Effect of Laser-Exposed Volume and Irradiation Position on Nonphotochemical Laser-Induced Nucleation of Potassium Chloride Solutions

Herein, we study the influences of the laser-exposed volume and the irradiation position on the nonphotochemical laser-induced nucleation (NPLIN) of supersaturated potassium chloride solutions in water. The effect of the exposed volume on the NPLIN probability was studied by exposing distinct milliliter-scale volumes of aqueous potassium chloride solutions stored in vials at two different supersaturations (1.034 and 1.050) and laser intensities (10 and 23 MW/cm2). Higher NPLIN probabilities were observed with increasing laser-exposed volume as well as with increasing supersaturation and laser intensity. The measured NPLIN probabilities at different exposed volumes are questioned in the context of the dielectric polarization mechanism and classical nucleation theory. No significant change in the NPLIN probability was observed when samples were irradiated at the bottom, top, or middle of the vial. However, a significant increase in the nucleation probability was observed upon irradiation through the solution meniscus. We discuss these results in terms of mechanisms proposed for NPLIN.


Laser Exposed Volume Dependency of DP Model
In line with the Dielectric Polarization model, the free energy change of cluster formation, ∆G, is made up of contributions from the free energy change of surface formation, ∆G s , the free energy change of phase transformation, ∆G v , and the change in free energy due to the introduction of an electric field, ∆G EF .In presence of an electric field and under the constraint that the ϵ p > ϵ s , the free energy change of cluster formation (∆G) is lowered by an amount proportional to −v(ϵ p − ϵ s )E 2 , where v is the volume of the cluster, ϵ p is the dielectric constant of a cluster of solute molecules, ϵ s is the dielectric constant of the surrounding medium and E is the electric field strength. 1or convenience, the electric field strength can be written in terms of intensity by the equation I = 1 2 ϵ 0 cE 2 , where I is the intensity of the light, ϵ 0 is the vacuum permittivity and c is the speed of light.Hence, by adding the contribution of the electric field induced by the light, the change in free energy becomes, where s and v are the surface area and volume of the precritical cluster, respectively, γ is the interfacial tension between the cluster and surrounding solution, ρ is the number of molecules per unit volume in the solid phase and ∆µ is difference in chemical potential of the substance in solution and in the crystal.The constant a contains the dielectric contrast between solute cluster and the surrounding medium, ?
Then, assuming the precritical clusters are spheres with a radius r, the free energy change of cluster formation becomes, ∆G(r, I) = 4πr where k B is the Boltzmann constant, T is the temperature of the solution and S is the supersaturation ratio of the solution.Analogously to the Classical Nucleation Theory it is possible to derive the critical radius, r c (I), and nucleation barrier height, ∆G c (I), under the influence of an electric field, Prior to laser irradiation when nucleation has not yet occurred, all clusters are said to be smaller than the critical radius (r < r c (0)).Upon laser irradiation, all clusters with size r c (I) < r < r c (0) become supercritical.The average number of precritical clusters (r < r c (0)) can be computed by, where N molecule is the number of solute molecules in the volume of the laser beam, and ⟨n⟩ is the average amount of solute molecules in a precritical cluster, which can be computed by assuming a Boltzmann distribution over the domain r ∈ [0, r c (0)], Hence, To obtain the average number of clusters that go on to form viable crystals after laser irradiation, N crystal , the average amount of precritical clusters is multiplied by the fraction of clusters that become supercritical, Hence, by combining equation 9 and 10 a function for the average amount of clusters that go on to form viable crystals is obtained, The amount of solute molecules per unit volume can be written in terms of the solute density, ρ = ρsN A M , where N A is Avogadro's constant and M is the molar mass of the solute.In addition, the amount of molecules in the volume of the laser beam can be written as, where V laser is the volume of the laser passing through the solution,ρ l is the density of the surrounding medium and W is the solute mass fraction.Therefore the average amount of clusters that go on to form viable crystals becomes, Under the constraints of constant intensity and supersaturation, the only dependent variable is V laser , and hence we can write N crystal as, where m(I, S) serves as a new, intensity and supersaturation dependent lability constant, By assuming a Poisson distribution it is possible to compute the probability that no crystals are observed in any of the repeated experiments, At last, the cumulative nucleation probability is computed as a function of intensity, supersaturation and laser exposed volume,

Laser Exposed Volume Calculation
A cross-sectional view in the xy-plane of the laser beam passing through the vial is shown in Figure SIA.The laser passes through three media with different refractive indices, resulting in two directional changes of the laser path in the y-direction.In order to calculate the volume of the beam, the angles of incidence and angles of refraction passing through the vial have to be calculated.The angle of incidence resulting from the change of medium from air to borosilicate glass, Θ 1 , can be computed by, where, r is the radius of the laser beam at a given cross-section and R 0 is the radius of the outer surface of the glass vial.Note that in Figure SIA, r is given as the radius of the in-cident beam, R L .The usefulness of this will become evident at a later stage of this derivation.Then, by implementing Snell's law, it is possible to compute the first angle of refraction, Θ 2 , where n 1 and n 2 are the refractive indices of the air and borosilicate glass, respectively.Further inspection of a zoomed in version of the triangle enclosed by the path of the laser through the glass wall, L L , and the outer and inner radius of the glass vial, R 0 and R 1 , respectively (see Figure SIB) reveals, where ∆Θ is the angle between R 0 and R 1 .In addition, it shows that, Hence, by combining equations 20 and 21 an equation is obtained to describe the outer radius of the vial as a function of ∆Θ, Equation 22 can be solved numerically to find the value for ∆Θ.Following a geometrical derivation, the second angle of incidence, Θ 3 , can be written as Θ 2 + ∆Θ.Then, using Snell's law again, the second angle of refraction, Θ 4 is obtained, From equations 18-23 all necessary angles can be computed in order to calculate the central angle, Θ s , and subsequently the area of the disc segment, A chord .The central angle is given by, and the area of the disc segment by, The cross-sectional area of the inner circle of the vial, A circle , can be written in terms of πR 2 1 .Hence, the area of the laser path through the solution, A laser , becomes, ) Subsituting equation 24 into equation 26 will give the area of the laser passing through the solution in terms of the second angle of refraction, Now that the cross-sectional area of a particular slice of the laser beam has been obtained, it is possible to compute the volume of the laser beam through the solution by summing all slices of the cross-sectional area.One particular slice is defined by half a chord length of the cross-sectional area of the incident laser beam (see Figure SII).Hence, Substituting equation 28 in equation 18 results in, Following this the area of the laser passing through the solution, A laser , becomes a function of ϕ.The laser exposed volume, V laser , can be computed by summing over all the slices, or, by substituting equation 28 and the derivative of equation 29, )) cos ϕdϕ (31) The integral in equation 31 has been solved numerically in order to find the volume of the laser beam passing through the solution.In addition, by recognizing that the cross-sectional area of the laser beam gets smaller due to the focusing effect in the y-direction, it is possible to calculate the change in intensity of the beam throughout the vial.Focusing the beam in solely the y-direction changes the profile of the beam from a circle to an ellipse.The radius of the ellipse in the y-direction is defined by the linear function of the laser beam through the solution, y = mx + b (see Figure SIII).The slope coefficient, m, can be computed by, At last, it is possible to compute the width of the ellipse at the end of the vial, R ellipse , This allows for the calculation of the intensity of the laser beam at the exit of the vial.? In the current study, the peak intensity, I peak is calculated by dividing the peak power, P peak , by the crosssectional area of the beam, A beam .
The peak power of the beam is obtained from the laser characteristics, where E pulse is the pulse energy of the laser and τ is the pulse width.Substituting equation 35 in equation 34 and assuming that the cross-sectional area of the fundamental beam is a circle, results in where R L is the laser beam radius.The borosilicate HPLC vials act as a cylindrical lens due to their geometry and difference in refractive indices of air (n 1 ), borosilicate glass (n 2 ) and potassium chloride solution (n 3 ).As a result, the peak intensity of the laser beam changes depending on the position within the vial.Because the vials only focus the beam in the ydirection, the resulting cross-sectional area of the beam within the vial can be described by that of an ellipse, and hence the peak intensity becomes where R ellipse (x) is the length of the semiminor axis of the ellipse, depending on the position, x, within the vial.Figure SIVA shows a cross-sectional top view in the xy-plane of the vial and corresponding fundamental laser beam with a diameter of 9 mm.The area of the crosssection of the laser beam is smallest at the back of the vial, and hence a maximum peak intensity, I max peak , is found at the position where the laser hits the inner walls at the back of the vial for the first time.
The first angle of incidence (Θ 1 ) changes depending on the laser beam diameter.As a result, the relative change of peak intensity throughout the vial is different for each beam diameter (shown in Figure SIVB).The peak intensity becomes increasingly larger when the laser propagates throughout the vial.This effect increases the larger the incident beam diameter.To account for this difference, a priori calculations have been performed to determine the necessary peak intensity of the laser beam in front of the vial to achieve a maximum peak intensity that is constant for all beam diameters.Values of 16.6 and 0.5 mm were assumed for the vial outer diameter and glass thickness, respectively.In addition, the refractive index of air and borosilicate glass are assumed to be 1.0000 and 1.5195, respectively. 3In absence of equipment to accurately record the refractive index of the potassium chloride solutions, an assumption has been made by calculating the refractive index using an empirical relation 4

Zemax simulation
The additional data presented here elaborates on the laser-meniscus interaction study conducted within an HPLC vial.This is further illustrated with extended visualizations from the CAD model, which specifically emphasize the laser's interaction with the meniscus.This interaction can be seen from both the side and top views as shown in Figures SVA and SVB.
In experiments, it has been observed that crystals always form at a point At this stage, to acquire a comprehensive understanding of the laser intensity distribution beneath the meniscus, a volumetric detector of dimensions 8*4*8 mm, color coded orange was placed directly below it.The volume detector was positioned based on the location of crystal formation observed experimentally on the meniscus following the laser shot.The positioning ensured that the bottom region of the meniscus was encompassed within the detecting volume.This detector was composed of 4 million voxels arranged in a 200 x 100 x 200 configuration, with each voxel designed to detect transmitted rays (flux) in MW/cm 2 .Each voxel has a dimension of 40×40×40µm.The power distribution within this 3D volume was visualized using 2D slices.
For clarity and illustrative purposes, the intensity distribution is represented across both parallel (XY) and perpendicular (YZ) planes, as depicted in Figures SVC & SVD.These images show the distribution at a fixed position of Z = 0 mm and a depth of X = 0 mm (axis numbers goes from -4 mm to 4 mm), respectively.For instance, at specific coordinates (highlighted as X in the figure), the detected laser peak intensity was approximately 17.18 MW/cm 2 .Such pronounced peak intensities, especially at the meniscus, could be attributed to the cylindrical nature of the vial, the total internal reflection induced by the meniscus, and the rear wall partial reflection.These factors may probably explain the high nucleation probabilities observed in laser irradiation experiments at the meniscus in the context of Dielectric Polarization model or Nanoparticle Heating mechanism.

Figure
Figure SI: (A) Cross-sectional area of the laser beam passing through a cylindrical vial.This figure is not to scale and thus the given angles are only a guide to the eye, (B) Zoomed in version of the triangle enclosed by the path of the beam and the outer and inner radius of the glass vial.

Figure
Figure SII: Cross-sectional area of the incident laser beam.

Figure SIII :
Figure SIII: Zoomed in version of the triangle enclosed by the chord formed by the path of the beam through the solution and the inner radii of the glass vial.

Figure
Figure SIV: (a) Top view of the laser path through a vial containing an aqueous potassium chloride solution (S = 1.034).A incident beam diameter of 9.0 mm is shown.(b) Plot showing relative change in peak intensity throughout the vial for different beam diameters (D beam ).

Figure
Figure SV: (A) Simulation in Zemax OpticStudio showcasing the behavior of light rays as they encounter total internal reflection when a laser illuminates the meniscus of a solution (side view), (B) Top view, (C) A representation of the peak intensity distribution across the XY plane at a depth of Z = 0 mm within the volume detector, (D) A representation of peak intensity distribution across the YZ plane at a position of X = 0 mm.