We outline techniques for the control and measurement of the nucleation of crystalline materials. Small angle x-ray scattering/wide angle x-ray scattering x-ray diffraction measurements are presented that demonstrate the impact of low power, continuous, non-cavitational ultrasound on the nucleation and crystallization of a wax—n-eicosane dissolved in a heptane/toluene solvent. A mathematical-physical approach based on the rectification of heat and mass transport by such a low power oscillating pressure field is outlined, and it is suggested that this approach be combined with dissipative particle dynamics computational modeling to develop a predictive method capable of modeling the impact of low power oscillating pressure fields (acoustics and ultrasonics) on a wide range of nucleating systems. Combining the ultrasound pitch and catch speed of sound measurements with low power harmonically oscillating pressure fields to monitor and control nucleation presents the prospect of entirely new industrially significant methods of process control in crystallization. It also offers new insights into nucleation processes in general. However, for the acoustic control technique to be widely applied , further theoretical and modeling work will be necessary since, at present, we are unable to predict the precise effect of low power ultrasound in any given situation.

Topics
Speed of sound, Ultrasonic phenomena, Computational models, Properties of nuclei, Crystallization, Crystallography, Mass diffusivity, Supersaturation, Thermal diffusivity, Fluctuation phenomena

Recently, it has been shown that a low power, non-cavitating, continuous, oscillating pressure field will change the manner of crystal nucleation and growth through a process of rectified heat transfer. The application and method are patented in Povey and Lewtas (2022),1 the theory of rectified heat transfer is described in detail in Povey (2016),2 and the applications of the technique are described in Povey (2017).3 Another approach is described in Haqshenas et al. (2016).4,5

The research questions that we answer in this study are (a) the effect of a harmonically varying low amplitude pressure field (not cavitating) on mass transfer to and from a crystal nucleus and (b) the experimental investigation of the practical effect of low power (sub-cavitational) pressure fluctuations on molecular and mesoscale ordering in both the unsaturated solution or pure melt and the saturated solution/undercooled melt, thereby modifying crystal nucleation and growth. About (a), we present in the next section a theoretical development of an expression for the effect of harmonic pressure fluctuations on the mass transfer between a suspending phase and crystal embryos/nuclei and draw tentative conclusions.

In Povey (2016),2 the generalized view of Threlfall and Coles6 was investigated using ultrasound measurements of the speed of sound in solutions of copper sulfate pentahydrate, in particular the events in the dead zone. In this study, we are interested in regions A, B, and C of Fig. 26 because we hypothesize that low power, non-cavitating oscillating pressure fields are capable of altering the molecular packing in the melt/solution in addition to the dead zone, thereby providing a means of controlling nucleation and polymorph, for example by favoring the 3L packing of triacylglycerols, which is a precursor for the more stable β form of the fat, rather than the 2L packing,7 which favors the less stable β′ form.

A new analysis of the data in Figs. 3 and 4 in Povey2 (Fig. 1) indicates that critical fluctuations in the dead zone can be detected by pulse-echo ultrasound techniques with a pulse width of the order of 5 µs. Each point plotted in Fig. 1 is the result of a single measurement lasting around 5 µs. The scatter in the data is far greater (by an order of magnitude, as can be seen from the scatter in the data obtained in the region of uncontrolled crystal growth, regions D and E6 in Threlfall and Coles for supersaturation >0.15) than the measurement error and indicates that the coming into existence and subsequent disappearance of solid material associated with crystal embryos in the dead zone associated with relative supersaturation between 0.072 and 0.15 can be detected and measured.

FIG. 1.
(Black filled squares) Crystal growth rate plotted against relative supersaturation (increasing supersaturation) for aqueous copper sulfate solution using “instantaneous,” i.e., not averaged over time, measurements of pulse-echo time-of-flight.9 (The red line) Instantaneous data smoothed using a 15 point Savitzky–Golay routine.
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(Black filled squares) Crystal growth rate plotted against relative supersaturation (increasing supersaturation) for aqueous copper sulfate solution using “instantaneous,” i.e., not averaged over time, measurements of pulse-echo time-of-flight.9 (The red line) Instantaneous data smoothed using a 15 point Savitzky–Golay routine.

FIG. 1.
(Black filled squares) Crystal growth rate plotted against relative supersaturation (increasing supersaturation) for aqueous copper sulfate solution using “instantaneous,” i.e., not averaged over time, measurements of pulse-echo time-of-flight.9 (The red line) Instantaneous data smoothed using a 15 point Savitzky–Golay routine.
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(Black filled squares) Crystal growth rate plotted against relative supersaturation (increasing supersaturation) for aqueous copper sulfate solution using “instantaneous,” i.e., not averaged over time, measurements of pulse-echo time-of-flight.9 (The red line) Instantaneous data smoothed using a 15 point Savitzky–Golay routine.

Close modal
FIG. 2.
Chemical potential of solid and solution phases in equilibrium.
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Chemical potential of solid and solution phases in equilibrium.

FIG. 2.
Chemical potential of solid and solution phases in equilibrium.
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Chemical potential of solid and solution phases in equilibrium.

Close modal

Finally, we present x-ray diffraction (XRD) studies of the crystallization of a wax (eicosane) from a solvent (heptane/toluene) in the presence and absence of an insonifying ultrasound field to investigate its effects both on the long-range order via small angle x-ray scattering (SAXS) and simultaneously the effect on molecular packing with wide angle x-ray scattering (WAXS). In comparison with the ultrasound data presented in Fig. 1, it takes around 100 ms to take an entire frame of SAXS/WAXS data, and consequently, we do not expect to see the critical fluctuations in the structure that are evidenced in the ultrasound data. However, we do see long time scale mesoscale effects due to insonification in the unsaturated solution, which are absent in the quiescent fluid.

We conclude that the time scale of measurements is an important factor in characterizing the structural dynamics of saturated solutions and suggest that changing (in particular, reducing) the measurement time window as is possible in pulse-echo ultrasound will provide quantitative information on the lifetime of crystal embryos, unlike even the most intense x-ray sources, such as the Diamond Light Source I22 beamline on which the experiments performed below were performed.

The treatment in Povey (2016)2 of rectified thermal diffusion under the influence of an oscillating pressure field neglects mass transfer, which we might also expect to be rectified.

While crystal nucleation theory (CNT) has come under multiple challenges,7–18 e.g., the multiple step nucleation theory,19, 20 the underlying physics as described by CNT is undeniable. Multiple step nucleation theories are microscopic, detailed theories addressing the attachment of individual molecules to surfaces, while CNT is a macroscopic theory, albeit applied to a relatively small object, e.g., a nucleus of ∼4 nm comprising a few hundred molecules. However, there is a great deal of evidence that CNT accurately describes crystal nucleation in the dispersed phase of emulsions.21, 22

Here, we provide a mathematical-physical description of mass transfer to and from an embryonic solid nucleus in the presence of a harmonically oscillating small pressure fluctuation and show that, despite the small pressure displacements involved, an impact on critical nucleus size can be expected. This treatment is consistent with the approach of Kashchiev in his book “Nucleation: Basic Theory with Applications” but differs in the respect that here the case of a harmonically oscillating small displacement pressure field is addressed.

First, consider an infinitely large solid crystal in equilibrium with a solution whose solute concentration is c0 at pressure P0 (Fig. 2). Thus, c0 defines the solubility of the solute molecules. At this concentration, the chemical potential in the solution will be the same as in the solid phase and is denoted by μ0.

What happens if the solid phase has a finite-sized nucleus of radius r? In that case, the chemical potential in the solid phase becomes
(1)
where γ is the interfacial tension between the solid and the solution and Vsolid is the molar volume of molecules in the solid phase. The solution that will be in equilibrium with this finite-sized nucleus will have a concentration that will ensure the same chemical potential as Eq. (1) in the solid. Denoting this as csolid0⁠,
μ0+RTlncsolid0c0=μ0+2γVsolidr,
where, throughout, RT is the product of the gas constant R and temperature T. Therefore,
(2)
What happens if we also increase the pressure from its original value, P0, to P = P0 + ΔP? In this case, the chemical potential in the solution μsolution will become
(3)
In addition, in the solid phase,
(4)
The symbol Vsolution in Eq. (3) represents the partial molar volume of the solute molecules in the aqueous solution. The concentration of solution adjacent to the nucleus and in equilibrium with it is obtained by equating Eqs. (3) and (4), with the result
(5)
Now, expand Vsolid and Vsolution to the first order term in ΔP,
(6)
(7)
If the crystal is pure, we note that ∂Vsolid∂P=Vsolid0K⁠, where K is the bulk modulus of the solid at P0. Similarly, we define β=∂Vsolution∂P so that using Eqs. (6) and (7), Eq. (5) becomes
2γVsolid0r+Vsolid0ΔP+12Vsolid0KΔP2=RTlncsolutionPc0+Vsolution0ΔP+12βΔP2.
Rearranging, we get
(8)

So far, the concentration adjacent to the crystal, and in equilibrium with it, is set as csolutionP⁠, while that in the bulk solution far away from the nucleus will be different, denoted as c∞. Therefore, there will be a diffusion flux of molecules from the solution toward the nucleus, or vice versa, if csolutionP≠c∞⁠.

Assuming a steady state flow, this mass flux is given by
(9)
where D is the diffusion coefficient of solute molecules in the solution and csolutionP̄ is the average value of csolutionP⁠. Rearranging Eq. (8),
(10)
csolutionP=c0⁡exp2γVsolid0RTrexpVsolid0−Vsolution0ΔPRT+12Vsolid0K−βΔP2RT.
In the case that the applied pressure fluctuation is small, i.e.,
Vsolid0−Vsolution0ΔPRT≪1
and
Vsolid0K−βΔP2RT≪1,
the second exponential term in Eq. (10) can be expanded as eθ≅1+θ+θ22!+⋯⁠, yielding
(11)
where we have only retained all terms up to OΔP2⁠. We can now determine the average flux from Eq. (9), by integrating over a single sinusoidal cycle period T of a harmonically oscillating pressure field applied to the solution. With ΔP = ΔPmax sin ωt and using Eq. (11),
j̄=1T∫0Tjdt=4πrDc∞−c0⁡exp2γVsolid0RTr−c0⁡exp2γVsolid0RTr×12RT2Vsolid0−Vsolution02+12RTVsolid0K−β×ΔPmax2T∫0T=2πωsin⁡ωt2dt
=4πrDc∞−c0⁡exp2γVsolid0RTr−c0⁡exp2γVsolid0RTrΔPmax24RT×Vsolid0K−β+Vsolid0−Vsolution02RT.
Note that the above expression is actually correct to (but not including) terms of OΔPmax4⁠, as all dependencies on odd powers of ΔP average out to zero for a sinusoidally applied field. More importantly, we observe that even when the bulk concentration in the solution c∞ is set to csolid0 corresponding to a concentration where r is the critical nucleus size, i.e., Eq. (2),
lncsolid0c0=2γVsolidRTr,
the average flux will still not vanish if
Vsolid0K−β+Vsolid0−Vsolution02RT≠0.
For such an otherwise critical nucleus (where the flux will normally be zero), the average flux is calculated to be
jcrit̄=−πrΔPmax2Dc0RTexp2γVsolid0RTr×Vsolid0K−β+Vsolid0−Vsolution02RT,
which can be positive if
(12)

and negative otherwise.

For the hexadecane example considered in Povey,2 the molar volume in the solid phase is 292 × 10−6 m3/mol (302 K).23 For liquid hexadecane at ambient pressure and at 303 K, the molar volume is 240 × 10−6 m3/mol11 and β≅310−295108 (Fig. 6 in Yurtseven and Tilki, 200624). We cannot find experimental data for the bulk modulus of solid hexadecane, but a plausible value is ∼1.4 × 109 Pa, which gives a value for the inequality Eq. (12) of −1.50 × 10−7 m3/mol/Pa, suggesting that the imposition of a harmonically oscillating low pressure field will tend to increase the size of a critical nucleus once it has appeared, thus effectively reducing the value of the critical radius. This effect has to be balanced against the thermal rectification effects discussed in Povey (2016),2 where we calculate that heat will flow into a nucleus under the conditions of an oscillating pressure field. An accurate resolution of this problem awaits further experimental investigation.

15% v/v of n-eicosane (C20, 99% pure) was made up in an 80/20 v/v heptane/toluene solvent (both 99% pure).

Measurements were conducted on the I22 SAXS/WAXS beamline at the Diamond Light Source, Rutherford Appleton Laboratory, Didcot, UK, using a specially designed acousto-optic cell (Fig. 3). The 16-mm diameter, 8-mm thick 840 material piezo-electric transducers (APC International, Ltd., USA) operated off-resonance at 2 MHz. The cover material is borosilicate glass slides, which are 1 mm thick. Experiments were also performed on anhydrous butter fat, both in the acousto-optic cell described here and in a 1 l sample, held in a 90 mm i.d. stainless steel pipe insonified magnetically, producing a well-defined uniform acoustic field verified using a hydrophone. XRD performed on samples removed from the 1 l sample agreed with the changes observed in the acousto-optic cell.

FIG. 3.
Left: Acousto-optical cell situated on the I22 beam line. Middle: Diagram of cell. Right: Off-line corroboration experiment setup—Cell dimensions 40 mm dia × 80 mm.
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Left: Acousto-optical cell situated on the I22 beam line. Middle: Diagram of cell. Right: Off-line corroboration experiment setup—Cell dimensions 40 mm dia × 80 mm.

FIG. 3.
Left: Acousto-optical cell situated on the I22 beam line. Middle: Diagram of cell. Right: Off-line corroboration experiment setup—Cell dimensions 40 mm dia × 80 mm.
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Left: Acousto-optical cell situated on the I22 beam line. Middle: Diagram of cell. Right: Off-line corroboration experiment setup—Cell dimensions 40 mm dia × 80 mm.

Close modal

The maximum flux at 4 W, assuming 5% conversion efficiency from electric excitation to acoustic output and a contact diameter of 1.5 mm with the 16 mm transducer, is (4 × 0.05)/(pi × (16/1.5)2 × 0.00152) = 0.25 kW/m2. This is around 100 times less than the cavitation threshold for water. The experiments were conducted in larger sample volumes externally to the acousto-optical cell used in the XRD experiments in order to corroborate visually and via speed of sound measurements the effects described below (Fig. 3 right).

Due to problems with the temperature controller, it was not possible to exactly reproduce the temperature–time profile in each run. In addition, the temperature oscillated around isothermal set points, so each run has a separate recording of the temperature, synchronized by timestamps to the diffraction data (Figs. 4–8).

FIG. 4.
Intensity (background subtracted) plotted against q for non-sonicated samples: [(a), the dashed line] 8.6 °C non-crystallized sample; [(b), the continuous line] 5 °C sample crystallized; and [(c), the dotted-dashed line] 3.8 °C sample crystallized.
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Intensity (background subtracted) plotted against q for non-sonicated samples: [(a), the dashed line] 8.6 °C non-crystallized sample; [(b), the continuous line] 5 °C sample crystallized; and [(c), the dotted-dashed line] 3.8 °C sample crystallized.

FIG. 4.
Intensity (background subtracted) plotted against q for non-sonicated samples: [(a), the dashed line] 8.6 °C non-crystallized sample; [(b), the continuous line] 5 °C sample crystallized; and [(c), the dotted-dashed line] 3.8 °C sample crystallized.
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Intensity (background subtracted) plotted against q for non-sonicated samples: [(a), the dashed line] 8.6 °C non-crystallized sample; [(b), the continuous line] 5 °C sample crystallized; and [(c), the dotted-dashed line] 3.8 °C sample crystallized.

Close modal
FIG. 5.
4 W insonified sample with intensity plotted against q with [(a), the solid line] 5 °C non-crystallized data; subsequent plots have the data in (a) subtracted; [(b), the dashed line] 4 °C; [(c), the dotted line] 1.3 °C; and [(d), the dotted-dashed line] 0.7 °C.
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4 W insonified sample with intensity plotted against q with [(a), the solid line] 5 °C non-crystallized data; subsequent plots have the data in (a) subtracted; [(b), the dashed line] 4 °C; [(c), the dotted line] 1.3 °C; and [(d), the dotted-dashed line] 0.7 °C.

FIG. 5.
4 W insonified sample with intensity plotted against q with [(a), the solid line] 5 °C non-crystallized data; subsequent plots have the data in (a) subtracted; [(b), the dashed line] 4 °C; [(c), the dotted line] 1.3 °C; and [(d), the dotted-dashed line] 0.7 °C.
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4 W insonified sample with intensity plotted against q with [(a), the solid line] 5 °C non-crystallized data; subsequent plots have the data in (a) subtracted; [(b), the dashed line] 4 °C; [(c), the dotted line] 1.3 °C; and [(d), the dotted-dashed line] 0.7 °C.

Close modal
FIG. 6.
Determination of the temperature at which an uncontrolled crystal growth occurs through the temperature dependence of the intensity of a single diffraction peak in the unsonicated sample. The red dashed line is the temperature, and the blue solid line is the intensity of the signal at q = 1.3345. In this case, the sample is crystallized at 6.05 °C.
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Determination of the temperature at which an uncontrolled crystal growth occurs through the temperature dependence of the intensity of a single diffraction peak in the unsonicated sample. The red dashed line is the temperature, and the blue solid line is the intensity of the signal at q = 1.3345. In this case, the sample is crystallized at 6.05 °C.

FIG. 6.
Determination of the temperature at which an uncontrolled crystal growth occurs through the temperature dependence of the intensity of a single diffraction peak in the unsonicated sample. The red dashed line is the temperature, and the blue solid line is the intensity of the signal at q = 1.3345. In this case, the sample is crystallized at 6.05 °C.
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Determination of the temperature at which an uncontrolled crystal growth occurs through the temperature dependence of the intensity of a single diffraction peak in the unsonicated sample. The red dashed line is the temperature, and the blue solid line is the intensity of the signal at q = 1.3345. In this case, the sample is crystallized at 6.05 °C.

Close modal
FIG. 7.
Data for samples insonified at <1 W crystallizing at 4.43 °C; the red dashed line is the temperature, and the blue solid line is the intensity for q = 1.3424.
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Data for samples insonified at <1 W crystallizing at 4.43 °C; the red dashed line is the temperature, and the blue solid line is the intensity for q = 1.3424.

FIG. 7.
Data for samples insonified at <1 W crystallizing at 4.43 °C; the red dashed line is the temperature, and the blue solid line is the intensity for q = 1.3424.
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Data for samples insonified at <1 W crystallizing at 4.43 °C; the red dashed line is the temperature, and the blue solid line is the intensity for q = 1.3424.

Close modal
FIG. 8.
Data for samples insonified at 4 W crystallizing at 4.06 °C. The red dashed line is the temperature, and the blue solid line is the intensity for q = 1.3424.
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Data for samples insonified at 4 W crystallizing at 4.06 °C. The red dashed line is the temperature, and the blue solid line is the intensity for q = 1.3424.

FIG. 8.
Data for samples insonified at 4 W crystallizing at 4.06 °C. The red dashed line is the temperature, and the blue solid line is the intensity for q = 1.3424.
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Data for samples insonified at 4 W crystallizing at 4.06 °C. The red dashed line is the temperature, and the blue solid line is the intensity for q = 1.3424.

Close modal

The straightforward conclusion from this and other published studies is that low power, continuous, non-cavitational ultrasound has a profound effect on nucleation and crystal growth in n-eicosane with a reduction in crystallization temperature from 6.05 °C in the case of the non-sonicated wax solution; sonicated at less than 1 W, 4.43 °C; and sonicated at 4 W, 4.00 °C.

However, there is much more to be explained. During the I22 experiments, we found that we could control the appearance and disappearance of crystallites and saw big changes in the electro-mechanical impedance of the transducer associated with the appearance and disappearance of diffraction spots on the 2D detector. The shifts in the peak positions as detailed in Table I require further analysis, as do the differences between our data and that of Doyle (Fig. 9). The differences are probably because Doyle et al., samples crystallized from the melt and from dodecane, whereas our samples crystallized from heptane/toluene. These solvent effects are interesting in themselves.

TABLE I.

Peak positions, intensities, widths, prominence, normalized intensity, assigned lattice parameters, calculated dhkl for assigned lattice parameters and relative to the non-sonicated sample: peak shift and inverse of peak broadening consequent to ultrasonication. These data were obtained using Matlab and are plotted in Fig. 11.

No. ultrasound, 4 °C (run 268)
PeakTcrystal = 6.05 °Cq = 2pi/dd (Å)pk intensityWidth (Å)ProminenceIntensity normalizedhklndhkl (Å)∆dhkl (Å)1/∆w
  0.4549 13.8117 0.1476 5.6451 0.0011 0.4520        
f  3.0780 2.0413 0.1507 0.0095 0.0011 0.4613        
m  2.6931 2.3330 0.1507 0.0116 0.0014 0.4616        
j  1.5191 4.1362 0.1522 0.0146 0.0014 0.4659        
e  2.7638 2.2734 0.1508 0.0052 0.0015 0.4617        
d  1.7939 3.5025 0.1510 0.0311 0.0015 0.4625        
 2.4065 2.6109 0.1533 0.0174 0.0040 0.4694        
g  1.5544 4.0422 0.1555 0.0128 0.0052 0.4762 1 0 1 1 30.45   
 1.3620 4.6133 0.1541 0.0171 0.0063 0.4719        
 1.3816 4.5477 0.1574 0.0388 0.0095 0.4821        
c  1.7350 3.6214 0.1661 0.0197 0.0162 0.5087 1 1 1  26.85   
b  1.6329 3.8478 0.1911 0.0249 0.0418 0.5851 0 1 1 1 24.16   
a  1.3424 4.6807 0.3266 0.0243 0.1793 1.0000 0 1 0 1 4.82   
No. ultrasound, 4 °C (run 268)
PeakTcrystal = 6.05 °Cq = 2pi/dd (Å)pk intensityWidth (Å)ProminenceIntensity normalizedhklndhkl (Å)∆dhkl (Å)1/∆w
  0.4549 13.8117 0.1476 5.6451 0.0011 0.4520        
f  3.0780 2.0413 0.1507 0.0095 0.0011 0.4613        
m  2.6931 2.3330 0.1507 0.0116 0.0014 0.4616        
j  1.5191 4.1362 0.1522 0.0146 0.0014 0.4659        
e  2.7638 2.2734 0.1508 0.0052 0.0015 0.4617        
d  1.7939 3.5025 0.1510 0.0311 0.0015 0.4625        
 2.4065 2.6109 0.1533 0.0174 0.0040 0.4694        
g  1.5544 4.0422 0.1555 0.0128 0.0052 0.4762 1 0 1 1 30.45   
 1.3620 4.6133 0.1541 0.0171 0.0063 0.4719        
 1.3816 4.5477 0.1574 0.0388 0.0095 0.4821        
c  1.7350 3.6214 0.1661 0.0197 0.0162 0.5087 1 1 1  26.85   
b  1.6329 3.8478 0.1911 0.0249 0.0418 0.5851 0 1 1 1 24.16   
a  1.3424 4.6807 0.3266 0.0243 0.1793 1.0000 0 1 0 1 4.82   
Ultrasound < 1W, 4 °C (run 267)Tcrystal = 4.43 °C
m  2.6971 2.3296 0.1502 0.0108 0.0008 0.6182      −0.0034 −1247.57 
l  0.4235 14.8362 0.1475 3.6454 0.0009 0.6068        
k  0.1447 43.4204 0.1476 1.4542 0.0009 0.6073        
j  1.5230 4.1256 0.1513 0.0276 0.0014 0.6224      −0.0107 77.03 
i  3.1879 1.9709 0.1511 0.0079 0.0016 0.6219        
h  3.8829 1.6182 0.1513 0.0038 0.0017 0.6227 1 0 2  57.78   
g  1.5622 4.0219 0.1516 0.0216 0.0018 0.6237 1 0 1  30.45 −0.0203 113.92 
f  3.0701 2.0466 0.1517 0.0059 0.0022 0.6241      0.0052 −274.73 
e  2.7599 2.2766 0.1522 0.0069 0.0029 0.6264      0.0032 585.52 
d  1.7979 3.4948 0.1576 0.0185 0.0083 0.6483      −0.0076 −79.00 
c  1.7311 3.6296 0.1750 0.0244 0.0255 0.7200 1 1 1  26.85 −0.0039 212.34 
b  1.6486 3.8111 0.2254 0.0190 0.0763 0.9276 0 1 1  24.16 −0.0367 −169.78 
a  1.3620 4.6133 0.2430 0.0181 0.0959 1.0000 0 1 0  4.82 −0.0675 −162.85 
Ultrasound < 1W, 4 °C (run 267)Tcrystal = 4.43 °C
m  2.6971 2.3296 0.1502 0.0108 0.0008 0.6182      −0.0034 −1247.57 
l  0.4235 14.8362 0.1475 3.6454 0.0009 0.6068        
k  0.1447 43.4204 0.1476 1.4542 0.0009 0.6073        
j  1.5230 4.1256 0.1513 0.0276 0.0014 0.6224      −0.0107 77.03 
i  3.1879 1.9709 0.1511 0.0079 0.0016 0.6219        
h  3.8829 1.6182 0.1513 0.0038 0.0017 0.6227 1 0 2  57.78   
g  1.5622 4.0219 0.1516 0.0216 0.0018 0.6237 1 0 1  30.45 −0.0203 113.92 
f  3.0701 2.0466 0.1517 0.0059 0.0022 0.6241      0.0052 −274.73 
e  2.7599 2.2766 0.1522 0.0069 0.0029 0.6264      0.0032 585.52 
d  1.7979 3.4948 0.1576 0.0185 0.0083 0.6483      −0.0076 −79.00 
c  1.7311 3.6296 0.1750 0.0244 0.0255 0.7200 1 1 1  26.85 −0.0039 212.34 
b  1.6486 3.8111 0.2254 0.0190 0.0763 0.9276 0 1 1  24.16 −0.0367 −169.78 
a  1.3620 4.6133 0.2430 0.0181 0.0959 1.0000 0 1 0  4.82 −0.0675 −162.85 
Ultrasound 4W, 4 deg C (run 269)Tcrystal = 4.00 °C
h  3.9732 1.5814 0.1504 0.0124 0.0008 0.8929 1 0 2  57.78   
k  0.1408 44.6315 0.1476 1.7598 0.0009 0.8765        
 0.4863 12.9196 0.1479 0.2290 0.0009 0.8786        
d  1.7900 3.5102 0.1501 0.0204 0.0009 0.8914      0.0077 −93.21 
e  2.7677 2.2701 0.1504 0.0050 0.0009 0.8929      −0.0032 −5036.18 
 0.7337 8.5636 0.1475 0.0499 0.0009 0.8758        
 1.7272 3.6378 0.1519 0.0060 0.0010 0.9019        
 1.8764 3.3486 0.1502 0.0116 0.0010 0.8918        
f  3.0740 2.0440 0.1507 0.0060 0.0011 0.8947      0.0026 −288.71 
 2.9327 2.1425 0.1507 0.0064 0.0012 0.8948        
j  1.5230 4.1256 0.1517 0.0348 0.0013 0.9007      −0.0107 49.42 
 2.7481 2.2864 0.1524 0.0065 0.0030 0.9050        
 0.2389 26.2953 0.1503 0.5942 0.0040 0.8926        
a  1.3541 4.6400 0.1522 0.0136 0.0051 0.9039 0 1 0   0.0268 −93.28 
b  1.6369 3.8386 0.1569 0.0138 0.0069 0.9320 0 1 1   −0.0092 −89.94 
g  1.5622 4.0219 0.1573 0.0329 0.0078 0.9343      −0.0203 49.74 
c  1.7390 3.6132 0.1684 0.0122 0.0193 1.0000 1 1 1   −0.0082 −132.34 
Lattice          
parameters          
according to          
Doyle et al.          
a (Å)  b (Å) c (Å) α (º) β (º) γ (º) V (Å3) dhkl λ (Å)      
4.282  4.818 27.412 85.586 68.279 72.607 501        
Ultrasound 4W, 4 deg C (run 269)Tcrystal = 4.00 °C
h  3.9732 1.5814 0.1504 0.0124 0.0008 0.8929 1 0 2  57.78   
k  0.1408 44.6315 0.1476 1.7598 0.0009 0.8765        
 0.4863 12.9196 0.1479 0.2290 0.0009 0.8786        
d  1.7900 3.5102 0.1501 0.0204 0.0009 0.8914      0.0077 −93.21 
e  2.7677 2.2701 0.1504 0.0050 0.0009 0.8929      −0.0032 −5036.18 
 0.7337 8.5636 0.1475 0.0499 0.0009 0.8758        
 1.7272 3.6378 0.1519 0.0060 0.0010 0.9019        
 1.8764 3.3486 0.1502 0.0116 0.0010 0.8918        
f  3.0740 2.0440 0.1507 0.0060 0.0011 0.8947      0.0026 −288.71 
 2.9327 2.1425 0.1507 0.0064 0.0012 0.8948        
j  1.5230 4.1256 0.1517 0.0348 0.0013 0.9007      −0.0107 49.42 
 2.7481 2.2864 0.1524 0.0065 0.0030 0.9050        
 0.2389 26.2953 0.1503 0.5942 0.0040 0.8926        
a  1.3541 4.6400 0.1522 0.0136 0.0051 0.9039 0 1 0   0.0268 −93.28 
b  1.6369 3.8386 0.1569 0.0138 0.0069 0.9320 0 1 1   −0.0092 −89.94 
g  1.5622 4.0219 0.1573 0.0329 0.0078 0.9343      −0.0203 49.74 
c  1.7390 3.6132 0.1684 0.0122 0.0193 1.0000 1 1 1   −0.0082 −132.34 
Lattice          
parameters          
according to          
Doyle et al.          
a (Å)  b (Å) c (Å) α (º) β (º) γ (º) V (Å3) dhkl λ (Å)      
4.282  4.818 27.412 85.586 68.279 72.607 501        
View Large
FIG. 9.
Comparison of Doyle et al.,25 data with the diffraction pattern obtained in I22 experiments. The intensities of the I22 data are as recorded, with cell and solvent background subtracted. The red large dashed line: Doyle et al. (a) The blue solid line: Doyle et al. (b) Doyle peak assignation associated with Doyle peaks; the green small dashes: no ultrasound 4 °C; the blue dotted line: <1 W ultrasound at 4 °C; the blue dashed-dotted line: 4 W ultrasound; the green solid line: 4 W ultrasound, 0.7 °C.
View largeDownload slide

Comparison of Doyle et al.,25 data with the diffraction pattern obtained in I22 experiments. The intensities of the I22 data are as recorded, with cell and solvent background subtracted. The red large dashed line: Doyle et al. (a) The blue solid line: Doyle et al. (b) Doyle peak assignation associated with Doyle peaks; the green small dashes: no ultrasound 4 °C; the blue dotted line: <1 W ultrasound at 4 °C; the blue dashed-dotted line: 4 W ultrasound; the green solid line: 4 W ultrasound, 0.7 °C.

FIG. 9.
Comparison of Doyle et al.,25 data with the diffraction pattern obtained in I22 experiments. The intensities of the I22 data are as recorded, with cell and solvent background subtracted. The red large dashed line: Doyle et al. (a) The blue solid line: Doyle et al. (b) Doyle peak assignation associated with Doyle peaks; the green small dashes: no ultrasound 4 °C; the blue dotted line: <1 W ultrasound at 4 °C; the blue dashed-dotted line: 4 W ultrasound; the green solid line: 4 W ultrasound, 0.7 °C.
View largeDownload slide

Comparison of Doyle et al.,25 data with the diffraction pattern obtained in I22 experiments. The intensities of the I22 data are as recorded, with cell and solvent background subtracted. The red large dashed line: Doyle et al. (a) The blue solid line: Doyle et al. (b) Doyle peak assignation associated with Doyle peaks; the green small dashes: no ultrasound 4 °C; the blue dotted line: <1 W ultrasound at 4 °C; the blue dashed-dotted line: 4 W ultrasound; the green solid line: 4 W ultrasound, 0.7 °C.

Close modal
FIG. 10.
Diffraction pattern, signal intensity, peak position, prominence, and width at 4 °C. [(a), top] The non-sonicated sample; [(b), middle] 4 W ultrasound; and [(c), bottom] <1 W ultrasound. Detailed data are provided in Table I.
View largeDownload slide

Diffraction pattern, signal intensity, peak position, prominence, and width at 4 °C. [(a), top] The non-sonicated sample; [(b), middle] 4 W ultrasound; and [(c), bottom] <1 W ultrasound. Detailed data are provided in Table I.

FIG. 10.
Diffraction pattern, signal intensity, peak position, prominence, and width at 4 °C. [(a), top] The non-sonicated sample; [(b), middle] 4 W ultrasound; and [(c), bottom] <1 W ultrasound. Detailed data are provided in Table I.
View largeDownload slide

Diffraction pattern, signal intensity, peak position, prominence, and width at 4 °C. [(a), top] The non-sonicated sample; [(b), middle] 4 W ultrasound; and [(c), bottom] <1 W ultrasound. Detailed data are provided in Table I.

Close modal

Increasing the acoustic power from <1 to 4 W causes a significant reduction in peak intensity. The application of ultrasound causes the main peak to shift to shorter spacing, less so in the case of a higher applied power.

Due to time limitations and limited access to the I22 SAXS/WAXS beam, we were unable to explore these phenomena further and are planning future experiments to examine the impact of insonifying frequency and power in greater detail. We are also planning to model the acoustic field in the sample cell; initial data suggest that the field distribution in the cell is very sensitive to the insonifying frequency.

We have explored the use of dissipative particle dynamics (DPD)26 in order to simulate the impact of a continuous oscillating pressure field on a box of wax molecules.27–29 Initial results suggest that the oscillating field significantly changes the packing of molecules in the liquid state and that switching off the oscillating field incurs a relatively slow relaxation back to the initial, unsonified state. Such changes in packing, for example from 2L to 3L, may be frozen in by cooling the sample, resulting in changes in the polymorph and morphology that we have seen in dairy fats. We would expect to see these effects in our SAXS measurements although the results presented below require further explanation.

Our SAXS measurements, carried out simultaneously with the WAXS measurements on the I22 beamline, indicate that insonification has a profound effect on the long-range order (Figs. 10 and 11).

FIG. 11.
SAXS plots of scattered intensity vs temperature (decreasing) and spacing. From top to bottom: (a) no ultrasound; (b) <1 W ultrasound; and (c) 4 W ultrasound.
View largeDownload slide

SAXS plots of scattered intensity vs temperature (decreasing) and spacing. From top to bottom: (a) no ultrasound; (b) <1 W ultrasound; and (c) 4 W ultrasound.

FIG. 11.
SAXS plots of scattered intensity vs temperature (decreasing) and spacing. From top to bottom: (a) no ultrasound; (b) <1 W ultrasound; and (c) 4 W ultrasound.
View largeDownload slide

SAXS plots of scattered intensity vs temperature (decreasing) and spacing. From top to bottom: (a) no ultrasound; (b) <1 W ultrasound; and (c) 4 W ultrasound.

Close modal

In the case of very low intensity ultrasound [Fig. 11(b)], the scattering intensity fluctuates wildly with temperature and the long-range order in the liquid state is very significantly depressed, but in the 4 W case [Fig. 11(c)], the long-range order, even in the melt, is significantly increased relative to the unsonicated sample. Once the system has crystallized, the long-range order is significantly increased by the application of ultrasound.

We show that ultrasound spectroscopy measurements can reveal quantitatively the behavior of embryos in the “dead zone” and suggest that the measurement time must be fast enough to capture the fleeting presence of the embryos. While the XRD measurement time frame is too long to detect critical behaviors, it is evident that low power insonification has dramatic effects on both short-range (molecular) and long-range (mesoscale) order.

We have outlined an approach that predicts the effect of a harmonically oscillating low power pressure field on critical nucleus size, which ultimately could permit the prediction of the effects of low power, continuous oscillating pressure fields on nucleation and growth in a variety of materials and fields. It is also suggested that DPD26 offers an approach that accounts for entropy effects and would consequently have greater generality. Unfortunately, we are unable to publish the DPD results for commercial reasons.

Our study of the crystallization of n-eicosane from the solution demonstrates that the application of low power, continuous ultrasound lowers the crystallization temperature and significantly alters the diffraction pattern. In particular, the influence of the solvent on the crystallization process merits further study. While this study reports only on data for n-eicosane, the authors have observed similar phenomena in anhydrous milk fat. We see remarkable effects of ultrasound on mesoscale order, including in the melt in the case of the highest powers used. These phenomena merit further investigation but are consistent with DPD results.

Combining the ultrasound pitch and catch speed of sound measurements with low power harmonically oscillating pressure fields to monitor and control nucleation presents the prospect of entirely new industrially significant methods of process control in crystallization. It also offers new insights into nucleation processes in general. However, for the acoustic control technique to be widely applied, further theoretical and modeling work will be necessary since, at present, we are unable to predict the precise effect of low power ultrasound in any given situation.

The full XRD dataset and analyses is available from the corresponding author upon reasonable request. The full dataset for the copper sulfate pentahydrate study is available in Fei Sheng’s thesis paper copy from the British Library.30

We are grateful to Dr. Nick Terrill and Dr. Tim Snow (Diamond Light Source) for their assistance and advice (Diamond I22 sm27656-2 Tuning crystal nucleation) and to Arla Foods (Dr. Ulf Andersen) for the support for a postdoctoral position. We acknowledge Lewtas Science & Technology, Ltd. for funding the preliminary DPD work at Hartree.

The authors have no conflicts to disclose.

Megan J. Povey: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Rammile Ettelaie: Formal analysis (equal); Writing – review & editing (equal). Ken Lewtas: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Writing – review & editing (equal). Andy Price: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Xiaojun Lai: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – review & editing (supporting). Fei Sheng: Investigation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

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Supplementary Material

Supplementary materials- zip file