation, EHD, Damme, Germany). A shielding (not shown in the Figure 4) guarantees that the light collected on the milky screen cannot directly enter the optics of the camera. The shearing conditions can be adjusted in a variety of ways. As can be seen from Figure 4, right, the width of the shear gap can be varied by an adequate distance ring. The position of the primary beam relative to the shearing cell can be changed in order to observe areas with different shearing rates and strains, for a constant gap width and rotation speed. The flexible part of the shearing cell, pivoted with two ball-bearings, can be driven either in continuous rotation mode or in oscillatory mode, using a frequency between 0 and 2 Hz. For the continuous rotation of the cell, the motion of the motor is simply transferred by a driving belt. The stability of the motor depends on its rotation speed; at high speeds the motor is more stable and vice versa. For this reason, the motor was adapted with a planetary gear box with a transmission ratio of 1:23 to be able to work with high rotation frequencies on the motor side. For the oscillatory mode, an eccentric shaft in combination with a symmetric system of levers was employed instead of the belt. The eccentric shaft and the levers rendered it possible for the motor to again be driven continuously and thus in a very precise and stable fashion. The symmetrically mounted system of levers made sure that all possible forces perpendicular to the cell axis were balanced. Thus, a possible tilt of the driven cylinder of the shearing cell could be minimized. One picture was recorded with the CCD camera approximately every second, and the chip of this camera consists of a square array of 512 × 512 pixels, with a size of 20 µm. To cover a wide enough dynamic range, a camera with a 16 bit integer data format was used.
fail to crystallize for any concentration. Under shear, it has been observed20-22 that a disorder-to-order transition can be induced for nearly monodisperse hard spheres at concentrations below 49.4 vol % as well as above 58 vol %. In the present study, oil-in-water emulsions with a polydispersity of either 10% or 18% and with concentrations between 46 and 70 vol % were investigated. The mean diameter was determined via static light scattering to 1.5 µm (see the Materials, Methods, Preparation, and Characterization section). Without shear, none of the emulsions were found to crystallize. Even under shear, the emulsion with the high polydispersity failed to crystallize, whereas a disorder-to-order transition could be induced in the emulsions with the lower polydispersity. The necessary shearing conditions depended on the concentration of the emulsions. Basically, oscillatory shear had to be applied below the glass transition, with a shear strain of 1-3 and a frequency of 1-2 Hz. Larger frequencies would possibly also induce crystallization in the emulsions; however, the highest available frequency of the described setup was here limited to 2 Hz. The width of the shear gap was chosen to be 0.07 mm, and similar shearing conditions were found for a lager gap of around 0.12 mm. For higher concentrations, above the glass transition, nonoscillatory shear had to be applied with angular frequencies between 2π and 4π rad/s (corresponding to a shearing rate γ˙ of 1600 to 3200 1/s) in order to create crystals. Nevertheless, it should not be excluded that also frequencies above 4π rad/s could provide a disorder-to-order transition. All the discussed shearing conditions were found experimentally. This means that, for an emulsion with a certain concentration, the shearing parameters were successively changed, until the optimal conditions were found. Alternatively, if such an experimental approach would fail, one could estimate the necessary shear rate according to the following equation:
Results As already discussed, monodisperse hard spheres crystallize in a concentration regime between 49.4 and 58 vol % without shear.5,6 These two limits are influenced by the polydispersity of the particlessfor a very large polydispersity, the system will
σR )
γ˙ ηER3 kBT
(2)
Here, σR refers to the so-called Pe´clet number, which depends on the shear rate γ˙ , the shear viscosity ηE, the particle radius
Melting Behavior of Shear-Induced Crystals
Figure 5. Scattering patterns of an oil-in-water emulsion with a concentration of 50 vol % (A) under shear, (B) directly after the shear was discontinued, (C) a few seconds later, and (D) after a few hundred seconds. The line in panel A indicates the orientation of the velocity vector of the shearing field. The “real” size of the pictures, respectively, the size of the used screen, is equal to 25 × 25 cm2.
R, the Boltzmann constant kB, and the temperature T. This Pe´clet number has to be larger than 1, in order to ensure that the influence of the shear exceeds that of the Brownian motion of the particles.20,46 But once again, an experimental input information is required, namely, the knowledge of the shear viscosity ηE as a function of the shear rate γ˙ . At first view, one could expect that either the whole emulsion inside the shearing cell would crystallize under the mentioned shearing conditions or that at least radially symmetric crystalline regions could be found. When the continuous shear was applied to the emulsions, a nice radially symmetric crystalline ring with a width of a few millimeters did indeed appear. On the other hand, when an oscillatory shear was applied to the emulsions, crystalline islands rather than radially symmetric regions were found in most of the cases. These crystalline islands, as well as the crystalline rings, can also be easily detected with a normal white light source, since the disordered emulsion appears turbid or milky, whereas the crystalline regions are quite transparent. Within the first 30 min of applying the shear, the crystalline areas grew; thus, the shearing history may be important. Figure 5 displays the two-dimensional scattering patterns of an oil-in-water emulsion with a concentration of 50 vol % under shear as well as some time after the shear. The grayscales correspond to different intensities; black represents a low intensity and white a high intensity. The black circle in the middle of the images originated from the beam stop used for the experiment. As can be seen from Figure 5A-C, four pronounced Bragg peaks and two tiny ones appeared under shear as clear indicators of the presence of orientated crystals in the sample. The tiny peaks were consistently orientated perpendicularly to the shearing direction, respectively, the velocity vector (the line in Figure 5A). This effect can be interpreted as a deformation of the crystal in the shear field. It has been theoretically47 and experimentally48 demonstrated that hexagonal layers are generated under shear and that these layers glide on each other along a zigzag path, resulting in the disappearance of two of the six Bragg peaks in reciprocal space. When the shear is stopped, the Bragg peaks become weaker with time, and the crystal melts. Finally, after sufficiently long periods of time, the peaks completely disappear, and one ends up with a radially symmetric ring in reciprocal space. Such a ring is always found in dense but isotropic colloidal systems. It originates from the interaction of the particles or droplets and is therefore called a structure factor peak or correlation ring. Of course, before the shear was applied to the emulsion, a similar ring was
J. Phys. Chem. B, Vol. 112, No. 40, 2008 12639
Figure 6. Scattering patterns of an oil-in-water emulsion with a concentration of 65 vol % (A) under shear, (B) directly right after the shear was stopped, (C) after 8000 s, and (D) after 45 000 s. The line in panel A indicates the orientation of the velocity vector of the shearing field. Note that after an extremely long time period of 45 000 s, the orientation of the crystals was somewhat changed (image D). We attribute this change to a slight evaporation of the solvent. Even at a measurement temperature of 18 °C in a humid atmosphere, a small amount of the solvent evaporated after 45 000 s in this example. The “real” size of the pictures, respectively, the size of the used screen, is equal to 25 × 25 cm2.
observed in reciprocal space, a typical sign of an isotropic system of interacting spheres. Partially, this ring was also present in the images A-C of Figure 5, indicating that the sample did not fully crystallize in this example. Interestingly, in the intermediate stages during the melting process, the correlation ring was deformed to a hexagon. Randomly orientated crystals would also provide a ring (Debye-Scherrer ring) rather than Bragg peaks. Such a ring can easily be distinguished from that of the fluid phase, because, as we observed (data not shown), the width of the crystalline ring is much smaller and the height much larger as compared to the correlation peak. For the sake of comparison, Figure 6 displays the scattering patterns of an oil-in-water emulsion with a concentration of 65 vol % during and after the applied shear. Under shear, orientated crystals were again found, indicated by four pronounced Bragg peaks and two tiny ones perpendicularly orientated to the shearing directions. In this case the crystals were extremely stable; even after 45 000 s the sample remained highly ordered. Directly after the shear was stopped, the two tiny peaks started to grow slightly, and after a sufficiently long period of time, all six Bragg peaks were of equivalent quality. This means that the shear deformation of the crystal is obviously able to relax in the course of time. Interestingly, the scattering intensity close to the beam stop under shear (Figure 6A), directly after the shear was stopped (Figure 6B), and after waiting for 8000 s (Figure 6C), was much higher than in the situation plotted in Figure 6D (45 000 s after the shear was discontinued). To clarify this observation, the scattering regime between the scattering angles 0 and ∼θBragg, the position of the first Bragg peak, is shortly discussed. A small crystal with the overall size x would provide a strong scattering signal between the scattering angles 0 and ∼θBragg/2. In comparison, a large crystal with the size y . x would provide much less scattering intensity for scattering angles below θBragg/2 but much more scattering intensity close to the scattering angle 0. In other words, as soon as the crystal size increases, the scattering intensity in the regime up to θBragg/2 is shifted to smaller scattering angles and possibly hidden by the beam stop. The scattering intensity of the Bragg peaks is influenced by two parameters, namely, the size of the crystal and its
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Figure 7. Melting behavior of emulsions with different concentrations (volume fractions). The different melting stages are characterized by the normalized area under the Bragg peaks.
degree of order. Large and highly ordered crystals provide pronounced Bragg peaks, whereas large crystals with low order provide weak Bragg peaks. Thus, the mentioned observation (see Figure 6D) concerning the disappearance of the scattering intensity close to the beam stop can be interpreted as growth of crystalline domains (a similar effect was also observed elsewhere49). Concurrently, the intensity of the Bragg peaks became lower (see Figure 7), and therefore the degree of order within these crystalline domains had to decrease. In the present case, the structure of the obtained crystals could not be determined. At best, three different orders of diffraction were observed; however, they were superimposed by the correlation ring, the form factor of the oil droplets, and distorted by the shearing field. In literature, possible crystal structures for hard sphere systems under similar shearing conditions have been discussed,50,51 and in these cases, the authors basically found stacks of hexagonally packed layers. In order to quantify the degree of crystallization for the different stages during and after shear, an evaluation procedure was developed to extract the peak height, the peak width, as well as the area under the peak, from the images of the CCD camera (see the Appendix). The results obtained through this evaluation technique are presented in the following. In Figure 7, the melting behavior of oil-in-water emulsions with a polydispersity of around 10% and different concentrations is presented. The results demonstrated that it was possible to induce crystallization under shear already at a concentration of 46%. The corresponding melting behavior is not shown in Figure 7, due to the crystalline order being very weak and the crystals completely disappearing within 1 s. The normalization factors were obtained by calculating the area under the peak of the images taken under shear. The peak in the melting curve for the 61 vol % sample was not reproducible and possibly originated from a slight evaporation of the solvent in this single example. In general, a similar behavior to that of emulsions with a concentration of 65 vol % was found. The typical melting time ranged from a few seconds to a few hours and increased with an increasing concentration. For concentrations below 60 vol %, the melting curves could be fitted via a double exponential
y ) A exp(-t/T1) + B exp(-t/T2) + y0
(3)
whereas the curve corresponding to the sample with a concentration of 65 vol % could be fitted with a single exponential
y ) B exp(-t/T2) + y0
(4)
The melting curves and the corresponding fit functions are presented in Figure 8, and the fitting parameters are summarized in Table 1.
Figure 8. Melting behavior of emulsions for different concentrations: the normalized area under the peak (open circles) in comparison with its fit functions (solid line).
TABLE 1: Fitting Parameters of the Melting Curves concn/% 48.5 51 54 56 65
A
T1/s
B
T2/s
y0
0.60 0.39 0.43 0.35
3.97 6.08 13.29 12.29
0.45 0.50 0.35 0.57 0.70
27.95 77.97 317.04 3012.95 25723.12
0.11 0.12 0.12 -0.04 -0.03
An explanation for the fast melting process during the first seconds, characterized by the parameter T1, could be that the shear induces a kind of flow in the emulsion. If this is true, it would take a few seconds until this stream of the oil droplets stops again. It is possible that the growth of the crystals takes place partially in the bulk (far away from the glass wall) of the emulsion and partially directly on the glass walls. Such “wall crystals” can be more stable as compared to those in the bulk, and thus, the slow melting process characterized by T2, possibly originated from the crystals attached to the wall. In a simple picture, melting can be described as a kind of reorganization process where the relative positions of all the particles change. The potential of the rearrangement of the particles must be related to their mobility. On the other hand, the mobility of the particles is reflected in their dynamical behavior characterized by the corresponding correlation function of a DLS experiment. Therefore, for the sake of comparison, the dynamics of the emulsions were investigated in a similar concentration regime by DLS.52 As can be seen from Figure 9, left, the typical decay time of the correlation function ranged from subseconds to several hours when the oil concentration was increased, agreeing, at least qualitatively, with the melting behavior.
Melting Behavior of Shear-Induced Crystals
Figure 9. Comparison between the dynamical behavior and the melting behavior of emulsions. Left: DLS intensity correlation functions for differentconcentrations.Right:meltingbehaviorfordifferentconcentrations.
The correlation functions presented in Figure 9, left, were measured in a kind of equilibrium. According to our experience, disturbing the emulsion, e.g., by filling it into the cell when preparing for a measurement, can give rise to a so-called aging process.53 In terms of DLS, the dynamical behavior can drastically change during a time period of 14 days. It can be assumed that filling the emulsion in the shear cell and shearing it for 30 min should also essentially change the dynamical behavior. A comparison between the dynamical behavior of emulsions in a quasi-equilibrium state and the melting behavior of a sheared emulsion can therefore only provide phenomenological information. It would thus be interesting to perform a time-resolved DLS experiment during the melting process, e.g., via the multispeckle method.54-58 Summary Oil-in-water emulsions with a polydispersity of around 10% were prepared, and it was demonstrated that the interaction potential could be reasonably well-described by a hard sphere structure factor model. Shearing experiments were performed on these emulsions with a new laboratorybuilt shearing device. For certain shearing conditions and oil concentrations (>46%), the oil droplets of the emulsions arranged themselves into a crystal lattice. The presence of oil droplet crystals was identified by light scattering, and pronounced Bragg peaks could be seen in the reciprocal space. Once the shearing was stopped, the crystals started to melt. Among other things, we think that this melting process is driven by the polydispersity of the oil droplets. Once monodisperse spheres are in a crystalline state, they would not melt (for conc > 49%). The melting behavior of the oil droplet crystals was followed by time-resolved static light scattering, and it was found that the typical melting time ranged from a few seconds to several hours (or even days) depending on the oil concentration. In dense systems, the crystals were much more stable than in their diluted counterparts, and for concentrations below 61%, the melting plot could be separated into two regimes. In the first few seconds the crystals melted very fast, after which the melting process slowed down. Thus, all of the melting curves could be characterized by two typical melting rates. For concentrations above 61%, the crystals became extremely stable, and even after several days, the crystals did not completely disappear. The results from the shearing experiments were compared with data from our DLS measurements performed on emulsions in a quasi-equilibrium state using the recently developed EchoDLS setup.52 We think that the melting behavior of the crystals is linked to the dynamics or mobility of the oil droplets.
J. Phys. Chem. B, Vol. 112, No. 40, 2008 12641 Qualitatively, low concentrated systems (46%) show rapid dynamics and the corresponding melting rate is high. The opposite is true for highly concentrated systems. As a perspective for the future, the dynamical behavior of the oil droplets during the melting could be studied via timeresolved DLS techniques such as the multispeckle method,54-58 in order to gain a deeper understanding of the melting process. The melting behavior of the oil droplet crystals can also be influenced by the viscosity of the oil used, since the elasticity of the oil droplets is obviously influenced by the oil viscosity. One can imagine that the potential of reorganization of easily deformable droplets should differ from that of stiff ones. Thus, it would be interesting to perform shearing experiments on emulsions with different species of oil. Moreover, it would of interest to study the dependence of the strength of an attractive interaction potential on the melting behavior. For the future it would be interesting to study the rheological behavior of concentrated emulsions in the context of shearinduced crystallization.59 Acknowledgment. The present research was supported by the Austrian Science Fund (FWF) under Grant P15698-N03 and by the Marie Curie Research Training Network under Grant MRTN-CT-2003-5047/2. Many thanks are expressed to Markus Pichler for his great job in designing and building the shearing device. Appendix This section deals with the extraction of the radial profile of the Bragg peaks and with the estimation of the degree of crystallinity from the two-dimensional images of the CCD camera. First of all, the data format of the pictures had to be converted from the so-called flexible image transport system (fits) to a text file including an intensity value between 0 and 65 536 for every pixel. This can be done via MATLAB or via a library for FORTRAN or C++. Here, MATLAB was used for this procedure. After the conversion, an adequate correction procedure can be applied to all of the images. The scattering intensity collected on the diffuse screen depends on the distance RD between the scattering volume and the various points of the screen according to 1/RD2. On the other hand, only a very thin ring from the whole image was used for the data evaluation. The main point of interest was not the exact profiles of the Bragg peaks but rather the relative intensities of peaks at different melting stages. In addition, the distance between the illuminated sample and the screen was sufficiently large to reduce the error of the measured intensity to a few percent (comparable to the intrinsic noise of the experiment). For this reason, the 1/RD2 dependence of the scattered light could be neglected for this specific case. Once the images were converted, the radial intensity profiles of the Bragg peaks could be calculated. For this purpose, a segment around the center (vector Rp) of the Bragg peaks was defined (see Figure A1). This segment was bordered by two circles with radii R1 and R2 as well as by two lines with an angle of 60° relative to each other and symmetrically placed around the center of the peaks. Finally, the mean intensity N
Im(r) )
∑
1 I (r) N j)1 j
(a1)
along a circular line, of radius r, centered in the peak, and consisting of N pixels inside the defined segment, was calculated.
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Freiberger et al. References and Notes
Figure A1. Sketch of a scattering pattern with four pronounced Bragg peaks and the definition of the segment for the radial averaging of the scattering intensity.
Figure A2. Example for the radial intensity profile of a Bragg peak, for different melting stages.
The described averaging procedure typically led to radial intensity profiles of the Bragg peaks plotted in Figure A2. The profile of a Bragg peak can include many interesting and difficult physics. For instance, the shape of the Bragg peaks in lamellar phases60-62 is qualitatively completely different from that in hexagonal phases.63-66 To our knowledge, a generally applicable model function for the profile of a Bragg peak of liquid crystals is not documented in literature, especially not for the case of a shear-deformed oil droplet crystal. In this special case, it turned out that most of the extracted intensity profiles could be reasonably wellfitted by the function
y ) a1 exp(-r/a2) + a3
(a2)
This permitted the evaluation of the peak height a1, the peak width a2, and the background level a3 () 8700-10 300 in this example). Alternatively, the correlation ring could be subtracted from all the images. It turned out that this procedure could falsify the result, due to the background level changing during the melting process. The subtraction of the correlation ring could lead to negative values, thus overestimating its level. In a first approximation, the decrease in scattering intensity of the Bragg peaks was accompanied by an increase in scattering intensity of the correlation ring. From this point of view, the background level had to be determined for every single picture of the melting process, as was done in the present study. In addition to the values a1, a2, and a3, the area under the peak was calculated, to characterize the particular degree of crystallinity, and thus the described procedure rendered it possible to follow the melting process in detail.
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