Controlling oleogel crystallization using ultrasonic standing waves

Oleogels are lipid-based soft materials composed of large fractions of oil (> 85%) developed as saturated and hydrogenated fat substitutes to reduce cardiovascular diseases caused by obesity. Promising oleogels are unstable during storage, and to improve their stability careful control of the crystalline network is necessary. However, this is unattainable with state-of-the-art technologies. We employ ultrasonic standing wave (USSW) fields to modify oleogel structure. During crystallization, the growing crystals move towards the US-SW nodal planes. Homogeneous, dense bands of microcrystals form independently of oleogelator type, concentration, and cooling rate. The thickness of these bands is proportional to the USSW wavelength. These new structures act as physical barriers in reducing the migration kinetics of a liposoluble colorant compared to statically crystallized oleogels. These results may extend beyond oleogels to potentially be used wherever careful control of the crystallization process and final structure of a system is needed, such as in the cosmetics, pharmaceutical, chemical, and food industries.

. (a) Finite element simulation of the ultrasonic standing field in the 2 MHz experimental chamber (top and bottom gray parts are the piezoceramic elements). (b) Schlieren imaging of the USSW field in the 2 MHz experimental chamber, with oil used as the medium. (c) Frontal view (XZ plane) of the particle movement simulation for 10,000 spherical particles of 100 µm diameter made of monostearin after 10 s in the 2 MHz USSW field. Boundary effects are visible at the lateral sides of the simulated experimental chamber. (d) Corresponding lateral view (YZ plane) of (c). (e) Photograph of 30 µm-175 µm long carbon rods dispersed in rapeseed oil and subjected to a 2 MHz USSW field. Scale bars represent 1 mm in the vertical direction. The arrow indicates the same pressure node across all images. (e) White intensity recorded in the Z direction (vertically) from a photograph as a function of the distance from the top transducer for the control sample (0 MHz) and the sonicated samples (1 MHz, 2 MHz, and 4 MHz). Curves were shifted vertically for clarity. The arrows indicate the presence of a crystalline band. (f) Thickness of a crystalline band measured on 15 points in the X direction (horizontally) from a photograph of the sonicated samples (1 MHz, 2 MHz, and 4 MHz). (g) Average crystalline band thickness and normalized number of crystalline bands per mm for the sonicated samples (1 MHz, 2 MHz, and 4 MHz) analyzed from photographs. Data is expressed as mean ± standard deviation based on n = 2 experimental replicates × 8-12 repeated measurements for the band thickness and n = 2 experimental replicates for the crystalline band frequency. Values with different uppercase or lowercase letters are statistically different (p < 0.05). In (a), (b), (c), and (d), oleogels contain 5% monoglyceride in rapeseed oil and are crystallized at 10 °C/min. The top and bottom dark areas are the piezoceramic transducers, and the scale bars are 1 mm (both vertical and horizonal directions). Figure S3. Analysis of simulation model. Theoretical estimated characteristic time for the rotation (orientation, trot) and translation (ttra) of disk-shaped monostearin crystals with a diameter of 1 µm, 10 µm, 50 µm, and 100 µm and an aspect ratio (d/h) equal to or greater than 10 in (a) a 2 MHz (2012 kHz) US-SW field and (b) a 4 MHz (4120 kHz) US-SW field. The highlighted area corresponds to the estimated pressure amplitude based on the time needed during experiments to form bands, taking into consideration platelets that are 50 µm-100 µm in size. Figure S4. Experimental data. Diffraction patterns expressed as intensity in the Qxy-Qz plane of (a) oleogel statically crystallized, control sample (0 MHz); (b) oleogel crystallized in a 1 MHz US-SW field; (c) oleogel crystallized in a 2 MHz US-SW field; and (d) oleogel crystallized in a 4 MHz US-SW field. All samples contain 5% monoglyceride in rapeseed oil and are crystallized at 1 °C/min. Q represents the reciprocal lattice vector; its modulus |Q| = 2π/d = (4π/λ)⸱sin(θ) is the reciprocal lattice spacing, where d is the lattice spacing, λ is the X-ray wavelength, and 2θ is the Bragg scattering angle. Figure S5. Experimental data. (a) X-ray intensity recorded on the Debye-Scherrer ring in the smallangle region corresponding to the 001 reflection of the monoglyceride lamellar structure (d001) as a function of distance during vertical scans for oleogel statically crystallized, control sample (0 MHz), and oleogels crystallized in a 1 MHz US-SW field and a 2 MHz US-SW field. Arrows indicate nodal (upward) and anti-nodal (downward) planes. (b) X-ray intensity recorded on the Debye-Scherrer ring in the small-angle region corresponding to the 001 reflection of the monoglyceride lamellar structure (d001) as a function of distance during horizontal scans on the crystalline bands showing the maximum intensity for (a) oleogels crystallized in a 1 MHz US-SW field and a 2 MHz US-SW field. All samples contain 5% monoglyceride in rapeseed oil and are crystallized at 1 °C/min.

Mathematical derivation of the dynamic behavior of crystalline platelets in oil subjected to a USSW field
The mathematic modelling of small particles immersed in an ultrasonic field is relatively straightforward. The particle behavior is defined by the acoustophoretic force and the drag force on the particle. We assume that the particles are much smaller than the wavelength of the ultrasonic standing wave (US-SW). By considering one crystal at a time and by ignoring boundary effects, we reduce the problem to one dimension, . By limiting ourselves to particles that are disk-shaped, the orientation of the crystal can be described with a single angle , which describes the rotational offset from the equilibrium orientation. The model considers neither the acoustic streaming nor the coupled flow of the fluid-crystal two-phase system. These phenomena require FEM modeling and are outside the scope of this analysis.
The equations of motion of the system are: 2 2 = drag + a and 2 2 = drag + a , with the first being for transport and the second for orientation. Due to their small size, the crystals exhibit negligible inertia compared to the viscous forces. Consequently, the equations reduce to: = sin 2 and = a sin 2 .
Here, and are the drag coefficient for transport and rotation, respectively. We write the orientation and position dependency on the acoustic torque and force. All coefficients have many dependent variables-crystal size, crystal shape, and standing wave pressure. The two equations are coupled because the acoustic force depends on the orientation and the acoustic torque and on the position in the wave. However, because these dependencies are weak and because the two phenomena occur at different time scales, we treat the two equations as separate. This way, we arrive at an effective approximation.
The detailed form of the acoustic force and torque on immersed ellipsoids is given by Silva and Drinkwater 1 and Fan, Mei, Yang and Chen 2 . Here, is the crystal volume, is the density of the medium, is the acoustic pressure where the particle is, is the density of the crystal, is the acoustic wavenumber, and is the speed of sound (phase velocity) in the medium. Factors 1 and 3 are values of added mass that describe the increase in the effective mass due to the inertia of the surrounding medium. Due to the assumption of a large aspect ratio, 1 ≫ 3 .
One sees that translational drag is proportional to the first power of the crystal diameter, whereas rotational drag is proportional to the third power. One also sees the different dependence on acoustic pressure for translation and rotation.
Once all coefficients are known, the differential equation can be solved. We focus on the rotational equation, as both equations feature a similar dependency. In a case where and stay constant over time, the equation has an analytical solution, ( ) = cot −1 ( −2 / ). This assumption holds when the rotation happens faster than the crystal growth, so the model only works for the shortest rotation times. If we assume a uniform distribution for the initial angle, the median time for a crystal to orientate to within 5 degrees from equilibrium is: rot = 2 ln ( tan 45 tan 5 ) ≈ 1.2 ⋅ ⁄ .
Because the translational equation has the same form, it has similar solution.
The model could be further developed by considering both equations simultaneously and by considering the crystal growth.

Symbols
acoustic pressure density of oil 910 g/l p density of crystal 1030 g/l speed of sound in oil viscosity crystal aspect ratio crystal volume coordinate along the standing wave crystal orientation, measured as deflection from equilibrium wave number