Polarimetric angle-resolved second-harmonic scattering (AR-SHS) is an all-optical tool enabling the study of unlabeled interfaces of nano-sized particles in an aqueous solution. As the second harmonic signal is modulated by interference between nonlinear contributions originating at the particle’s surface and those originating in the bulk electrolyte solution due to the presence of a surface electrostatic field, the AR-SHS patterns give insight into the structure of the electrical double layer. The mathematical framework of AR-SHS has been previously established, in particular regarding changes in probing depth with ionic strength. However, other experimental factors may influence the AR-SHS patterns. Here, we calculate the size dependence of the surface and electrostatic geometric form factors for nonlinear scattering, together with their relative contribution to the AR-SHS patterns. We show that the electrostatic term is stronger in the forward scattering direction for smaller particle sizes, while the ratio of the electrostatic to surface terms decreases with increasing size. Besides this competing effect, the total AR-SHS signal intensity is also weighted by the particle’s surface characteristics, given by the surface potential Φ_{0} and the second-order surface susceptibility $\chi s,22$. The weighting effect is experimentally demonstrated by comparing SiO_{2} particles of different sizes in NaCl and NaOH solutions of varying ionic strengths. For NaOH, the larger $\chi s,22$ values generated by deprotonation of surface silanol groups prevail over the electrostatic screening occurring at high ionic strengths; however, only for larger particle sizes. This study establishes a better connection between the AR-SHS patterns and surface properties and predicts trends for arbitrarily-sized particles.

## I. INTRODUCTION

Second harmonic generation (SHG) is a nonlinear optical process in which two photons with the same frequency ω interact with a nonlinear medium; the two photons are combined and generate a new photon with twice the frequency 2*ω*. Within the electric dipole approximation, this optical process is only allowed in non-centrosymmetric media. At an interface, the symmetry is broken by morphological discontinuities, and an SH response can be detected. SHG is, therefore, a method of choice to probe complex interfaces on the molecular level, for example, the electrical double layer (EDL) formed at solid surfaces in contact with aqueous solutions.^{1–5} For studies that involve turbid samples such as nano- or microparticles, SHG has to be adapted to detect scattered light from the sample.^{6} Second harmonic scattering (SHS) of particles dispersed in solution was first demonstrated in 1996,^{7} and since then SHS has been applied to systems of colloidal polystyrene beads,^{8–13} dielectric and metallic nanoparticles,^{14–17} clay particles,^{18} liposomes and oil-water droplets,^{19–22} and cellular membranes.^{23,24} To increase the signal-to-noise ratio, SHS is most often performed in a resonant configuration, with the aid of second harmonic (SH) labels adsorbed at the surface.^{25–30} In 2013, our group reported a polarimetric angle-resolved SHS instrument (AR-SHS) with a throughput that was 2–3 orders of magnitude higher than standard systems at the time,^{31} enabling high signal-to-noise ratios for the measurement of non-resonant SHS, opening the possibility to measure SHS of unlabeled interfaces of nano- and micro-sized objects in solution with angular resolution.

For centrosymmetric particles in water, the SHS signal originates at the interface between the scattering particles and the water, where the centrosymmetry is broken. In the absence of SH labels, the dominant contribution to the signal is due to water dipoles aligned near the scatterer/water interface.^{10,32} When a surface electrostatic field is present in the interfacial region, water dipoles further away from the interface can be aligned by electrostatic interactions.^{1,33,34} This effect is responsible for an additional contribution to the SH signal as it produces additional centrosymmetry breaking. Both the electrostatic response and the surface response, where the latter is defined as the signal from the aqueous interfacial layer closest to the surface (with a thickness of a few water molecules), contribute to the total SHS signal. Because the optical beams associated with each contribution may not originate at the same point in the solution, the propagation of the beams in the solution and their corresponding phase variation may result in interferences. The penetration depth, or probing depth, of the technique will, therefore, be modulated by these interference effects originating from water molecules oriented by either the surface or the field. Previous studies have investigated these effects on planar surfaces^{4,35–38} and colloidal particles.^{10–12,39} Our group has derived a theoretical expression in the scattering geometry allowing for the determination of the probing depth as a function of ionic strength.^{40} While the formalism has been established, several factors need to be taken into account when practical experiments are performed. In the case of the scattering geometry, how are interference effects depending on the diameter of the nano- or micro-sized objects? How are the surface charge density of the studied material and the composition of the aqueous environment influencing the probing depth? While the effect of particle size on the total SHS signal has been previously examined,^{27,41,42} the effect of the size, of the material and of the electrolyte, have yet to be addressed on the individual surface and electrostatic contributions. To answer the above questions, a detailed analysis combining the geometric effects and the surface properties needs to be performed. Here, we first review the theoretical expression for the geometrical form factors contributing to nonlinear scattering^{43–45} and the expression of the SH signal intensity in the scattering configuration derived in our previous work.^{40} The nonlinear form factor functions are size- and scattering angle-dependent and represent the nonlinear scattering efficiency of an object of a certain size. We then calculate geometrical form factors associated with the surface response and with the electrostatic response as a function of the scattering angle for three different particle diameters (100, 200, and 300 nm), allowing for a direct comparison of the size effect on each form factor function. We further examine the effect of particle size directly on the shape and on the relative intensity of the SHS patterns by calculating the ratio of the electrostatic to the surface response as a function of the scattering angle for the three different sizes. This ratio does not include any specific material’s properties and, therefore, represents the effect of inserting an object of a certain size in an aqueous solution of defined ionic strength. We then apply our theoretical predictions to a case study on 100, 200, and 300 nm diameter SiO_{2} particles in NaCl and NaOH solutions. We show that the competition between the electrostatic response and the surface response, which influences the total AR-SHS intensity, is weighted by the material’s surface properties. Finally, we provide an area chart to predict the dominating response (whether the surface or electrostatic response) for arbitrarily-sized objects in an aqueous solution.

## II. THEORETICAL BACKGROUND OF AR-SHS

In a non-resonant AR-SHS experiment, a femtosecond laser at frequency *ω* is focused in an aqueous solution containing nano- or microparticles, inducing oscillating molecular dipole moments emitting a frequency component 2*ω*. This emission will cancel out for centrosymmetrically distributed molecular dipoles, but not in the case of noncentrosymmetrically-distributed ones, such as molecular dipoles oriented at the particle surface (provided the particle size is on the order of the wavelength of the SH light or a fraction of it).^{6} As discussed in the introduction, we consider here the oriented molecular dipoles to correspond to water molecules in the interfacial region, as interfacial water molecules largely outnumber surface groups. The sum of all the generated molecular dipoles is represented by a macroscopic parameter, the second harmonic polarization $P22\omega $,^{33,39,46}

where *ɛ*_{0} is the vacuum permittivity, $E\omega $ is the incoming electric field with frequency *ω*, and $\chi s2$ is the second-order surface susceptibility, quantifying the SH response of the medium to the electric field. Early studies showed that, at charged interfaces, there could be an additional contribution to the SH signal, namely an effective third-order contribution $\chi 3\u2032$.^{1,33} The effective third-order polarization is then

where **E**_{DC} represents the electrostatic field generated by surface charges, which extends into the bulk liquid. The effective third-order susceptibility $\chi 3\u2032$ mainly results from the polarization of bulk water molecules in the presence of a charged interface. Indeed, **E**_{DC} contributes to breaking the centrosymmetry of bulk water by orienting the molecular dipoles along the electrostatic field. $\chi 3\u2032$ is representative of all processes that lead to emission at 2*ω* and that require interaction with **E**_{DC}; as such, it is not a pure third-order process and is, therefore, stated as an effective third-order, indicated with a prime.

The static electric field **E**_{DC} is a function of *r*, which is the direction normal to the surface. It is connected to the surface potential Φ_{0} with respect to the bulk water or bulk electrolyte by integration along *r*, given as $\Phi 0=\u222br+\u221eEDCrdr$. Therefore, the total scattered SH intensity is

Equation (3) can also be rewritten in terms of the electric field contributions. In this case, $I2\omega $ is proportional to the squared modulus of the sum of the surface response $E22\omega $ and the electrostatic response $E3\u20322\omega $

where $n2\omega $, *ɛ*_{0}, and *μ*_{0} are the refractive index, vacuum permittivity, and vacuum permeability, respectively.

In the scattering geometry, within the Rayleigh–Gans–Debye approximation, one can define an effective particle susceptibility **Γ** that combines the geometric properties of the scatterer together with the incoming electromagnetic field and either the second-order surface susceptibility $\chi s2$ or the third-order effective susceptibility $\chi 3\u2032$. In this context, Eq. (4) can be expressed as

where $\Gamma 2$ and $\Gamma 3\u2032$ are the effective second- and third-order susceptibility, respectively. From Eqs. (4) and (5), we have $E22\omega 2\u221d\Gamma 2R,\chi s2,\theta 2$ and $E3\u20322\omega 2\u221d\Gamma 3\u2032R,\chi 3\u2032,\theta \Phi 02$. It can be further derived that the scattered intensity $I2\omega $ from a sphere or shell in the two independent polarization combinations PPP and PSS, normalized by the bulk water signal, can analytically be expressed as^{40,47}

*P* corresponds to a light polarization parallel to the scattering plane, while *S* corresponds to a light polarization perpendicular to the scattering plane. The first letter corresponds to the polarization of the outcoming second harmonic beam, while the two last letters correspond to the polarization of the incoming excitation beam. $\beta \u0304(2)$ is the effective averaged hyperpolarizability of water molecules, *N*_{p} is the density of particles, and *N*_{b} is the density of the bulk water (3.34 × 10^{28} molecules/m^{3}), so that *N*_{b}/*N*_{p} is the number of bulk water molecules per particle. *θ* is the scattering angle, which is defined as the angle between the incoming and outgoing wavevectors of the electrostatic field. The effective susceptibility elements $\Gamma 12$, $\Gamma 22$, and $\Gamma 23\u2032$ are going to be defined in the following. Note that we have omitted the scattering angle-dependent proportionality coefficients present in Eq. (6) [that is, $cos\theta 23$, $cos\theta 2$, and $2\u2061cos\theta +1$] from Eq. (5) for simplicity.

Having defined that the total SH signal intensity is dependent on both the surface response and the electrostatic response, which are represented by the effective susceptibilities $\Gamma 2$ and $\Gamma 3\u2032$, respectively, in Eq. (5), we will now discuss in more details the expression of the effective susceptibilities to address the size dependence of both $\Gamma 2$ and $\Gamma 3\u2032$. The tensor $\Gamma 2$ is defined as^{40}

where **q** is the scattering wave vector, *ijk* is the coordinate system with respect to **q**, and $\chi s,abc2$ is the second-order surface susceptibility of the spherical scatterer with the polarizations of the electric fields written in spherical coordinate *abc*. The integration is performed by summing up the phases of all the SH photons generated on the surface Ω of the scatterer. $\chi s,abc2$ is a tensor with 27 elements; because of the isotropy in the lateral dimensions of a particle surface, this number reduces to four, leaving only the components $\chi s,12,\chi s,22,\chi s,32$, and $\chi s,42$,^{45} where the numbers in the subscript describe different combinations of *abc* (see supplementary material). Assuming non-resonant interactions and Kleinman symmetry, we obtain $\chi s,22=\chi s,32=\chi s,42$,^{40,45,48} and thus there are only two independent elements of the second-order surface susceptibility. By performing the geometrical transformation shown in Eq. (7), we can specify the relationship between the tensor elements of $\chi s2$ and $\Gamma 2$ as follows:

where the geometrical transformation form factors are $F1qR=2\pi R2isinqRqR2\u2212cosqRqR$ and $F2qR=4\pi R2i3sinqRqR4\u22123cosqRqR3\u2212sinqRqR2$, where *R* is the radius of the particle, $q=q$, and $\chi s,12\u2032\u2032$ and $\chi s,22\u2032\u2032$ are the second-order susceptibility tensor elements corrected for changes in the refractive index between the medium and the particle as indicated in the supplementary materials. From the relationships stated in Table S1 and assuming an orientationally broad water distribution^{49} (which makes $\chi s,12$ vanish), one can find that $\chi s,12\u2032\u2032=3np/nH2O4\u22121\chi s,22\u2032\u2032$, with *n*_{p} and $nH2O$ the refractive index of the particle and the water. In this case, Eq. (8) can also be written as

Here $a=3np/nH2O4\u22121$. We have used *n*_{p} = 1.46 as the refractive index of SiO_{2}. All the form factors and effective susceptibilities are summarized in Table I for clarity. For the effective third-order susceptibility, the tensor $\Gamma 3\u2032$ is defined as^{40}

The integration includes the effect of the electrostatic field E_{DC} following the definition given in Eq. (2). It sums up the phases of all SH photons generated due to the electrostatic response at the surface and farther away in the EDL. The subscript *abc*, ⊥ stands for the polarizations of the electric fields in the spherical coordinate system, with ⊥ referring to the direction of E_{DC}, which is normal to the particle surface. Applying symmetrical properties to $\chi 3\u2032$, we can get $\chi 13\u2032=0,\chi 23\u2032=\chi 33\u2032=\chi 43\u2032$ (the relevant polarization combinations can be found in the supplementary material). Since $EDCr=\u2212d\Phi rdr$, we take $\Phi r=\Phi 0Rre\u2212\kappa r\u2212R$,^{50} with *κ* being the inverse Debye length, as the functional form for the surface potential. As the analytical expression for $\Phi r$ only contributes several nanometers away from the surface, the exponential term can be used without loss of generality.^{40} The result of the integration in Eq. (10) gives us the expression of the only remaining independent component $\Gamma 23\u2032$,

where the geometrical form factor $F3qR,\kappa R=2\pi R2iqR\u2061cosqR+\kappa R\u2061sinqRqR2+\kappa R2$ and $\chi 23\u2032\u2032$ is the effective third-order susceptibility corrected for the changes in the refractive index (see supplementary material). Note that F_{3} depends on the Debye length.

Form factors q = scattering wave vector modulus, R = particle radius, and κ = inverse Debye length
. | Effective susceptibilities with $a=3np/nH2O4\u22121$ . |
---|---|

$F1qR=2\pi R2isinqRqR2\u2212cosqRqR$ | $\Gamma 12=a2F1\u22125F2\chi s,22\u2032\u2032$ |

$F2qR=4\pi R2i3sinqRqR4\u22123cosqRqR3\u2212sinqRqR2$ | $\Gamma 22=2F1+aF2\chi s,22\u2032\u2032$ |

$F3qR,\kappa R=2\pi R2iqR\u2061cosqR+\kappa R\u2061sinqRqR2+\kappa R2$ | $\Gamma 23\u2032=2\Phi 0F1+F3\chi 23\u2032\u2032$ |

Form factors q = scattering wave vector modulus, R = particle radius, and κ = inverse Debye length
. | Effective susceptibilities with $a=3np/nH2O4\u22121$ . |
---|---|

$F1qR=2\pi R2isinqRqR2\u2212cosqRqR$ | $\Gamma 12=a2F1\u22125F2\chi s,22\u2032\u2032$ |

$F2qR=4\pi R2i3sinqRqR4\u22123cosqRqR3\u2212sinqRqR2$ | $\Gamma 22=2F1+aF2\chi s,22\u2032\u2032$ |

$F3qR,\kappa R=2\pi R2iqR\u2061cosqR+\kappa R\u2061sinqRqR2+\kappa R2$ | $\Gamma 23\u2032=2\Phi 0F1+F3\chi 23\u2032\u2032$ |

The geometrical form factors connect $\Gamma 2$, $\Gamma 3\u2032$ and $\chi s2$, $\chi 3\u2032$ by considering the interference effect between photons that are generated at different positions on the surface or in the bulk of the EDL. Then further coordinate transformations have to be performed on $\Gamma 2$ and $\Gamma 3\u2032$ to transform the coordinate with reference to **q** to the coordinate of the incoming electric fields, which gives rise to the trigonometric functions in Eq. (6). The detailed coordinate transformations can be found in Ref. 51. In the following, we will analyze the angular dependence of $\Gamma 22$ and $\Gamma 23\u2032$ and their dependence on the particle size. Note that we neglect the contribution of $\Gamma 12$ in Eq. (6), as it contributes ten times less to the surface response than $\Gamma 22$. Additionally, throughout the text, we will use $\chi s,22$ and $\chi 23\u2032$ to represent the refractive index-corrected version of the relevant nonlinear susceptibilities for simplicity (omitting the double prime).

## III. MATERIALS AND METHODS

### A. Chemicals

Sodium hydroxide (NaOH, >99.99% trace metals basis, Sigma-Aldrich) and sodium chloride (NaCl, >99.999%, Sigma-Aldrich) were used as received. SiO_{2} colloids of different nominal sizes (100, 200, and 300 nm) were purchased from Polysciences/Bangs Laboratories Inc. The 100 nm diameter SiO_{2} nanoparticles were received in solution (Polysciences, catalog number 24041-10, 5.9% w/w, mean diameter reported by the manufacturer: 0.1 ± 0.03 *μ*m). The 200 nm diameter SiO_{2} nanoparticles were also received in solution (Bangs Laboratories Inc., catalog number SS02000, 10.26% w/w, mean diameter reported by the manufacturer: 0.177 *μ*m), while the 300 nm diameter SiO_{2} nanoparticles were received as a powder (Bangs Laboratories Inc., catalog number SSD2001, mean diameter reported by the manufacturer: 0.3 *μ*m). The measured hydrodynamic diameter was found to be, in some cases, slightly different from the nominal size and closer to the mean diameter reported by the manufacturer upon receipt (see Sec. III C).

### B. Sample preparation

All procedures described hereafter used ultrapure water (Milli-Q, Millipore, Inc., electrical resistance of 18.2 MΩ × cm). For 100 nm SiO_{2} samples, the purchased stock solution was sonicated for 10 min (35 kHz, 400 W, Bandelin) and vortexed for 2 min prior to usage. Then the stock was diluted in water to 0.5% w/w, sonicated again for 3 min, and vortexed for 2 min. To remove residual ions from the synthetic procedure, the 0.5% w/w solution was centrifuged for 10 min at 7800 rpm (5430R, Eppendorf). Then, 9 ml of the supernatant were removed, and the pellet was resuspended in the same volume of MilliQ water by vortexing and ultrasonication for 3–5 min. The conductivity of the washed solution was measured using a conductivity meter calibrated with the appropriate buffer solutions (HI 5522 pH/ISE/EC bench meter and HI 76312 conductivity electrode, Hanna Instruments) to ensure that the initial ionic strength was as low as possible (≤5 *μ*M). The SiO_{2} particle suspensions were further diluted to 0.05% w/w solutions containing the desired amount of NaOH or NaCl, corresponding to ∼2.3 × 10^{11} particles/ml and a total surface area of ∼1.2 × 10^{−2} m^{2}/ml, using these estimations the hydrodynamic diameter measured by dynamic light scattering (see Sec. III C). The pH or ionic strength of the solutions was adjusted with 0.01 M solutions of NaOH or NaCl. The samples without any additional ions added were labeled 5 *μ*M throughout the text to account for the presence of residual ions from the synthetic process and the presence of HCO_{3}^{−} ions from the dissolution of atmospheric CO_{2}. The actual ionic strength used in the fitting procedure is indicated in the supplementary material. Corresponding water references at the same pH and the same ionic strength were prepared for each SiO_{2} sample. All preparation steps and measurements were performed at room temperature.

To allow for comparison between AR-SHS patterns of different nanoparticle sizes, the particle concentration of both the 200 and 300 nm samples was adjusted to ensure a similar total surface area as the 100 nm sample. The preparation procedure for the 200 and 300 nm SiO_{2} samples was similar to the one for the 100 nm samples. For 200 nm samples, the stock solution was first diluted to 0.7% w/w and then to 0.07% w/w samples containing the desired amount of NaOH or NaCl. The corresponding particle concentration and total surface area are ∼1.3 × 10^{11} particles/ml and ∼1.2 × 10^{−2} m^{2}/ml. For 300 nm particles, 100 mg of SiO_{2} nanoparticles were first dispersed in 1 ml of ultrapure water, sonicated for 15 min, and then diluted to 10 ml with ultrapure water to prepare a 1% w/w solution. After being centrifuged and sonicated, particles were further diluted to 0.1% w/w, corresponding to a particle concentration of 3.3 × 10^{10} particles/ml and a total surface area of ∼1.0 × 10^{−2} m^{2}/ml. A slightly lower surface area was used in the case of the 300 nm particles with respect to the 100 and 200 nm samples in order to remain in the linearity range of the SH signal. Indeed, for each sample size, the linearity of the SH signal with particle concentration, indicating the absence of multiple scattering events, was ensured in separate dynamic light scattering experiments (i.e., the range of linearity of the scattered signal as a function of particle concentration was determined). A particle concentration of 0.05% w/w for 100 nm particles, 0.07% w/w for 200 nm particles, and 0.1% w/w for 300 nm particles fulfilled both the criteria of similar surface area as well as the absence of multiple scattering events.

### C. Sample characterization

For each sample, the particle size distribution was measured by dynamic light scattering (DLS), and the zeta potential (ζ) was measured by electrophoresis. The DLS measurements were performed at a wavelength of 633 nm, and the size was obtained as the Z-average parameter (i.e., the intensity-weighted mean hydrodynamic size) of the measurement in the backscattering configuration (174°). The intensity-weighted mean hydrodynamic diameters measured by DLS (Zetasizer Ultra, Malvern) for 100, 200, and 300 nm samples were ∼120, 177, and 310 nm, respectively (for details, see the tables in Results and Discussions), with a narrow distribution [for most samples, the polydispersity index (PDI) was <0.1]. Electrophoretic mobilities were converted to ζ using Ohshima’s approximation^{52} as already discussed in our previous studies.^{53,54} The pH of the samples was measured using a pH meter (HI5522 pH/ISE/EC bench meter and HI1330 pH electrode, Hanna Instruments) calibrated with the appropriate buffer solutions. The conductivity of the solution was obtained using the conductivity meter mentioned in Sec. III B. From the measured conductivity σ, the average ionic strength, which is equivalent to the ionic concentration for monovalent ions, can be calculated as shown in our previous work.^{32,48}

### D. AR-SHS measurements

Second harmonic scattering measurements were performed on the same AR-SHS setup as described in Ref. 31. In an AR-SHS measurement, a 1032 nm fundamental beam is generated by a mode-locked Yb:KGW laser (Pharos-SP, Light Conversion) with a 190 fs pulse duration and a 200 kHz repetition rate. The polarization of the fundamental beam is controlled by a Glan–Taylor polarizer (GT10-B, Thorlabs) and a zero-order half-wave plate (WPH05M-1030) to be either horizontal (P, parallel to the scattering plane) or vertical (S, perpendicular to the scattering plane). The beam is further filtered using a long-pass filter (FEL0750, Thorlabs) and then focused into the cylindrical glass cuvette containing the sample (LS Instruments, 4.2 mm inner diameter) with a plano-convex lens (f = 7.5 cm). The beam power at the sample was set to 62 mW, corresponding to a fluence at the focus of ∼3.4 mJ/cm^{2}. The 516 nm SH signal is scattered from the SiO_{2}/water interface, collected and collimated with a planoconvex lens (f = 5 cm), polarization-analyzed by a Glan-Taylor polarizer (GT10-A, Thorlabs), and filtered by a 516 ± 10 nm filter (CT516/10bp, Chroma) before being focused into a gated photomultiplier tube (H7422P-40, Hamamatsu). The acceptance angle was set to 3.4° for scattering patterns. Patterns were obtained in steps of 5° from θ = −90° to θ = 90° with 0° being the forward direction of the fundamental beam. The signal was acquired with a gated photon counter (SR400, Stanford Research Instruments). The acquisition time of the photon counter was set to 1.5 s. Each data point was recorded as an average of 20 measurements. To correct for incoherent hyper-Rayleigh scattering (HRS) from the solvent phase, both the SHS response from the sample solution *I*_{PPP}, _{sample}(*θ*) and the HRS response from a solution *I*_{PPP}, _{solution}(*θ*) of identical ionic strength but without nanoparticles are collected. The HRS is subtracted from the SHS signal of the sample, and the obtained difference is then normalized to the isotropic SSS signal of pure water to correct for differences in the beam profile on a day-to-day basis,

Here, the normalized signal of the sample *S*(*θ*)_{PPP} is given for AR-SHS in the PPP polarization combination. The normalization procedure was applied in the same way for AR-SHS measured in the PSS polarization combination.

## IV. RESULTS AND DISCUSSION

### A. Theoretical part: The angular dependence of the geometrical form factors and of the ratio of the pure electrostatic to surface contributions and the influence of the particle size

Figures 1(a)–1(c) plot the calculated geometrical form factors as a function of the scattering angle for three different particle diameters (100, 200, and 300 nm): F_{1} (gray line), aF_{2} + 2F_{1} (purple line), and F_{3} + F_{1} at 5 *μ*M (blue line). These nonlinear form factor functions are a measure of the nonlinear scattering efficiency of the second-harmonic wavelength by an object of a certain size. As detailed in Sec. II, the form factor aF_{2} + 2F_{1} contributes to the surface response [Eq. (9)]. aF_{2} + 2F_{1} increases with increasing particle diameter, although with a different angular dependence. On the other hand, the form factor F_{3} + F_{1} contributes to the electrostatic response [Eq. (11)]. While its maximum increases with increasing particle diameter, the values at large scattering angles decrease significantly with size. With increasing particle diameter, aF_{2} + 2F_{1} prevails over F_{3} + F_{1}, and the relative differences between aF_{2} + 2F_{1} and F_{3} + F_{1} decrease at small scattering angles while they increase at large scattering angles.

Figures 1(d)–1(f) plot the geometrical form factor F_{3} + F_{1} [Eq. (11)] normalized to the maximum value of F_{1} as a function of scattering angle for different ionic strengths and different particle diameters. This normalized form factor decreases with increasing ionic strength, as expected for shorter Debye lengths. The normalization helps to visualize that the relative changes between different ionic strengths decrease with increasing particle size. It is evident from Fig. 1 that the intensity and the angular response of both form factors, aF_{2} + 2F_{1} and F_{3} + F_{1}, are strongly dependent on the particle diameter. Their size dependence, together with the decrease in F_{3} + F_{1} with increasing ionic strength, will, therefore, influence the total AR-SHS signal [Eq. (5)].

As mentioned in Sec. II, in order to know how the form factors contribute to the AR-SHS pattern, an additional coordinate transformation with respect to the scattering angle *θ* needs to be performed to transform the coordinate of the nonlinear susceptibilities to that of the polarization of the incoming electric fields, given as the trigonometric functions shown in Eq. (6). To further examine the size effect on the AR-SHS patterns after this coordinate transformation and to compare between the surface and electrostatic contributions, we calculate the ratio of the pure electrostatic contribution at each angle, which we define here as $E3\u2032\theta 2$ divided by $\chi 23\u2032\u22c5\Phi 0$, and of the pure surface contribution at each angle, given by $E2\theta 2$ divided by $\chi s,22$, in one specific polarization combination. We define $RES\u2032\theta $ as the ratio of the pure electrostatic effect to the pure size effect,

We later omit the angular term $\theta $ for better readability of the equations. The division of the electric field contributions by $\chi 23\u2032\u22c5\Phi 0$ and $\chi s,22$ ensures that the ratio $RES\u2032$ is independent of the material’s properties and of its surface charge. It, therefore, allows us to discuss the effect of different particle diameters in solutions of varying ionic strength independently of the nature of the particle itself. This parameter separation is mathematically feasible because $\chi s,22$ and $\chi 23\u2032\u22c5\Phi 0$ can be found as separate coefficients in Eqs. (9) and (11) and they do not depend on the scattering angle. We calculated $RES\u2032$ in the PPP polarization combination for 100, 200, and 300 nm particle diameters and plotted the logarithm of $RES\u2032$ in Fig. 2. Plots for the PSS polarization combination have similar trends and are shown in the supplementary material.

It can be seen that for 100 nm particles, the ratio $RES\u2032$ decreases with increasing ionic strength. For 200 and 300 nm particles, $RES\u2032$ also decreases with increasing ionic strength in the forward scattering direction (0°–∼40°). At higher angles for these two sizes, the trend becomes more complicated, reflecting the trends of the geometrical form factor F_{3} + F_{1} in Fig. 1, where the values at large scattering angles and low ionic strength decrease significantly with size. The values of $RES\u2032$ as well as the relative difference between $RES\u2032$ for different ionic strengths drop going from 100 to 300 nm. All the values of $RES\u2032$ for 100 nm are above 1; for 200 nm the ratio for 1 mM ionic strength is almost 1, while that for 300 nm is even below 1. The trend of $RES\u2032$ with size reveals that the contribution of the pure surface response $E22/\chi s,22$ increases with increasing particle size, whereas the opposite is the case for the pure electrostatic response $E3\u20322/\chi 23\u2032\u22c5\Phi 0$. As previously mentioned for the form factors, apart from the size effect, there is also an angular dependence to the values of $RES\u2032$. For each size, the larger the angle, the smaller the ratio $RES\u2032$ (except for a very small increase above 80–85° for the 200 and 300 nm cases). This angular dependence shows that, at larger angles, the surface contribution increases while the electrostatic contribution decreases, which is consistent with Figs. 1(a)–1(c). This phenomenon is particularly remarkable for lower ionic strengths (≤100 *µ*M), while at higher ionic strengths (500 *µ*M and 1 mM), the ratio $RES\u2032$ has little to no angular dependence.

From Figs. 1 and 2, it can be concluded that the contribution due to pure electrostatic terms is stronger for smaller particle sizes and in the forward scattering direction (0°–40°) but will nonetheless always decrease with increasing ionic strength due to the screening effect of electrolyte charges. At larger particle sizes, surface terms will increasingly play a role. The total AR-SHS intensity signal will, therefore, be influenced by the competition between these two opposite effects, as it will be shown in Sec. IV B. We will additionally show that these effects are weighted by the material’s properties, particularly its surface characteristics, represented by the surface potential Φ_{0} and the surface susceptibility $\chi s,22$.

### B. Experimental part: The influence of surface properties and electrolyte solutions

#### 1. The case of NaCl

We now present a case study for 100, 200, and 300 nm SiO_{2} particles as a function of NaCl concentration. The measured AR-SHS patterns (PPP polarization combination) normalized to the isotropic SSS signal of pure water as well as the fitting curves obtained following Eq. (6) are shown in Figs. 3(a)–3(c). The resulting fitting coefficients Φ_{0} and $\chi s,22$ extracted by applying the procedure described in detail in Refs. 32, 40, and 48 are listed in Table II. The parameters used for the fitting are given in the supplementary material. The sample nominally called 5 *μ*M does not contain any intentionally added NaCl, but the effective ionic strength of the solution may be slightly higher due to residual ions from the synthetic procedure, hydrogenocarbonate ions originating from the dissolution of atmospheric CO_{2}, and possibly, at these very low ionic strengths, ions leaking from the quartz cuvette (see Sec. III and supplementary material).

(nm) . | Ionic strength (µM)
. | R (nm) . | ζ (mV)
. | Φ_{0} (mV)
. | $\chi s,2(2)$ (10^{−22} m^{2}/V)
. |
---|---|---|---|---|---|

100 | 5 | 65 ± 15 | −82 ± 74 | −138 ± 30 | −2.7 ± 0.8 |

100 | 61 ± 14 | −60 ± 38 | −66 ± 30 | −3.1 ± 0.3 | |

500 | 59 ± 13 | −55 ± 35 | −282 ± 160 | 7.0 ± 1.6 | |

1000 | 59 ± 14 | −65 ± 46 | −379 ± 400* | 7.4 ± 3.9 | |

200 | 5 | 90 ± 21 | −58 ± 42 | −103 ± 10 | −1.6 ± 0.1 |

100 | 88 ± 19 | −52 ± 35 | −3 ± 10 | −2.2 ± 0.1 | |

500 | 86 ± 19 | −41 ± 13 | −203 ± 30 | 3.9 ± 0.1 | |

1000 | 86 ± 20 | −34 ± 32 | −495 ± 20 | 6.6 ± 0.1 | |

300 | 5 | 160 ± 37 | −63 ± 32 | −44 ± 10 | −1.7 ± 0.1 |

100 | 149 ± 35 | −47 ± 30 | −18 ± 10 | 2.0 ± 0.1 | |

500 | 148 ± 31 | −42 ± 24 | −196 ± 30 | 3.3 ± 0.4 | |

1000 | 145 ± 34 | −39 ± 26 | −222 ± 30 | 3.3 ± 0.5 |

(nm) . | Ionic strength (µM)
. | R (nm) . | ζ (mV)
. | Φ_{0} (mV)
. | $\chi s,2(2)$ (10^{−22} m^{2}/V)
. |
---|---|---|---|---|---|

100 | 5 | 65 ± 15 | −82 ± 74 | −138 ± 30 | −2.7 ± 0.8 |

100 | 61 ± 14 | −60 ± 38 | −66 ± 30 | −3.1 ± 0.3 | |

500 | 59 ± 13 | −55 ± 35 | −282 ± 160 | 7.0 ± 1.6 | |

1000 | 59 ± 14 | −65 ± 46 | −379 ± 400* | 7.4 ± 3.9 | |

200 | 5 | 90 ± 21 | −58 ± 42 | −103 ± 10 | −1.6 ± 0.1 |

100 | 88 ± 19 | −52 ± 35 | −3 ± 10 | −2.2 ± 0.1 | |

500 | 86 ± 19 | −41 ± 13 | −203 ± 30 | 3.9 ± 0.1 | |

1000 | 86 ± 20 | −34 ± 32 | −495 ± 20 | 6.6 ± 0.1 | |

300 | 5 | 160 ± 37 | −63 ± 32 | −44 ± 10 | −1.7 ± 0.1 |

100 | 149 ± 35 | −47 ± 30 | −18 ± 10 | 2.0 ± 0.1 | |

500 | 148 ± 31 | −42 ± 24 | −196 ± 30 | 3.3 ± 0.4 | |

1000 | 145 ± 34 | −39 ± 26 | −222 ± 30 | 3.3 ± 0.5 |

As can be seen from the experimental patterns in Figs. 3(a)–3(c), the AR-SHS intensity decreases with increasing salt concentration, no matter the particle size, although for the smallest particles (100 nm) at high concentrations (500 *μ*m and 1 mM), the difference is very small due to the low signal-to-noise ratio. The situation is now more complex than in the case of the form factors, as we now need to take into account the effect of the particle’s surface characteristics. In order to further visualize the material- and electrolyte-specific effects, we decompose the total SH signal intensity I_{tot} (I_{tot} being the result from the fitting procedure applied to the experimental patterns in Fig. 3) into the intensities associated with the surface contribution $E22$ (dotted lines) and the electrostatic field contribution $E3\u20322$ (dashed lines), as described in Eq. (4). Figures 3(d)–3(f) display a low salt concentration case (5 *μ*M), while a higher salt concentration case (1 mM) is presented in Figs. 3(g)–3(i). It is important to note that here $E22$ and $E3\u20322$ are weighted by the experimental values of the surface potential Φ_{0} and of the surface susceptibility $\chi s,22$, which are representative of the material’s surface properties in a specific electrolyte solution. The values of Φ_{0} and $\chi s,22$ extracted from the fitting procedure are listed in Table II. Note that the computed patterns for the individual contributions do not add up because we plot the intensities of the bulk and surface contributions while omitting the cross product, which only affects I_{tot}.

From the calculated patterns [Figs. 3(d)–3(i)], one can see that when the particles are in low ionic strength conditions, e.g., 5 *µ*M, intensities from the surface contribution and the electrostatic contribution are constructively interfering: $Itot>E22$ and $Itot>E3\u20322$. For SiO_{2} particles at 5 *µ*M, $E3\u20322$ is dominant for 100 nm particles, in agreement with the trends observed for the form factors aF_{2} + 2F_{1}, F_{3} + F_{1}, and the ratio $RES\u2032$. However, the $E3\u20322$ component decreases going from 100 to 300 nm (the difference between 100 and 200 nm is very small on the scale of the graph and is more pronounced at larger angles). Knowing that F_{3} + F_{1} at 5 *µ*M increases with particle size (see Fig. 1), the decrease in $E3\u20322$ can be explained by the decrease in Φ_{0} from 100 to 300 nm (at 5 *µ*M, −138 mV for 100 nm particles, −103 mV for 200 nm particles, and −44 mV for 300 nm particles). Additionally, the shape of the $E3\u20322$ contribution for the 300 nm particles reflects the very small contribution of F_{3} + F_{1} at large angles (>40°).

On the other hand, we observe an increase of the $E22$ contribution with size [panels (d)–(f)], in agreement with the trends of aF_{2} + 2F_{1}. However, at first, this seems to be in contradiction with the lower values of $\chi s,22$ for larger-diameter particles (at 5 *µ*M, −2.7 · 10^{−22} m^{2}/V for 100 nm particles, −1.6 · 10^{−22} m^{2}/V for 200 nm particles, and −1.7 · 10^{−22} m^{2}/V for 300 nm particles). Here, one has to examine more closely the ratio $RES\u2032\theta $. From Fig. 2, we obtain that the ratio in the PPP polarization combination $RES\u2032=E3\u2032/(\Phi 0\u22c5\chi 23\u2032)E2/\chi s,222=E3\u2032\u22c5\chi s,22E2\u22c5(\Phi 0\u22c5\chi 23\u2032)2$ is close to 10 at 20° for a 300 nm sample in a 5 *µ*M salt solution. Rearranging, Eq. (13) becomes $E3\u2032E22=RES\u2032\theta \u22c5\Phi 0\u22c5\chi 23\u2032\chi s,222$. Because the dominant $E22$ component noticeable in Fig. 3(f) indicates that $E3\u2032E22<1$, the ratio $\Phi 0\u22c5\chi 23\u2032\chi s,222$ must be small enough to compensate for a large $RES\u2032$—in other words, the effect of $\chi s,22$ must be large and a small decrease in its value with increasing size will not significantly influence the ratio, given the fact that Φ_{0} is also decreasing with increasing size. We conclude that at low ionic strength, the decrease in $E3\u20322$ with size is compensated by a larger increase in $E22$ with size, therefore, explaining why the total signal I_{tot} increases going from 100 to 300 nm at 5 *µ*M.

At higher ionic strength (1 mM), the F_{3} + F_{1} factor is expected to have a smaller contribution than at 5 *µ*M due to charge screening, which explains the smaller $E3\u20322$ contribution at 1 mM for 100 nm particles [compare panels (d)–(g)]. Despite the value of Φ_{0} being relatively high for the 1 mM case, the weighting by $\Phi 0\u22c5\chi 23\u2032\chi s,222$ is not sufficient to compensate for the decrease in the pure electrostatic contribution for this size. On the other hand, both the 200 and 300 nm particles shows a stronger $E3\u20322$ contribution at 1 mM than at 5 *µ*M [compare panels (e)–(h) and (f)–(i)]. This can be explained both by the fact that the relative decrease in F_{3} + F_{1} with ionic strength decreases with increasing size [see Figs. 1(d)–1(f)] and by the weighting by $\Phi 0\u22c5\chi 23\u2032\chi s,222$, which is larger at 1 mM than at 5 *µ*M due to the large increase in Φ_{0} for the 200 and 300 nm samples (Table II). This large increase in Φ_{0} is related to the formation of a condensed layer of hydrated ions at the surface; for an in-depth discussion of the physical meaning of Φ_{0} and its trend with ionic strength, see Ref. 48. We also observe that the $E22$ contribution is larger for the 200 and 300 nm particles with respect to the 100 nm ones, highlighting the fact that the pure surface contribution plays a major role in the $E22$ contribution despite the lower values of $\chi s,22$ for larger-diameter particles. Contrarily to the 5 *µ*M case, this time $E22>E3\u20322$ for all sizes, indicating that the $\Phi 0\u22c5\chi 23\u2032\chi s,222$ ratio is small enough and can compensate for the cases in which $RES\u2032$ is slightly above 1 (100 and 200 nm particles at 1 mM, Fig. 2). Another notable effect is the fact that, while the $E22$ component is dominant at all sizes, the total signal I_{tot} is much smaller than at lower ionic strengths. This destructive effect is related to inversion in the sign of the surface susceptibility (see Table II). In the patterns displayed in Figs. 3(g)–3(i), this flip in sign corresponds to a destructive interference between $E22$ and $E3\u20322$, and thus $Itot<E22orE3\u20322$ [see Eq. (4)]. Previous measurements performed in our laboratory showed that the change in sign of the surface susceptibility $\chi s,22$ indicates a flip in the average orientation of interfacial water molecules due to the formation of a layer of hydrated counterions close to the surface.^{32,48} The change in sign as well as increase in $\chi s,22$ at higher ionic strength will then generate the dominant contribution to the signal; however, this contribution is now directly destructively interfering with the electrostatic one, therefore, explaining the very low total signal I_{tot} at 1 mM.

An analogous analysis can be applied to the patterns measured in the PSS polarization combination, whose trends appear similar to those in the PPP polarization combination. The AR-SHS patterns in the PSS polarization combination are presented in the supplementary material.

#### 2. The case of NaOH

We finally treat the case of 100, 200, and 300 nm SiO_{2} particles as a function of NaOH concentration (pH 7, 10, and 11). Figures 4(a)–4(c) show the measured AR-SHS patterns (PPP polarization combination) as well as the corresponding fitting curves according to Eq. (6). No NaOH was added to the sample at pH 7, which is an identical case to the 5 *µ*M sample presented in Fig. 3. The samples at pH 10 and 11 only contain NaOH and, therefore, have an ionic strength of 100 *µ*M and 1 mM, respectively. We find that the relative intensity at pH 10 and pH 11 increases with size, and a crossover in the order of the patterns for the largest size—the 300 nm particles—can be observed.

Figures 4(d)–4(f) show the total SH signal intensity I_{tot} from the fitting procedure, the decomposed surface contribution $E22$ (dotted lines) and the electrostatic field contribution $E3\u20322$ (dashed lines) for the SiO_{2} samples at pH 11. Between samples in NaCl solutions (case 1) and NaOH solutions (case 2), the geometric form factors aF_{2} + 2F_{1} and F_{3} + F_{1} remain the same when comparing similar ionic strengths and similar sizes, as they only depend on the size and the ionic strength. Therefore, to explain the increasing relative intensity and the crossover in the AR-SHS patterns, we must consider the weighting effect of $\chi s,22$ and Φ_{0}. We compare the obtained parameters between the 300 nm NaCl and NaOH samples (shown in Tables II and III, respectively) and find that surface potentials are also quite similar within error in both cases (−222 mV at 1 mM and −204 mV at pH 11). However, the increase in $\chi s,22$ values at higher ionic strength is larger for NaOH (5.3 ± 0.4 · 10^{−22} m^{2}/V) than for NaCl (3.3 ± 0.5 · 10^{−22} m^{2}/V). The larger $\chi s,22$ values are responsible for the larger $E22$ contribution, as shown in Figs. 4(d)–4(f).

(nm) . | pH . | R (nm) . | ζ (mV)
. | Φ_{0} (mV)
. | $\chi s,2(2)$(10^{−22} m^{2}/V)
. |
---|---|---|---|---|---|

100 | pH 7 | 65 ± 15 | −87 ± 41 | −139 ± 20 | −2.4 ± 0.7 |

pH 10 | 60 ± 15 | −56 ± 46 | −8 ± 40 | 5.2 ± 0.2 | |

pH 11 | 58 ± 13 | −41 ± 48 | −226 ± 190 | 7.4 ± 1.7 | |

200 | pH 7 | 90 ± 21 | −56 ± 41 | −89 ± 10 | −2.2 ± 0.1 |

pH 10 | 89 ± 21 | −70 ± 45 | −4 ± 10 | 4.2 ± 0.2 | |

pH 11 | 86 ± 19 | −58 ± 48 | −492 ± 40 | 9 ± 0.7 | |

300 | pH 7 | 153 ± 46 | −47 ± 31 | −39 ± 10 | −1.7 ± 0.1 |

pH 10 | 153 ± 32 | −55 ± 33 | −44 ± 10 | 3.5 ± 0.2 | |

pH 11 | 148 ± 35 | −56 ± 27 | −204 ± 60 | 5.3 ± 0.4 |

(nm) . | pH . | R (nm) . | ζ (mV)
. | Φ_{0} (mV)
. | $\chi s,2(2)$(10^{−22} m^{2}/V)
. |
---|---|---|---|---|---|

100 | pH 7 | 65 ± 15 | −87 ± 41 | −139 ± 20 | −2.4 ± 0.7 |

pH 10 | 60 ± 15 | −56 ± 46 | −8 ± 40 | 5.2 ± 0.2 | |

pH 11 | 58 ± 13 | −41 ± 48 | −226 ± 190 | 7.4 ± 1.7 | |

200 | pH 7 | 90 ± 21 | −56 ± 41 | −89 ± 10 | −2.2 ± 0.1 |

pH 10 | 89 ± 21 | −70 ± 45 | −4 ± 10 | 4.2 ± 0.2 | |

pH 11 | 86 ± 19 | −58 ± 48 | −492 ± 40 | 9 ± 0.7 | |

300 | pH 7 | 153 ± 46 | −47 ± 31 | −39 ± 10 | −1.7 ± 0.1 |

pH 10 | 153 ± 32 | −55 ± 33 | −44 ± 10 | 3.5 ± 0.2 | |

pH 11 | 148 ± 35 | −56 ± 27 | −204 ± 60 | 5.3 ± 0.4 |

From a chemical perspective, the larger $\chi s,22$ is consistent with the presence of a higher proportion of deprotonated surface silanol groups upon addition of NaOH, which leads to a stronger surface contribution to the total AR-SHS intensity (at pH 11, one can expect a deprotonation slightly above 25%, based on the results of Brown *et al.*,^{55} who calculated a 25% surface deprotonation for silica particles at pH 10 in 0.1 M NaCl). Figure 5 shows a schematic view of a SiO_{2}/water interface for the high concentration of salt (left) and the high concentration of base (right). The mean orientation of water molecules is given by the net dipole moment of water molecules schematically marked as the blue arrow. As shown by the figure, the NaOH case has a larger net dipole moment compared with the NaCl case. The sign of the dipole moment is indicated in agreement with Tahara^{56} and our previous work.^{32,48} This larger net dipole moment can be rationalized by a larger number of water molecules being oriented with their hydrogens toward the surface within the surface and the compact layer of hydrated counterions. The mechanism at the origin of this reorientation is likely to be associated with a change in the structure of the H-bond network in the very first interfacial layers of water. Based on this discussion, while often the decrease in the AR-SHS signal intensity with increasing ionic strength can be attributed to a lower number of aligned water molecules in the EDL due to charge screening,^{48,57} this is only true for specific sizes and specific surface charge densities. At larger sizes and for higher surface charge densities, the signal originating from the orientation of water molecules in the very first layers close to the surface may dominate the total AR-SHS signal intensity.

### C. Predicting the dominating contribution for an arbitrarily-sized particle sample

In the area chart shown in Fig. 6, the solid lines represent equal contributions from the surface response and the electrostatic response at the scattering angle of 40°, that is, $E3\u2032E22=RES\u203240\xb0\u22c5\Phi 0\u22c5\chi 23\u2032\chi s,222=1$. The y axis represents the change of the ratio $\Phi 0\u22c5\chi 23\u2032\chi s,22$, which describes the material’s properties at the interface, while the x axis represents the particle radius, considering $RES\u2032$ is size dependent (see Fig. 2). The dependence on ionic strength of $RES\u2032$ is illustrated by different area colors: 5, 100 *µ*M, and 1 mM are shown as examples. The solid line separates the area where the electrostatic response dominates (above the line) from the area where the surface response dominates (below the line, filled with specific patterns as shown in the legend). From the plot, it can be seen that for each ionic strength, the value of $\Phi 0\u22c5\chi 23\u2032\chi s,22$ on the curve increases with the increase of particle radius. This indicates that particles with a larger size are more likely to have a dominating surface response, while particles with a smaller size are more likely to have a dominating electrostatic response. Furthermore, for particle radius <160 nm, a larger surface response dominating area can be expected with increasing ionic strength due to the relative decrease of the electrostatic response. This map can be used to predict the dominating contribution of a particle sample with a particular size and particular surface properties at a specific ionic strength. We indicate our experimental data for 100, 200, and 300 nm SiO_{2} colloidal silica samples with no added ions, 100 *µ*M and 1 mM NaCl, as well as 100 *µ*M and 1 mM NaOH, respectively, on the map. We take half of the measured hydrodynamic diameter as the particle radius of the sample (see Tables II and III). For samples with no ions added (green squares), only the data point of the 300 nm sample falls into the surface response dominating area (green, 5 *µ*M), which is consistent with the decomposition shown in Figs. 3(d)–3(f). For samples with 1 mM NaCl (red square) and 1 mM NaOH added (open red triangles), all the experimental data points fall into the surface response dominating area (red dotted, 1 mM). Additionally, the 1 mM NaOH data points always display smaller values of $\Phi 0\u22c5\chi 23\u2032\chi s,22$ than the 1 mM NaCl data points for all three sizes, with the largest difference for the 300 nm sample. This is consistent with the larger effect of $\chi s,22$ associated with a higher proportion of deprotonated surface silanol groups, which increases with increasing particle size. Note that an angle of 40° is selected here because the ratio $E3\u2032E22$ at this angle can nearly reflect that of the whole pattern. However, since the contributions are angular dependent, the result may change for larger or smaller angles.

## V. CONCLUSION

In this work, we focused on the experimental parameters influencing the probing depth of second harmonic generation in the scattering geometry. The angle-resolved second harmonic scattering (AR-SHS) signal intensity is modulated by the interference between surface- and electrostatic-induced contributions. These contributions will largely depend on the size of the scatterer and on the chemical nature of the surface, together with the composition of the aqueous environment. First, we theoretically show how a particle diameter increasing from 100 to 300 nm modifies the nonlinear scattering form factors contributing to the surface and electrostatic response. The intensities of both the form factor for the surface term ($aF2+2F1$ and the one for the electrostatic term (F_{3} + F_{1}) are size dependent. With increasing particle size, their maximum values increase, with eventually aF_{2} + 2F_{1} prevailing over F_{3} + F_{1}; for the latter, while the normalized function decreases with increasing ionic strength, the relative changes between different ionic strengths decrease when particle size increases. To further examine the size effect on the AR-SHS patterns, we define $RES\u2032$, which is the angular-dependent ratio of the pure electrostatic contribution to the pure surface contribution, that is, the ratio expression is independent of the material’s properties and of its surface charge. Similarly, we observe that $RES\u2032$ decreases with increasing size. For most ionic strengths, $RES\u2032$ also decreases with increasing angle; this phenomenon is particularly remarkable for lower ionic strengths (≤100 *µ*M). Together with the trend of the form factors, it reveals that the contribution due to electrostatics is the strongest for smaller particle sizes (e.g., 100 nm diameter) and in the forward scattering direction (0°–40°). In other words, the AR-SHS signal from smaller particles is due in large part to water dipoles aligned farther away from the surface by the surface electrostatic field. Nevertheless, such pure electrostatic contribution always decreases with increasing ionic strength due to the screening effect of electrolyte charges. On the other hand, at larger particle sizes (e.g., 300 nm diameter), the pure surface contribution will increasingly play a role and provide signal intensity at larger scattering angles (20°–60°). In a molecular-level picture, this indicates that the AR-SHS signal may here be due to a large extent to the very first interfacial water layers, which are likely chemically associated with the surface, such as with H-bonds. In practice, the total AR-SHS intensity signal will be influenced by the competition between these two opposite effects at different sizes, and whether one or the other dominates will be dictated by the material’s surface properties, represented by the surface potential Φ_{0} and the second-order surface susceptibility $\chi s,22$, which is a measure of interfacial water orientation.

This weighting effect of the material’s surface properties is illustrated by a case study for 100, 200, and 300 nm SiO_{2} particles as a function of NaCl concentration and as a function of NaOH concentration. The weighting effect is defined by the ratio $\Phi 0\u22c5\chi 23\u2032\chi s,222$. We show that a dominating surface contribution corresponds to this ratio being small enough to compensate $RES\u2032\theta $, which requires either a larger particle size (smaller $RES\u2032\theta $) or a larger $\chi s,22$. We also observe that the sign inversion of $\chi s,22$ at higher ionic strengths, representative of a change in the average orientation of the interfacial water, results in a destructive interference between the surface contribution and the electrostatic one. The weighting effect of $\Phi 0\u22c5\chi 23\u2032\chi s,222$ is well reflected by the difference of the AR-SHS patterns between the NaCl and NaOH cases. Because of the larger $\chi s,22$ due to the presence of a higher proportion of deprotonated surface silanol groups, the NaOH samples have a stronger surface contribution, which results in a larger AR-SHS intensity than for the NaCl samples at the same ionic strength and for the same particle size.

Analysis of the factors influencing the AR-SHS signal can be effectively used to predict the dominating contributions for an arbitrarily-sized particle. In Sec. IV C, we propose an area chart to predict the dominating response (whether the surface or electrostatic) for an arbitrarily-sized objects in aqueous solution with the ratio $\Phi 0\u22c5\chi 23\u2032\chi s,222$ varying in a certain range. Such predictions allow for better connections between the behavior of the AR-SHS patterns at different ionic strengths, different electrolyte solutions, and the surface properties of a particle. The interpretation of the AR-SHS signal will assist in obtaining a better understanding of the structure of EDL and of oriented interfacial water molecules. The different angular dependences of the electrostatic and surface contributions open possibilities to determine the surface potential and the surface susceptibility individually.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the relevant constants in the AR-SHS model and theory, the experimental parameters used for fitting the AR-SHS patterns, the logarithm of the calculated $RES\u2032$ in the PSS polarization combination for 100, 200, and 300 nm particle diameters, and the AR-SHS patterns of 100, 200, and 300 nm diameter SiO_{2} particles in NaCl solutions as a function of ionic strength and in NaOH solutions as a function of pH.

## ACKNOWLEDGMENTS

This work was supported by the Julia Jacobi Foundation, the Swiss National Science Foundation (Ambizione Grant No. PZ00P2_174146 to A.M.), and the Marie Skłodowska-Curie Actions Innovative Training Network (Grant No. H2020-MSCA-ITN-2019, proposal 860592, PROTON) to B.C. and S.R. We thank Dr. Marie Bischoff for her help during the preliminary AR-SHS measurements and Dr. Alex G. F. de Beer for helpful discussions on the theoretical model.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

A.M. and S.R. conceived and designed the work. B.C. carried out the theoretical calculations and performed the SHS experiments with the help of A.M. All authors discussed the results and contributed to the manuscript.

**Bingxin Chu**: Formal analysis (lead); Investigation (lead); Writing – original draft (equal); Writing – review & editing (equal). **Arianna Marchioro**: Conceptualization (lead); Formal analysis (supporting); Funding acquisition (supporting); Investigation (supporting); Methodology (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). **Sylvie Roke**: Conceptualization (supporting); Funding acquisition (lead); Methodology (supporting); Supervision (supporting); Validation (supporting); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).

## DATA AVAILABILITY

The raw data sets generated and analyzed during the current study are available from the corresponding author upon reasonable request.