A new method for estimating zeta potential of carboxylic acids’ functionalised particles

Zeta potential is an important colloid parameter that dictates many intriguing phenomena such as colloid stability/aggregation, electrophoretic deposition and particle separation technology. Zeta potential is commonly used to characterise colloid particles’ electrostatic interactions coupled with other forces such as van der Waals attraction, hydration and steric forces. However, the measurement of zeta potential involves complicated sample preparation and instrument/theory selection, often resulting in inaccurate measurement. Here, we developed a simple method that combines Stern–Grahame and Gouy–Chapman theories, for estimating zeta potential using parameters such as acid site pKa, acid site density and particle size. The method is applied to a wide range of carboxylic acids’ functionalised colloid particles with various pKa. Results show that the new method is capable of capturing zeta potential dependence on pKa and pH. The new method can be applied to particles with complicated or multiple acid/base sites. Without the need for sample preparation/instrument, this new method is believed to save efforts for estimating zeta potential with good accuracy. GRAPHICAL ABSTRACT


Introduction
Carboxylic acid functionalised nano-particles are a class of important materials in applications ranging from heavy metal ion removal and drug delivery to biomass heterogeneous catalysis [1][2][3][4].When analysing particles' performance interacting with other materials, typically in an aqueous solution, an electrostatic force is universally present and it is a crucial force in so-called DLVO analysis, a theory for colloidal interaction [5][6][7].The electrostatic force dictates many phenomena such as colloid stability/aggregation, ions-induced protein misfolding and particle flocculation [8,9].Therefore, understanding carboxylic acid functionalised particles' surface charge is crucial for designing materials [10,11].
Surface potential is a fundamental electrical property to characterise particle surface charge.However, CONTACT Ziyang Zhang zzhang10@wpi.eduSupplemental data for this article can be accessed online at https://doi.org/10.1080/00268976.2023.2260014.
given that surface potential is the potential difference between the particle surface and steric layer, the steric layer is packed with a high density of counterions, making the measurement inaccessible.Instead, zeta potential is defined as the potential difference between the slipping plane and the particle surface and the slipping plane is more accessible because of the loose diffusion layer [12,13].Zeta potential is defined as the potential at the slipping plane where ions have higher mobility than inner layer ions [14].Zeta potential can be affected by many factors, including pH, ionic strength, particle surface functional group, ion mobility and the rheology of solution (e.g.viscosity, thickness).Hence, measuring zeta potential has to be conducted with careful sample preparation, liquid medium condition control and proper theory selection [15].Of various methods used to measure zeta potential, they all bear limitations [16,17].Kelvin Probe Force Microscopy (KFM) is a direct method for measuring surface potential.KFM measures the contact potential difference between the probe tip and solid particle surface, and it also creates highresolution topological images [18][19][20].Despite that KFM has high sensitivity and excellent spatial resolution, KFM needs to re-calibrate each time using it, and its measurement is highly sensitive to surface roughness, meaning that repeatability and accuracy are compromised [21].Streaming Potential Measurement is another widely used technique for measuring zeta potential.In this method, particles or fibrous samples are immobilised onto substrates, and a flow containing electrolyte is forced to flow through a capillary tube and the current difference is measured.The electrophoresis approach is well established to measure the mobility of charged particles upon applying an electrical field, and then particle mobility data are converted into zeta potential results through Smoluchowski equations [22,23].Despite their simplicity, those methods often require exhaustive sample preparation, and a special set-up that includes tube/capillary can significantly reduce the efficiency of measurement.The theoretical model has limitations that can only measure zeta potential smaller than 50 mV [21,24].
Zeta potential is a complicated property that is related to many factors as mentioned above, and yet sometimes it is so important that fast measurement/estimation is needed to justify particle surface modification.In fact, when preparing samples for zeta potential measurement, unsuitable ionic strength can cause coagulation, resulting in inaccurate measurement.The optical transparency of the cuvette can also mislead the measurement because of the optical variations [25,26].A fast method that can quickly estimate zeta potential is needed.Computational estimation can be a useful tool for compiling and computing electrical properties.For example, Nduna et al. [27] developed a Matlab-based model for estimating zeta potential dependence on pH.However, the model only applies to oxide and hydroxide particles due to the incorporation of crystallographic data.Keesom et al. [28] derived an ionogenic model for computing membrane zeta potential upon adsorption of ionic surfactants.The model has good agreement with streaming potential measurement, but the model is specifically designed for the surfactant-adsorption type of interactions due to its use of Langmuir adsorption isotherms.
This study particularly revolved around developing a simple yet accurate model to estimate and compute the zeta potential of carboxylic acid functionalised nano-particles using common parameters such as ionic strength, carboxylic acid pKa, acid site density, pH, particle size and temperature.The new method can estimate the zeta potential of nano-particles functionalised with common monocarboxylic acid, benzoic carboxylic acid and dicarboxylic acid.The new method can be applied for a wide range of acid pKa between 2.5 and -6 and pH conditions of 2-7.

Theory
Micro-/nano-particles suspended in a liquid medium obtain electrical charges through the contact of dissimilar materials, ion bombardment and induction [29][30][31].Among these three mechanisms, ion bombardment is the most common mechanism which includes ion adsorption or chargeable group dissociation.For particles bearing carboxylic acid groups (-COOH) or similar ionisable functional groups, they gain charges mainly by deprotonation, e.g.P-AH P-A − + H + where P-AH denotes acid functionalised particles, P-A − represents negatively charged attached particles while H + represents free state protons in a liquid [32].
Generally, for particles functionalised with more than one acid site, a resemble dissociation equation can be written as follows: P-XH P-X − + H + where XH denotes any acid functionalised on particles.The deprotonation proceeds until its reaches the equilibrium state.Further protonation would only occur in extremely acidic conditions.The particle surface charge relies on dissociated site density (e.g.X − ) and any counterions with low mobilities that are trapped within steric layers.For simplification purposes, any other electrolytes due to impurity are omitted.Therefore, the only counterions are acidified water (e.g.H 3 O + ) which has high mobility according to the Grotthuss mechanism [33], leaving the only contributor of particle surface charge as P-X − .
For generalisation purposes, XH will be used for the following method development.Acid strength (k X ) and total chargeable site ( X t ) balance can be expressed as follows: [34] where k X is the acid strength or acid dissociation constant, X − is the head group surface coverage (e.g.mol/m 2 ) that dissociates and XH is the head group surface coverage that does not dissociate.[H + ] 0 is the proton concentration or activity around particles.The chargeable and unchargeable site density can be correlated as follows: where X t is the total surface coverage.Use equations the proton activity around the surface.Protons are concentrated around particles due to the attraction between protons and negative charge surfaces, and the concentration drops as the distance apart from particles increases.It can be estimated using proton activity in the bulk phase (or pH) using d Boltzmann distribution [35]: where [H + ] b is the proton concentration in bulk liquid, e is the unit electron charge (1.6 × 10 −19 C), T is the system temperature, ψ 0 is the surface potential and k b is the Boltzmann constant (1.38 × 10−23 J • K −1 ).After solving X − using equations (1-4), chargeable site density can be expressed in terms of a combination of variables including pH, XH , k X , T and ψ 0 .Among those variables, the only unknown is ψ 0 .If multiple acid groups are functionalised on particles, or the acid has multiple ionisable sites such as dicarboxylic acids, similar routines may apply for any other acids, then total surface charge density can be added into equation ( 5): where σ 0 is the total particle surface charge density, n represents the number of acid groups that contribute to the overall surface charge, e is the electron charge and N A is the Avogadro constant.
Combining Equations (1-5), the only hurdle to solving zeta potential is that intermediate surface potential must be solved.Compared to zeta potential, surface potential is a direct measure of particle surface charge.However, the measurement of surface potential remains one of the most challenging works for several reasons.Firstly, the inner layer or the Stern layer of particles contains densely packed ions, and it is challenging to insert a probe into the Stern layer; Secondly, it is challenging to define the actual 'surface' as most surfaces have roughness.Therefore, measuring or estimating zeta potential is more practical compared to measuring surface potential.Assume you have zeta potential data under some known conditions, e.g.ionic strength or pH, the surface potential can be estimated by the Gouy-Chapman model [36]: where ψ 0 is the surface potential (V), kb is the Boltzmann constant, T is the temperature (K) and z is the valance.In a 1:1 electrolyte, z = 1.Z is the zeta potential (V) and d is the distance between the particle surface and the slipping plane, often taking the value of 5-6 Å, κ: inverse Debye length (m).Substituting equation (1-7) into Equation ( 5), an expression for surface charge density can be obtained which reveals the chemical information of surface charge density.
The main contribution of the Gouy−Chapman model is the introduction of an ion diffusion layer in which ions can be stripped away from particle proximity due to thermal motion and viscous action of the fluid.However, the Gouy−Chapman model failed to accurately estimate particle surface charge because of its simplistic description of the ions-solid bonded surface region.
Instead, the Stern-Grahame model further divides the ions adsorbed layer into the inner Helmholtz plane (IHP) and outer Helmholtz plane (OHP).Each layer plays an important role in the net surface charge, depending on the thickness and ion density of each layer.For particles with ka > 0.5, its Stern-Grahame model can be expressed as follows [29,37]: where ε 0 is the dielectric constant of vacuum, ε 3 is the dielectric constant of water and a is the particle radius.
Here, Equation (8) will be used to solve surface charge for two reasons: (i).The model only uses the Gouy-Chapman model to get expression for zeta potential.Surface charge derived from equation (1-7) will be substituted by Equation ( 8); (ii).The new method eliminates concerns about the inaccuracy of the Gouy-Chapman model and uses a more accurate Stern-Grahame model to compute surface charge.
Figure 1 shows the schematics of newly developed methods for estimating zeta potential.The new method combines liquid medium environmental variables with particle surface properties for estimating zeta potential.

Results and discussions
Zeta potential can be estimated using chemistry parameters including acid site density, pKa, pH and ionic strength (I).Those parameters are easily accessible.The first objective is to validate the theory by calculating zeta potential dependence on pH and pKa, the two most common factors affecting zeta potential.This will give confidence in the robustness of our model.We then apply our model to calculate a library of commonly used monocaboxylic acid, benzoic acid and dicarboxylic acid in a wide range of pH values.

Zeta potential depenence on pH and pKa
pH and pKa control particle surface charge density as pH is the liquid medium variable and pKa is the particle surface functionality [38,39].Understanding zeta potential dependence would guide both environmental (electrolyte concentration, pH and temperature) and compositional design (the surface acid group, pKa, particle size) of colloid particles [40][41][42].Therefore, Figure 2 shows the estimated zeta potential for two particles functionalised with acids of pKa ranging from 3 to -4.5 and the modelled solution pH is changing from 2 to 4.5.Particle radius is assumed to be 100 nm, acid site density is assumed to be 0.05 nm −2 (Figure 2(a) left) and 0.1 nm −2 (Figure 2(b) right) respectively and ionic strength is set as 0.1M.
It is worthnoting that the presented model only considers deprotonation, meaning carboxylic acid functionalised particles are entirely negatively charged.In practical application, especially in cases where low pH (e.g.pH < 1.5) conditions are required [43], acid groups with high pKa may be protonated, resulting in surfaces obtaining a positive charge.In this case, equation (1-5) can be modified so that the isoelectric point (IEP) can be estimated.
In both cases, zeta potential decreases when pH increases, due to continuous deprotonation of acid sites.At lower pH (e.g.pH = 3), zeta potential is insensitive to pKa because of the protonation of acid sites; at high pH around 4, zeta potential decreases from −8 mV to −35 mV when pKa decreases from 4 to 2, due to the increase of acid strength and proton dissociation.Comparing 0.05 nm −2 with 0.1 nm −2 , acid site density has a huge impact on zeta potential change.For example, when pH = 4 and pKa = 2, zeta potential drops from −24 mV to −45 mV when acid site density doubles, which agrees with literature-reported findings [39,44].Alignate (pKa = 3.2) has zeta potential around −21 mV when pH is 4, agreeing with the new model's estimation (−19.8 mV) [39].Overall, the proposed model captures pKa, pH and acid site density on zeta potential estimation.This new method is also capable of estimating the pKa of unknown chemicals if zeta potential is measured under a series of pH, simplifying experimental efforts [39,45].In this case, equation (1-5) can be modified and set pKa as unknowns and all other parameters are known.A regression model can capture the trend and estimate the pKa that gives the best fit.

Monocarboxylic and benzoic acid
Building a computational database that compiles all zeta potential estimations for commonly used carboxylic acid can help researchers promptly get a sense of surface charge.Therefore, the dissociation constant (pKa) for commonly used monocarboxylic acid and benzoic acid at ambient temperature are extracted from the literature as listed in Table 1 [46][47][48][49][50]. Selected monocarboxylic acid has pKa ranging from 3.74 to -4.9, representing a group of common weak acids while benzoci acid pKa ranges from 2.97 for 2-Hydroxy-benzoic acid to 4.80 for 3-Hydroxy-benzoic acid.Selected carboxylic acid and their respective pKa represent a single complete molecule.Those groups are often attached to large molecules.However, the data in this paper only show the pKa of each type of carboxylic acid.Using Equations (1-7), zeta potential Figures 3 and  4 shows predicted zeta potential of nanoparticles functionalised with those selected carboxylic acid with a site density of 0.1 nm −2 and a particle size of 100 nm and an ionic strength of 0.1 M. The present study only concerns surface charge rising from the dissociation of functionalised carboxylic acid.The substrate material, e.g.silica, activated carbon or metal oxide may also obtain charges via ion adsorption, but it is not considered in Figures 3  and 4.
Figure 3 shows estimated zeta potential drops in a nonlinear fashion with respect to pH increasing from 2 to 7, a commonly used pH range to deprotonate carboxylic acid.In all cases, zeta potential reaches around −63 mV at high pH due to fixed acid site density (0.1 nm −2 ).The trend of zeta potential with respect to pH is pKa dependent.Carboxylic acid with low pKa (strong acid) drops zeta potential faster than carboxylic acid with high pKa (low strength).This indicates that weak acid functionalised particles have relatively stable and low surface charge when pH is low.(e.g.pH < 2.5).For strong acids, pH < 2.5 can not prevent protons from departing the particle surface and hence zeta potential drops.For example, the zeta potential of particles with formic acid functionalisation drops to −4 mV at pH = 2.5 and continues to decrease to around −63 mV at pH = 6.Acid with high pKa, e.g.2-methylpropanoic acid or nonanoic acid, starts dropping zeta potential at delayed pH around 3-4 and reaches the minimum value (−63 mV) at pH = 6-7.Clearly, pKa can drastically shift the zeta potential-pH dependence curve and the model can capture this observation.
Benzonic carboxylic acid is a class of surface modifiers that is widely used in catalysis and pharmaceutical drug application due to its high van der Waals interaction and π interaction with target molecules [51,52].Benzonic carboxylic acids have a wide range of acidity because of their π structure.Hence, various benzonic carboxylic acid pKa listed in Table 1 are extracted and zeta potential is estimated under the same parameters.We notice a similar trend as monocarboxylic acid in that strong acid zeta potential starts decreasing at low pH and reaches equilibrium at relatively low pH.For example, in Figure 4, a particle functionalised with 2-Hydroxy-benzoic acid has zeta potential around −6.6 mV at pH = 2 and it decreases to −65.2 mV at pH = 5.5 because of its deprotonation.Similarly, 2-Phenylbenzoic acid functionalised nanoparticle has a zeta potential of −2.6 mV, and it decreases to −65.3 mV at pH = 5.Conversely, for weakly charged benzoic carboxylic acids such as trans-p-Methyl  cinnamic acid, zeta potential has not started to drop until pH increases to 3.5 and zeta potential still decreases at pH = 7, suggesting the weak acid has a delayed response to pH change.Overall, the new method predicts zeta potential trend with pH change agrees with the literature trend [43,53].

Dicarboxylic acid zeta potential estimation
Dicarboxylic acids are a class of important organic acids in catalyst applications such as cellulose hydrolysis [54].
Having two carboxylic acid groups enhances particles' binding and catalysing ability [55].Those unique acids have high proton concentrations and typically complicated surface morphologies, depending on the pKa of individual acid groups.Different acid groups may contribute to various functionalities such as binding or catalysing [55].Table 2 shows pKa values for common dicarboxylic acids.In general, dicarboxylic acid with a shorter carbon chain has weaker acidity compared to long-chain carboxylic acids.For example, ethane dioic The method of incorporating extra acid or basic groups is similar to monocarboxylic acid where only equation ( 5) needs to be updated to accommodate extra acid or base groups.Zeta potential for dicarboxylic acid is presented in Figure 5. Again, total acid site density is assumed to be 0.1 nm −2 and the particle size is set to 100 nm in diameter.Interestingly, for ethane dioic acid, zeta potential drops from around 0 mV to −95 mV within the pH range of 2-4.This is because of its low combined pKa.cis-Butene dioic acid and propane dioic acid have three phases of zeta potential changes: at low pH < 2.5, zeta potential stays relatively constant due to its weak acid group; when pH increases to between 2.5 and 5, zeta potential drops linearly with respect to pH due to the deprotonation of the secondary acid group; when pH > 5, zeta potential stays relatively constant and reaches a saturation value.For acids that have two weak acid sites (e.g.octane dioic acid, nonane dioic acid), the shape of zeta potential vs pH is similar to the weak acid in which zeta potential drops linearly with pH.
The substrate-carboxylic acid interactions can affect zeta potential measurement, depending on the substrate type and interaction type.Metal oxide tends to form hydroxyl groups on the oxide surface and the proton transfer from carboxylic acid to the hydroxyl group may form a positive electrical field, causing zeta potential increase [56].Therefore, tuning acid-based equilibrium on the oxide surface may influence zeta potential estimation using the existing model.Polymer substrates with non-polar surface groups may adsorb long-chain nonpolar carboxylic acid groups through hydrophobic interaction, increasing zeta potential.Therefore, molecular interaction theory has to be considered to account for non-covalent or non-ionic bonds.In either case, the presented model has to be modified to fit specific applications.

Conclusions
Colloidal particles functionalised with carboxylic acid have broad applications in drug delivery, heterogeneous catalysis and biological applications.Estimating or measuring zeta potential is crucial for understanding its interactions with itself or other chemicals.However, zeta potential measurements typically involve complex sample preparations and instrument operations and questions regarding the accuracy and precision of those measurements are raised due to complicated electric double layers combined with surface morphologies.Therefore, a computational model that incorporates particle characteristics and liquid medium properties is needed to estimate zeta potential.
Here, a new model that consists of several explicit equations has been developed for estimating the zeta potential of carboxylic acid functionalised particles with a handful of inputs including acid pKa, acid site density, particle size and ionic strength.The proposed new model combines the advantages of the Gouy-Chapman model and the Stern-Grahame model: use Gouy-Chapman for introducing the diffusion layer and the Stern-Grahame model for accurate surface charge calculations.The new model also incorporates the chemical properties of the acid head group including acidity/basicity, adsorption equilibrium constant and liquid properties such as pH and ionic strength.We first applied the model to estimate zeta potential as a function of pKa and pH, and it can estimate zeta potential and predict a trend that is similar to the literature.Estimated zeta potential agrees with the literature-reported values because it is challenging to compare to literature values given multiple variable complicacy.Then we build a database of zeta potential estimation for monocarboxylic, benzoic carboxylic and dicarboxylic acid functionalised particles with varying pH, at fixed particle size, surface acid group and ionic strength.The model is useful to establish a zeta potential database of not only covalent-bonded acid groups but is also capable of capturing physical adsorption-induced surface charge by incorporating the Langmuir isotherm equilibrium.

Figure 1 .
Figure 1.Schematics of the proposed model for estimating zeta potential.

Figure 2 .
Figure2.Estimated zeta potential carboxylic acid functionalised nanoparticles and its dependence on pH and pKa.The left plot has a surface acid coverage of 0.05 nm −2 and the right plot has a surface acid coverage of 0.1 nm −2 .Particle radius is assumed as 100 nm and the temperature is 298 K, the ionic strength is 0.1M.

Figure 3 .
Figure 3.Estimated Zeta Potential of Carboxylic Acid as a Function of pH.Particle Size is set as 100 nm and acid site density is assumed as 0.1 nm −2 .

Figure 4 .
Figure 4.Estimated Zeta Potential of Carboxylic Acid as a Function of pH.Particle Size is set as 100 nm and acid site density is assumed as 0.1 nm −2 .

Figure 5 .
Figure 5.Estimated Zeta Potential of Carboxylic Acid as a Function of pH.Particle Size is set as 100 nm and acid site density is assumed as 0.1 nm −2 .