Impact of Solution {Ba2+}:{SO42–} on Charge Evolution of Forming and Growing Barite (BaSO4) Crystals: A ζ—Potential Measurement Investigation

The impact of solution stoichiometry on formation of BaSO4 (barite) crystals and the development of surface charge was investigated at various predefined stoichiometries (raq = 0.01, 0.1, 1, 10, and 100, where raq = {Ba2+}:{SO42–}). Synthesis experiments and zeta potential (ζ-potential) measurements were conducted at a fixed initial degree of supersaturation (Ωbarite = 1000, where Ωbarite = {Ba2+}{SO42–}/Ksp), at circumneutral pH of ∼6, 0.02 M NaCl, and ambient temperature and pressure. Mixed-mode measurement–phase analysis light scattering (M3-PALS) showed that the particles stayed negative for raq < 1 during barite crystal formation and positive for raq > 1. At raq = 1, two populations with a positive or negative ζ-potential prevailed for ∼2.5 h before a population with a circumneutral ζ-potential (−10 to +10 mV) remained. We relate the observations of particle charge evolution to particle size and morphology evolution under the experimental conditions. Furthermore, we showed that the ζ-potential became more negative when the pH was increased for every raq. In addition, our results demonstrated that the type of monovalent background electrolyte did not influence the ζ-potential of barite crystals significantly, although NaCl showed slightly different behavior compared to KCl and NaNO3. Our results show the important role of surface charge (evolution) during ionic crystal formation under nonstoichiometric conditions. Moreover, our combined scanning electron microscopy and ζ-potential results imply that the surface charge during particle formation can be influenced by solution stoichiometry, besides the pH and ionic strength, and may aid in predicting the fate of barite in environmental settings and in understanding and improving industrial barite (surface chemistry) processes.


I ---Appropriate Physicochemical Conditions for ζ-Potential Measurements
The initial Ωbarite was chosen carefully with regards to the accuracy of the M3-PALS measurements.Trialand-error optimization was used to find this initial Ωbarite (effectively defining what the expected potential particle concentration would be at equilibrium), which is a trade-off between minimizing aggregation and agglomeration conditions (i.e. a concentration too large) and the signal-to-noise ratio (i.e. a too low concentration).A concentration too large is accompanied with increasing blackening of the electrodes, affecting the field strength and the measurement adversely (i.e.drifting ζ-potential values), and the loss of measurable particles that became too large due to agglomeration and ultimately sank to the bottom of the cell.Clogston & Patri (2011) 1 mentioned that samples should be optically clear and not turbid for M3-PALS.
We believe that the used initial Ωbarite-value leads to an intermediate particle concentration that provided sufficient signal while minimizing aggregation and agglomeration.The challenge of investigating barite formation is the very low solubility (Ksp = 10 -9.99 ), which meant we had to work with a high degree of supersaturation to create enough material to obtain an appropriate signal-to-noise ratio.
Automatically, this brings questions to the prerequisite that the particles in solution should only be brought into motion by the applied electric field.Anything else that would affect the motion affects directly the apparent ζ-potential.That means that particle-particle interactions (i.e.agglomeration and long-lasting structuring) should have preferentially been minimized, so that particles could behave as if they were in an infinite medium/solvent.A high degree of supersaturation is therefore not ideal.However, this had to be balanced by the practical limits of the scattered light's detection, which was defined by the system we investigated.Most likely, aggregation took place before the initial ζ-potential measurements in each of our experiments based on our backsacttering detection angle (BSD) particle size observations (i.e.SI-X and SI-XI), but was most likely negligible during the measurements.Also, we estimated by DLVO calculations that agglomeration is negligible (SI-VI).In addition, long-lasting structuring of the particles was negligible (SI-VI).
Lastly, although CO2 dissolution is supposedly slow in alkaline environments, 2 we could not rule out that small amounts of CO2 may have been converted into CO3 2-in solutions nos.3.4 -3.6 (Table 1), which may

S2
have led to witherite (BaCO3) formation alongside barite.In that case, witherite has a different refractive index than barite (1.504 -1.529 compared to ~ 1.64) 3 and forms more needle-like (or acicular) structures comparable to that of aragonite (CaCO3) and has different crystallization kinetics than barite. 4Consequently, this may have led to two different types of crystals with a dissimilar ζ-potential, causing the interference as observed in Figure S16i.This would have been most prominent at raq = 100, where the relatively large concentration of Ba 2+ ions increased the ionic activity product (i.e.{Ba 2+ }{CO3 2-}) to form BaCO3.If the 'noise' was caused by co-existing witherite formation, then the noise disappeared in the second half of the experiment, because witherite was at least more than 2 orders of magnitude less supersaturated than barite.
Consequently, barite crystals were present at much larger amounts than witherite toward equilibrium and dominated the ζ-potential signal.However, our SEM results at pH = 10 (i.e. Figure 1j-l) showed that no witherite crystals were formed and our measured pH-values did not drop notably at those conditions, which would be expected in the case of witherite formation.Based on hydration and hydroxilation rates of CO2 at pH 10, 2 a maximum of 19% CO2 saturation was reached after 48 hours in our prepared growth solutions.Chemical speciation calculations show that a 19% CO2 saturation (i.e.19% of the current pCO2 of 400 ppm) leads to a slight supersaturation of witherite only for conditions of pH = 10 and raq = 100 (i.e. initial Ωwitherite = 4.3), while calculations showed undersaturation with respect to witherite for all other conditions.However, at such low supersaturation, nucleation of witherite is very unlikely and likely outcompeted by barite nucleation and growth.

II ---Evolution of Ionic Strength and Debye Length during Experiments
We have calculated the ionic strength at the initial and equilibrium conditions among the different predefined solutions, to check if that the ionic strength did not change significantly during our experiments.The Debye Length was calculated by the following equation: 5 where εr is the fluid's dielectric constant [-], ε0 the permittivity of free space [C 2 s 2 kg -1 m -3 ], kB the Boltzmann constant [m 2 kg s -2 K -1 ], T the absolute temperature [K], NA Avogadro's constant [mol -1 ], e the elementary charge [C] and I the ionic strength [M].The relationship between the Debye Length over the range 1 mM < I < 1 M is shown in Figure S1a. Figure S1b displays the change in I as BaSO4 formed during batch experiments.Figure S1c shows the resulting change in the Debye Length at the surface of the BaSO4 particles.The Debye length varied from ~2.1 nm to ~2.3 nm (Figure S1c).Therefore, our electrical double layer (EDL) stayed fairly constant throughout the experiment for each condition.When performing ζ-potential measurements during particle formation, it is important to validate if the Smoluchowski limit (i.e.f(κα) = 1.50) of Henry's function (i.e.Equation S2) can be used.Henry's function depends on the geometry of the particle and on the product between the particle size radius and inverse Debye length (κa). 6Furthermore, Henry's function is restricted in the sense that the (absolute) potential of the surfaces should be approximately less than 25 mV and the distortion of the electrical double layer, by retardation and relaxation, should be minimal during electrophoresis (i.e.strongly determined by I).Though it is expected that retardation of the (charged) particles movement would occur due to the ion-containing fluid surrounding the particles in our chemical solutions, this is corrected for in the Henry function by the measurement of electro-osmosis. 6Relaxation effects where ions have to constantly rearrange themselves due to the charge disbalance in the ion-containing fluid surrounding the particles of interest were considered negligible, as the Debye length in our investigated solutions is expectedly short (I = 0.02 M).

S4 S5
Nonetheless, we investigated how reasonable it is to assume that the Smoluchowski limit is always applicable for our measurements.To calculate at which particle sizes the Smoluchowski limit is valid, the following approximation proposed by Ohshima 7 for the original Henry equation has been used, because the original Henry equation has a hiatus for κa-values between 5 and 25: 8 Note that a = rp = 0.5dp = particle radius.Equation S2 leads to a deconvolved sigmoidal-like relationship between f(κa) and κa (Figure S3) and has a maximum of 1% relative error.The relationship in Figure S3 is only valid for | ζ | < 25 mV, because relaxation effects become more significant at higher ζ-potential values and affect the value for Henry's function substantially. 9In our experiments, where I is ~ 0.02 M, κa is ~ 0.07 (i.e.(0.3 nm/2) / 2.2) at dp = 0.3 nm, while κa is ~ 2300 at dp = 10000 nm. Figure S3 shows that we cannot

S7
apply the Smoluchowski limit over the entire dp range, (note that 4κα ~ dp (and 2κα ~ rp), because (dp / 2)(κ -1 ) ~ dp / 4) and that a different value of f(κα) should be considered in the interpretation of our results.However, our measurements in the forward detection angle (FWD) show dp-values ≥ 200 nm and Figure S3 shows that the Smoluchowski limit of the Henry function is accurate for dp ≥ 300 nm, for which f(κα) ≥ 1.45.Based on Figure S3, all ζ-potential values for particles > ± 100 nm are valid with the use of the Smoluchowski limit with a maximum error of 26.5% for f(κα) for particles of 100 nm.
Another challenge is the interpretation of the ζ-potential and what that means for the surface charge of the measured particles.The use of the Smoluchowski-limit of the Henry function comes with the assumption that particles behave as ideal hard spheres.The hard-sphere approach for ions is mostly used for inorganic mineral systems, where the anion and cation are represented as hard spheres and assumed to be in contact, although oversimplified by the fact that the size in the hard-sphere model is assumed to be constant with a defined charge and co-ordination number and considers that the attractive forces between anion and cation with identical opposite charge is constant. 10Lang & Smith (2010) 10 showed that in a soft-sphere model, where some overlap exists between the anion and cation, internuclear distances for Ba 2+ with most other monatomic anions match better with experimental values compared to the hard-sphere model.However, the choice for the hard-sphere model is still preferred as a more reliable estimate of the surface charge density can be made, especially when the heterogeneity is large with regards to the particle size (polydispersity) and differences in morphology of the particles among different experiments occur. 11

V ---Influence of Surface Conductivity on the Smoluchowski Limit of Henry's Function
Surface conductivity is also neglected with Henry's function.For I > 0.1 M, surface conductivity is usually negligible compared to the bulk water conductivity 9,12,13 and, therefore, the Dukhin number Du, which relates surface to bulk dispersant conductivity, is ~ 0. In this case, the measured or apparent ζ-potential is roughly equal to the ζ-potential corrected for surface conductivity.However, when Du >> 0.1, surface conductivity may play an important role and nonlinear electrokinetic phenomena need to be considered. 14,15Therefore, we calculated Du with a set of fixed ζ-potential values, Debye length, ionic strength, the translational diffusion coefficient and particle size (Equation S3).Du is defined as 16,17 : where κ σ is the surface conductivity and Cf the bulk fluid electrical conductivity.The surface conductivity κ σ is defined as 18 : where z is the ion valency, Ci the ionic concentration of the bulk fluid [M] and m the dimensionless electroosmosis contribution to the ion motion in the double layer 14 and is defined as: Note that the ion size is important for κ σ , as it determines the magnitude of DTrans and that the ionic strength influences the value of κ.Equation S3 -S5 assumes that the electrolyte is symmetrical, where the respective diffusion coefficients are equal.This is not entirely true (i.e.1.693x10 -9 and 2.130x10 -9 m 2 s -1 for Ba 2+ and SO4 2-respectively, according to Kadhim & Gamaj (2020), 19 but so far gives the best approximation.Dukhin number is lower than 0.05 (see also Figure 2b), which was the case for our rapidly developing systems.

VI ---Influence of Collision Particle-Particle Interactions based on DLVO-Calculations
Besides the importance of κa (SI-IV) and the surface conductivity (SI-V), Henry's function also ignores the Van der Waals interactions and electrostatic repulsion (relaxation effects).To include the relaxation effects, one could use the Modified Booth Equation, which evolved from the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, 23 and solves the ζ-potential numerically for any set of rp, ion valences, concentrations and limiting ion conductivities. 9The numerical solution uses 4 coupled equations that include the Henry function plus 3 relaxation correction functions.By using a 3 rd -order polynomial approach to describe these functions, a graph similar to Figure S3 is obtained with a regression coefficient greater than 0.997. 6,9Another way to obtain information about the importance of the relaxation effects in our systems is to use the extended-DLVO theory, which originates from the 90s [24][25][26] and describes the stability of suspensions via several established explicit analytical equations.Although, given the results of Deshiikan & Papadopoulos (1998), 6 this effect should not be significant in our case as our | ζ | is rarely is beyond 25 to 30 mV, for completeness we have investigated the relaxation effects with the use of the extended-DLVO theory.
According to the extended-DLVO theory, the key parameter is the total energy of interaction between two solid phases separated by a liquid medium (ΔGSLS).ΔGSLS is the sum of electrostatic forces (ΔGSLS,EL), van der Waals interaction (ΔGSLS,vdW) and hydration forces originating from the solvation of the solid (ΔGSLS,AB).
The latter can be understood as electron-acceptor/electron-donor polar interaction contributions due to hydrophilicity or hydrophobicity.8][29] This implies the following equation: If the ζ-potential has the same value between two distinct particles of the same size, then ΔGSLS, EL can be related to ζ-potential according to the following equation [30][31][32] :

S14
where with κ as the inverse of the Debye Length (Equation S1) and where H is the distance between the surfaces of two particles.

S15
where for P0 < 0.5, and for 0.5 < P0 < ∞, where λc in Equation S11 is the wavelength of intrinsic oscillations of atoms and typically has a value of 10 -7 m. 41 A in Equation S10 is the so-called Hamaker constant and is defined as 33,42,43 : where H0 is the minimum equilibrium contact distance between the BaSO4 particles, for which Gallardo et al.
(2000) 33 used a value of 1.58 Å and van Oss et al. (1990) 44 a value of 1.63 Å.If we use H0 = 1.58 Å, then we obtain A = 1.17x10 -20 kg m 2 s -2 .Such a value, between 10 -20 and 10 -19 kg m 2 s -2 (J) is expected for single material in solid-liquid systems. 42,45SLS, AB is calculated by the following equation 46,47 : where h0 is the correlation length of water molecules and varies between ~ 0.2 nm (for nonhydrogen bonded water molecules) 48 and 13 nm. 49A reasonable value is 1 nm. 25,33,50,51ΞAB in Equation S15 contains the acid/base components of the surface free energy of both the BaSO4 and UPW phases and is defined as: From the experimental values found by Gallardo et al. ( 2000) 33 (see their page 13), the interfacial energy (γ) between the UPW phase and BaSO4 phase is calculated by using Good's Equation 52: If we insert the values for γL,vdW, γL -, γL + ,γS,vdW, γS -and γS + , then we obtain a value of -22.6 mJ m -2 for γ.A value with that order of magnitude is quite common among oxides or ionic crystals, like BaSO4, immersed in water that only have electron donor properties. 53,54Consequently, it is easier for water to penetrate between those surfaces and hence, lead to repulsion between BaSO4 particles.This would suggest that the interaction energy ΔGSLS (Equation S4.1) should be rather low.Vijayabaskar & Vishveshwara (2010) 55 regarded values of 0 -5 kJ/mol for ΔGSLS as low energy interactions and ≤ -20 kJ/mol as high energy interactions.All in all, the consideration of low and high interaction energies by Vijayabaskar & Vishveshwara (2010) 55 and all the parameter values previously addressed, we are now able to solve Equation S7: where εr, ε0, κ, H0, γS,vdW, γL + , γL -, γS + , γS -and h0 are (assumed to be) constants, f(P0) is a semi-continuous function, ζ and rp are controlled variables, H the independent variable and ΔGSLS the dependent variable.
Figure S6a displays the contributions of ΔGSLS,EL, ΔGSLS,vdW and ΔGSLS,AB to ΔGSLS.while Figure S6b and Figure S6c illustrate the relationship between ΔGSLS and H for various rp and ζ-potential, respectively.
It is difficult to predict the average H in our suspensions, but existing literature in the field of electrokinetics describing definitions for dilute, semi-dilute and concentrated systems gives us an indication.Suspensions having a solid volume percentage of 10% or larger are often categorized in the concentrated regime (Posner (2009) 57 and references therein).8][59] Between solid volume percentages of 0.1% and 10%, the semi-dilute regime persists.Our investigated suspensions at initial Ωbarite = 1000, show a maximum solid volume percentage of ~ 0.013% at equilibrium, meaning that all investigated suspensions fall into the dilute regime.Whereas in concentrated and semi-dilute regimes the distances between particles are much smaller than the particle size and approximately in the range of the particle size, respectively, the distances between particles in the dilute regime is much larger than the particle size. 56,60,61

S18
According to Figure S6, the total surface free energy is negligibly negative for H > 30 nm, independent of ζpotential and rp.A simple equation for monodisperse spheres has been derived to estimate the average separation distance between two particles in suspension 62,63 : where øs is the solid volume percentage.Using a value for rp of 100 nm (smallest observed particle size during batch experiments in FWD) and 0.013 for øs (the largest equilibrium solid volume fraction we used), then we estimate that H is approximately 400 nm (i.e.413 nm).Larger rp would result in larger H. Huang et al. ( 2010) 64 showed, with the use of the extended-DLVO theory, that differences in crystal morphology may generally affect the particle-particle interactions more than the differences in surface roughness.However, in our case, that would not compensate enough such that the interparticle distance would be < 30 nm for particle sizes > 200 nm and would still lead to the conclusion that the Smoluchowski limit of the Henry function was evidently valid for our batch ζ-potential experiments.

S19
In summary, irrespective of high and low interaction energies, we considered the relaxation effects for our barite suspensions negligible as the calculated average separation distance is far from the position of the well (Figure S6).

VII ---Influence of Long-Lasting Particle-Particle Interactions
Besides the likelihood of agglomeration in our systems (SI-VI), we also calculated if the position of particles in solution are influenced by their neighboring particles (i.e.long-lasting particle-particle interactions due to packing) and determined therefore the structure factor.If the structure factor is not equal or close to unity, particles are not randomly positioned in the solution.Ultimately, this has an effect on the scattering signal during light scattering measurements. 65The absolute total scattering intensity Itot is defined as: where I0 is the incident beam intensity, Kc a constant and a factor related to the measurement set-up (i.e. it includes wavelength and scattering volume-detector distance among others) [m -6 ], N is the total number of particles, Vp the particle volume, P(q) the form factor of a single particle and S(q) the inter-particle structure factor.The 'contrast factor' is the difference of a specific physicochemical property between the solid and liquid (e.g. the difference in density (Δρ) betwe ) between the solid ρ) betwes and the liquid ρ) betwe l or the difference in refractive index (Δη) betwe ) between the solid η) betwe s and the liquid η) betwe l).Note that when P(q→0) = 1, P(q→∞) = 0 and S(q→∞) = 1. 66In Equation S20 (and the following ones), q is the wave vector and is defined as: where η) betwe l is the refractive index of the medium, λ0 the wavelength of the light in the medium and θ the angle of light diffraction.
The shape factor P(q) is defined as (i.e. in the Porod-regime): where rp is the particle radius.When P(q→0) = 1 and the gyration radius is used (i.e.Rg 2 = 3rp 2 /5), then the Guinier approximation applies: The Ornstein-Zernike integral equation 67 is used to describe the inter-particle interactions and is defined as: where c(rid) represents direct interactions among two particles and g(rid) indirect interactions by other particles in the surrounding.The problem with Equation S24 is that c(rid) and g(rid) are both unknowns and represent distances and therefore cannot be negative.For that reason, a so-called 'closure'-relationship is needed to the Ornstein-Zernike integral equation.The simplest closure relationship, which is able to describe a colloidal system well 68 and which provides an analytical solution, 69 is that of Percus-Yevick 70 : where dp is the particle diameter.Then the analytical solutions for c(rid) are the following: where Φs is the solid/particle volume fraction [-] and the dimensionless λ1 and λ2 parameter are defined as: (S27) The analytically derived Fourier transform of the direct correlation function (Equation S26.b) is the following: 2 [ (qd p ) 4 cos(qd p ) − 4(qd p ) 3 sin(qd p ) − 12(qd p ) 2 cos(qd p ) + 24(qd p )sin(qd p ) + 24 cos(qd p ) − 24 where N is the number density of particles and C a material constant, which is defined by the density and refractive index among others, 66 but is not important to know for the solution of the structure factor.The structure factor for perfect hard spheres is defined as: Using Equations S21, S22, S27-S29, Figure S7a-d was obtained, where the structure (S(q)) and shape factor (P(q)) were plotted against qrp for BaSO4 particles in aqueous suspensions at λ0 = 632.8nm.Therefore,

S23
Figure S7a-d is independent of particle size.Figure S7a and S7b show the trend on a linear scale, while Figure S7c and S7d show the same trend on a logarithmic scale.Φs = 0.00013 corresponds roughly with conditions of initial Ωbarite = 1000 and raq = 1 and it shows that the structure factor does not take any role (i.e.S(q) = 1), except at qrp ~ 0 (Figure S7b and S7d).Contrary, at Φs = 0.13, meaning that there is 1000x more solid material present in an equal volume than can be formed in our experiments, it shows that for qrp < 8, S(q) ≠ 1.In Figure S7a and S7b, it seems that P(q)S(q) = 0 at larger qrp, but on a logarithmic scale one can observe that that is not the case.
To understand better what Figure S7 entails with regards to light scattering measurements, one can plot P(q) and S(q) as a function of θ in polar coordinates (Figure S8).At Φs = 0.00013, the structure factor is at almost every angle equal to unity (Figure S8a).The 'gaps' at certain θ-values, where S(q) < 0.99 are about 0.17.In total, this equates to a total angle of about 9.6, where S(q) < 0.99.At this solid volume fraction, S(q) = 0.99959 at the forward scattering detection angle (i.e.θ = 12.78°), ), S(q) = 1.00001 at the side scattering detection angle (i.e.θ = 90.00°), ) and S(q) = 0.99999 at the back scattering detection angle (i.e.θ = 174.70°),).Conversely, at Φs = 0.13 (i.e.1000x more solid volume than can be formed in our systems), the structure factor is never reaching a constant period where S(q) is between 0.99-1.01,which would mean that structuring of particles is a significant factor (Figure S8b).To conclude, it is safe to assume particle structuring did not occur in our systems/was not observable at the angles we measured.

S25 S26
Figure S8: Shape factor P(q) and structure factor S(q) versus the scattering angle θ in polar coordinates.P(q) shows a maximum value at minima of S(q) and vice versa.Besides the changes of ζ-potential due to evolving surface chemistry, as discussed in Section 3.6., the changes of ζ-potential during non-equilibrium conditions may also be associated to one or any combination of the following processes: Aggregation and agglomeration, sedimentation and arising differences in crystal surface structure. 71Before discussing these processes and how they may have affected the ζ-potential into more detail, it is important to note that changes in the particle size do not affect ζ-potential directly (Equation 3).
Aggregation and agglomeration could have influenced the ζ-potential indirectly, not because of the sudden particle size increase, but because of different electrokinetic properties of two or more particles that collide with each other (a phenomenon known as electroflocculation). 72 In addition, preferential aggregation or agglomeration of particles with favourable electrokinetic properties may have expounded the impact on ζpotential.However, Ruiz-Agudo et al. (2014) 73 and Seepma et al. ( 2023) 74 showed that aggregation behaviour of barite crystals mainly occurred among particle sizes < 100 nm.So, we assumed little to no aggregation in the batch experiments occurred, as in most cases, the first DLS measurements for each experiment showed a particle size of 200 nm or larger.In addition, based on our extended-DLVO calculations (see Section 3.2, the discussion in SI-V and calculations in SI-VI), we believe that agglomeration in our very dilute samples, with a maximum solid volume fraction of ~ 0.013%, was negligible, as the average particle separation distance is ought to be in the range that is larger than the average particle size.Furthermore, Hang et al. (2009) 75 found that for similar NaCl background electrolyte concentrations in suspensions of barite particles are electrostatically stable at pH < 6, thus supporting our case.
An increase in average particle size may have promoted sedimentation, where sedimentation could have occurred when particles reached a size of about 700 nm. 74Figure 3a-e shows strong evidence that sedimentation occurred, because, during each experiment, the absolute count rates decreased again (preceded by an initial increase), illustrated by the intensity of the yellow color.However, it is difficult to assess to

S27
what extent this affected the apparent ζ-potential in our experiments.For example, if multiple populations of particles existed in the sample, but with a similar ζ-potential (i.e. with similar crystal surface structure and composition), then the apparent ζ-potential is unaffected by sedimentation.Causative is that the number of charges on the particle's surface driving the particle forward (in the electrical field) and the viscous drag on the particle are proportional to the surface area. 9In other words, only the (average) particle size changes.
However, if these settling particles are within the measurement window, then there is the possibility that they create an additional (velocity) vector other than the one induced by the electrical field and this may influence the apparent ζ-potential.In our types of samples, the apparent ζ-potential can also be affected by sedimentation as long as there were multiple populations present (i.e. at raq = 1; Figure 4c) with a different ζpotential (Figure 3c) and where only one population may have grown larger than the other(s).Though we have no reason to assume that such a process took place, as neither of the two distinct ζ-potential distributions disappear suddenly (Figure 3c), differences in ζ-potential among the different particle size populations may have caused, for example, preferential aggregation and/or agglomeration and subsequent sedimentation of the circumneutral particles.
The crystal surface structure is important to translate the apparent ζ-potential to the potential at the surface.If the dominant crystal surface structure, especially its (micro-)roughness, changed as the system went to equilibrium, then the apparent zeta-potential, as Bikerman (1941) 76 pointed out, would need a correction, which would be different for each of the time steps (see also Wolthers et al. (2012)  77 ).The crystal surface roughness of barite, caused by surface steps and other types of defect structures, leads to free energies that are significantly larger for such sites than those of perfect crystal faces.Past research has shown that the barite (001) surface displays a degree of heterogeneity, with the presence of defect structures like steps and etch pits, and cause an accumulation of ion species, which may alter the surface potential Ψs, [78][79][80][81] but cannot cause sign reversal. 81Although the influence of surface roughness on the Ψs has not been investigated for barite, Na et al. (2007)  82 investigated this effect for cleaved (1014) calcite surfaces.Using surface potential microscopy (i.e. a variant of non-contact atomic force microscopy), they showed that the Ψs is on average about 137 mV larger for nanostructure terraces of about 2 nm that exist on the substrate terraces.However, they observed about twice as much substrate terrace compared to nanostructure terrace (i.e.relative

S28
frequency difference), even though they chose to measure along a 'xy'-line with a much larger nanostructure terrace density, because from their topography images it is clear that about 80% is substrate terrace.Due to the technique they used and their sample preparation, this large offset could be observed.The effect of surface roughness on colloidal particles immersed in an aqueous solution with a background electrolyte is much less (see Gan et al. (2012)  83 ; i.e. their figure 8a), whom used numerical computations of the zeta potential using Monte Carlo simulations.For monovalent background electrolytes, with a concentration five times higher than our concentration (i.e.0.1 M versus our 0.02 M), they showed that the difference is < 5 mV over a wide range of surface charge densities for a height difference of 0.4 nm.We consider that the extreme case presented by Na et al. ( 2007) 83 may only be important in systems where Ωbarite is continuously large, so that (2D-)nucleation is dominant.Under these conditions, adhesive growth on newly nucleated particles could create large differences in surface roughness.However, Kuwahara et al. ( 2016) 84 showed that Ωbarite needs to be larger than 1000 for 2D-nucleation to occur in the [001] direction (i.e. the crystal face that is most expressed) on a large scale.When Ωbarite < 1000, the edge pits, kink sites and steps are filled with growth units or ions, so that the surface roughness disappears.Although, our BE concentration was less and their choice of the position of the slipping plane (i.e. about one ion away from the surface) is slightly different compared to our systems, it can be envisaged that the differences in ζ-potential was not more than 10 mV.Therefore, the differences in our case, potentially caused by surface roughness, were most likely captured in the distribution and error of the ζ-potential measurements themselves and most likely did not cause another distribution to appear in our measurements.
In our experiments, the average particle size was already far beyond the critical nucleus size of ~ 10 nm 74 and they grew most likely toward 600 -700 nm by spiral growth. 84While the role of surface roughness on ζpotential was likely negligible in our experiments, crystal morphology cannot be ignored so easily.6][87] Also, among different BEs and at the same raq-values, different morphology is developed. 88However, it is less likely that in one particular experiment, more than one type of crystal morphology existed at the same time. 85In that view, ζ-potential evolution in the experiments of Figure 3 could not be explained by crystal morphology and High quality data of the ζ-potential includes "smooth" Doppler phase curves, where the fast and slow field reversal are well distinguishable.This is the case for raq ≠ 1 (Figure S9a, b, d and e).At raq = 1, there are some Doppler phase curves, which show "spaghetti-like" behaviour.These "spaghetti-like" curves represent the circumneutral population in Figure 3c.We expected that the ζ-potential at raq = 1 should be the closest to neutral compared to raq ≠ 1 and, therefore, the particles have negligible preferences in moving direction upon the initiation of the electric field.In other words, this data is not necessarily poor in quality as this was something we expected to happen for raq = 1.
The Voltage-Current plots (Figure S9f-j  between 100 and 1000 nm; Figure 3f-j) and not in the very precise hydrodynamic size.Besides the very first measurement (t ~ 0), which was dominated by the highly stochastic nature of crystal nucleation, the size range for t ≠ 0, dominated by crystal growth, was indeed in the same size range as was measured by FWD without an applied electric field and we assumed that the applied electric field did not significantly influence the particles in the investigated systems.

XI ---Particle Size Distributions for Other Conditions
Similarly to Figure 4, Figure S15 shows the same trend in intensity for backscattering detection angle (BSD) measurements; sometimes a particle size distribution at < 10 nm at t = 0 seconds was observed, meaning conditions of nucleation and growth occurred.

Figure S1 :
Figure S1: The dependency of the Debye length on the lower region of ionic strength (a), the drop in ionic strength during batch DLS measurements when the system approached equilibrium (b) and the associated change in Debye length (c).

Figure S3 :
Figure S3: The relationship between Henry's function, which is an approximation proposed by Ohshima, and κa.The a.The particle size diameter dp is about 4.3 times as large as κa.The a.The DLS particle size range is between the red-dotted lines.The Debye-Hückel limit is accurate for dp ≤ 11 nm (i.e.f(κa.The a) ≤ 1.05) and the Smoluchowski limit for dp ≥ 300 nm (i.e.f(κa.The a) ≥ 1.45).dpvalues that fall in between would require a correction.

S11Figure S4 :Figure S5 :
Figure S4: The relationship between the Dukhin number and the particle size at specific ζ-potential values (i.e. 5, 15, 25, 50, 75 and 100 mV).The red-dotted line represents the condition of Du = 0.1 and specifies more or less the threshold at which surface conductivity should not be ignored.

Figure S6 :
Figure S6: Contributions of the electrostatics (EL), van der Waals (vdW) and hydration (AB) to the total surface free energy (TOT) with H (the distance between particles) for rp = 1000 and ζ = 25 mV (a), the influence of rp on the total surface free energy with H for ζ = 25 mV (b) and the effect of ζ-potential on the total surface free energy with H for rp = 1000 nm (c).

Figure S7 :
Figure S7: Shape factor P(q), structure factor S(q) and the product of both (i.e.P(q)S(q)) as a function of qrp for Φs = 0.13 and Φs = 0.00013 on both a linear and a logarithm scale; linear scale for Φs = 0.13 (a), linear scale for Φs = 0.00013 (b), logarithmic scale for Φs = 0.13 (c) and logarithmic scale for Φs = 0.00013 (d).Note: There are tiny 'gaps' in the calculated data.Causative is q, because of its sinusoidal nature; With increasing θ, from 0 to 360, q continuously increases and decreases, causing the data to be non-ordered with respect to qrp.The calculated data was obtained by steps of θ = 0.01 (i.e.36,001 data points were acquired).

S29
we have discussed changes in ζ-potential during batch experiments in light of surface chemistry development.S30 IX ---Doppler Phase Shift & Voltage-Current Logging during Measurements with Varying r aq

S32Figure S9 :
Figure S9: Doppler phase plots, showing the phase shift between the measured beat frequency (scattered beam) and the reference frequency as a function of time (a-e) and the Voltage-Current logging (f-j), coinciding with the experiments presented in Figure 3 (i.e. initial Ωbarite = 1000, varying raq).raq increases from top to bottom; 0.01, 0.1, 1, 10 and 100.

X ---BSD Size Measurements Performed Routinely in-between ζ-Potential Measurements Some
89ze measurements for every raq were performed during the batch ζ-potential experiments (FiguresS10-S14).These size measurements were performed to check whether the applied electric field caused unusual changes in dp.The application of an electric field may have a variety of consequences during crystal formation.Examples include: Reduced nucleation times, increased nucleation rates, less crystals formed but with enlarged sizes, different orientation of crystals, increased crystallization yield and alteration of transformation processes affecting polymorphism (see review of Alexander & Radacsi (2019)89and references therein).The DLS measurements in between ζ-potential measurements were performed in BSD, because it was more suitable to detect different size populations in the smaller range of dp and we were only interested if the dp was in the range that we measured in FWD where no electric field was imposed (i.e.