The effect of cation-π interactions in electrolyte/organic nanofiltration systems

Gang YangYu Ma ,Weihong Xing

Nanjing Tech University,Nanjing 210009,China

1.Introduction

Numerous works published in recent years have been pointed out nanofiltration(NF)technology as a downstream operation in organic acid production process is expected to be a large application field[1-4].Organic acids are mainly produced by fermentation,which generates a broth containing the dissociated form of the acid and different impurities,such as mineral salts.However,the coexistence of salts and organic matter(OM)in NF usually causes the rejection of both of them to decrease in an obvious manner(especially in the system of π bonds contained organics and mineral salts[5]).The following proposals were suggested to explain these decreases,and the main points were summarized:

(1)The partial dehydration of the OM occurs in the presence of salts(i.e.,the Hofmeister effect)[5-8].The smaller effective OM volume leads to increasing transfer through the pores of the membrane.

(2)More counterions converge in the membrane pores as the bulk salt concentration increases.The electrical double layer at the intrapore wall is thus compressed,resulting in stronger electrostatic repulsion(i.e.,the swelling effect)[5,6].The expanded pores let the salt pass through easily.

(3)A membrane pore size distribution exists.When the solute concentration is high enough,the solute flux through the small pores decreases and can even become blocked(i.e.,the electroviscous effect)[9],which leads to an increase in the average pore size and a decrease in the steric effect.

(4)The viscosity at the membrane surface increases with the OM concentration.The back diffusion in the polarization layer is hindered[10],which results in a higher membrane surface concentration and a lower solute retention(for both the OM and the salt,if they coexist).

(5)If the functional group in the OM tends to polarize,the OM interacts with the charged membrane.The OM retention is then affected by the charge properties of both the membrane charge and the OM[11,12].

The above proposals can be categorized as follows:

(1)The effects of salting-out,pore swelling and electroviscosity reduce the OM rejection;these phenomena require a higher electrolyte concentration.

(2)The rejection change,which is caused by viscosity,is related mainly to the differences between forward diffusion and back diffusion.There seems to be a critical viscosity below which the retention is barely dependent on the organic concentration.

(3)The electrical effect,which is pronounced in the presence of OM,is weak on the membrane.Therefore,the negative rejection of the organics should be small enough that it can be ignored.

As indicated above,changes in solute rejection under conditions of low salt concentration and unaltered viscosity solutions are worth discussing.In addition,a previous report suggested that in sodium lactate(NaLac)/glucose solutions,the rejection of NaLac changed(2%-20%)[5].However,in the previous study,the changes in OM rejection in the NF system were their primary target,so they did not take the changes in the NaLac rejection into consideration.

In this article,we suggest that the change in NaLac rejection cannot be ignored.This change in the NaLac rejection is the emphasis of the research into this phenomenon.Lactate is a type of carboxylate radical that contains the hydroxyl group;the three sp2hybrid orbitals combine with two oxygen atoms and one carbon atom,ultimately generating three σ bonds.The p orbital that does not participate in the orbital hybridization combines with another p orbital on the membrane to form a π bond,and the p orbital of the oxygen atom in the hydroxyl,which consists of the unshared electron pair,can p-π-conjugate with the πbond on the carbonyl.Therefore,the phenomenon that occurred in report[5]might be the result of a cation-π interaction that occurred between Na+and Lac-.Cation-π interactions are a type of electrostatic attraction between a cation and a large π bond.The six(four)Cδ--Hδ+bond dipoles of certain molecules such as benzene(ethylene)combine to produce a region of negative electrostatic potential on the surface of the π system.Simple electrostatics facilitates the natural attraction of cations to the surface[13-15].Because the p-π-conjugate occurs in Lac-,this effect enhances the interactions between the cation and the large π bond.Previous research showed that cation-π interactions in aqueous salt solutions can be characterized based on the2H nuclear magnetic resonance(NMR)spin-lattice relaxation time[16],so the existence of cation-pi interactions in solutions is unquestionable.In previous cases,when glucose was added,Na+would partially participate in the Hofmeister effect with glucose,which disrupts the original interaction balance of a single component system and leads to a large change in the rejection of NaLac.

In this study,experiments were used to demonstrate these assumptions.Therefore,we reasonably simplified the experiments:the NF systems consist of benzyl alcohol-KCl or benzyl alcohol-NaCl solutions.The rationale for the simplification consists of two parts: first,benzyl alcohol is soluble and forms a π bond in aqueous solution;second,in aqueous environments,due to the lower solvation energy of the alkali,the screening effect of the hydration shell is weaker than the screening effect of the other multivalent cations.As previously reported[13-15],K+and Na+can form a relatively strong cation-π interaction with OM.

In specific experiments,other factors have been added to analyze the influence of these factors,and the rejection differences of the electrolyte between these two systems should not be ignored.In this research,the velocity variation method(VVM)was used to extrapolate the rejection from observed rejection to intrinsic rejection.We also use the DSPM-DE model to program and fit the intrinsic rejection of KCl.Using these methods,the changes in the Stokes radius of an ion can be calculated and used for demonstration purposes.Finally,we concluded that both the steric effects and charge densities of the electrolytes(which are generated by the dehydration of the cation)are important.

2.Theory

2.1.Model description

The DSPM-DE model[17]was used here to model the rejection decrease of KCl.However,the variations in the steric and electrical effects,which are caused by the changes in the radius of the cation,are the main reasons for the change in KCl rejection in mixed solutions.Therefore,this model cannot be used accurately to fit multiple experimental points in mixed solutions because there is only one experimental point corresponding to one value for the steric and electrical effects.So the best way to fitting experimental points in this situation should coupling use DSPM-DE model and quadratic criterion,which required five steps:

1.The parameterrpwas determined from glucose real retention which would be discussed in Section 4.1;ri,s(0)in this step was fixed,which value is 0.125 nm.

2.DSPM-DE model was utilized to regression analysis the rejection data for KCl in pure KCl solutions to obtainXd(0).

3.Values ofXd(0)were fixed in mixed solutions which were obtained from step 2,then the parameterri,s(1)was adjusted to fulfill the minimum result of Eq.(1).

4.Fixri,s(1)which was determined from step 3,then DSPM-DE model was repeatedly used in regression analysis the rejection data of KCl in mixed solutions to obtainXd(1).Eq.(1)was reused to fit a line and determine the minimum value.

5.Steps 2 and 3 were repeatedly used untilXd(n-1)-Xd(n)≤0.01 andri,s(n)-ri,s(n-1)≤0.001.

whereQis the quadratic criterion,nis the number of experimental points andmis the number of ions.Ri,expandRi,modare the experimentally obtained real rejections(viathe VVM)and the values predicted by the model,respectively.

The variation in the average pore dielectric constant was estimated as proposed by Bowen and Welfoot[18]:

The use of the above equation implies that the pores consisted of one layer of oriented water molecules where the dielectric constant at the pore wall approached the high frequency limit of the dielectric constant.

The Stokes-Einstein equation follows:

whereDiis the diffusion coefficient of the ion,rsis the Stokes radius,η is the dynamic viscosity,andNAis the Avogadro constant.

In low concentration solutions,because co-ion adsorption does not dominate the Donnan effect,the change in the Stokes radius of a cation also affects its concentration in the membrane[28],and the increased value of the membrane charge density(Xd)can be described as follows:

2.2.Concentration polarization

Because we consider only the situation of single salts,the calculation ofci,win both charged and uncharged nanofiltration systems utilizes the velocity variation method(VVM)[19,20],which extrapolates observed rejections to real rejections.It is expressed as follows:

In Eq.(5),to calculate the real rejection,the method consists of determining the observed rejectionRi,obsfor various increasing tangential velocities and extrapolating these values to infinity.The obtained limit value is the real rejectionRi,realthat would correspond to a virtual situation where no polarization layer exists(that is,ci,wwould be equaltoci,f)[19].The mass-transfer value can then be derived from an empirical Sherwood relationship:

Introducinginto this equation leads to

At 25°C,the values of the coefficientsa,bandcare 0.023,0.875 and 0.25[21],respectively.

3.Experiment Section

3.1.Membrane,chemicals and apparatus

The experiments were performed using a Desal DK membrane(GE Osmonics),which is a flat thin film composite membrane with a polyamide top layer on a polyester support.The effective area is 80 cm2.The molecular weight cutoff of the membrane is in the range of 150 to 300 g·mol-1(according to the supplier).The nanofiltration experiments were performed using aqueous KCl solutions and aqueous KClbenzyl alcohol solutions with KCl concentrations ranging from 5.5 mol·m-3to 12.5 mol·m-3.The aqueous NaCl solutions and the aqueous NaCl-benzyl alcohol solutions had NaCl concentrations that ranged from 3.7 mol·m-3to 7.5 mol·m-3.The solutions were prepared from pure analytical-grade salts and Milli-Q quality water(conductivity ＜ 1 μs·cm-1).We used an inductively coupled plasma emission spectrometer(ICP)(ICPS-7510,Tsushima,Japan)and a DDS-308A conductivity meter(ShangHai LEI CI Chemical Reagent Company)to determine the concentrations of the KCl on the retentate and permeate sides of the membrane.The dynamic viscosities were measured using an NDJ-1E Brook field viscometer(Shanghai,China).

3.2.Experiments

A custom-designed laboratory-scale filtration apparatus was used to perform our experimental work.A plunger pump pushed the fluid into the membrane module,which tangentially flows through the membrane surface.The average tangential velocities were controlled by a frequency converter.The values were 1.63 m·s-1,1.94 m·s-1and 2.49 m·s-1at frequencies of 20 Hz,25 Hz and 30 Hz,respectively.The range of applied pressures used in the experiments was 0.4 MPa to 1.2 MPa,and the temperature of the system was maintained at(25±0.5)°C,controlled by a cooling circulating pump.The retentate and permeate streams were recycled into the feed tank to hold the feed concentration constant.Samples were collected after sufficient time elapsed to reach steady-state conditions.Before each experiment,the pilot was cleaned with demineralized water.The mixed solutions were shaken for 20 min,heated in a water bath for 24 h at 40°C and allowed to stand for another 5 days to ensure full cation-π bonding.In each of these experiments,the membrane was cleaned with ethanol.Next,the membrane was compacted with demineralized water at 2 MPa for half an hour.In addition,the flux was monitored to detect membrane deterioration.

4.Results and Discussion

4.1.Structure characterization

4.1.1.Concentration polarization

Nanofiltration was performed under different average tangential velocities(controlled by a frequency converter)and different cross membrane pressure to derive every single rejection data.Then VVM was used to extrapolate the observed rejectionRi,obsto the real rejectionRi,real[19].

As shown in Fig.1,each line was derived from fit experiment data which have the same cross membrane pressure,average tangential velocity makes them different from higher to lower rejections.According to Eqs.(5),(10)and(11),the coefficientKand the limit of the real rejection,Ri,real,can be determined graphically by plottingas a function of.Kis assessed from the slope of the curves and from the intercept coordinate,and finally,we obtained a value forKof 1.989×10-8.

Fig.1.Determination of the mass-transfer coefficient k and the real rejection Ri,real using the velocity variation method(for a 2 g·L-1 glucose solution).

4.1.2.Pore radius and effective membrane thickness

The determination of the mean pore radius is required to adjust the value of the pore radius and effective membrane thickness and fit the real rejection curve of glucose,which is shown in Eq.(12):

Pedenotes the Peclet number inside the membrane defined as:

whereci,pandci,w,are the concentrations of the permeate side and the concentrated side,respectively.Δx/Akis the effective membrane thickness(Akis the porosity).Ki,cis the hindrance factors for convection;Ki,dis the hindrance factors for diffusion.Jvis permeate volume flux of the solution.

Serenaet al.[22]attempted to obtain an accurate evaluation of the pore radius.The use of neutral solutes that are not strongly rejected is recommended,i.e.,asymptotic rejections lower than 90%-95%.The concentration of glucose does not influence its rejection significantly[5],so we use only 2 g·L-1glucose to calculate the pore radius,and it is obvious that the rejection of glucose can fulfill this requirement.The asymptotic rejection is depicted in Fig.2;the radius is 0.53 nm.

Fig.2.The real rejection of a 2 g·L-1 glucose solution vs.the permeate volume flux of the solution.

The hydraulic permeabilityLpis obtained from the evolution of the water flux with pressure(Fig.3).AnLpvalue of 1.56×10-11m·s-1was found,which is in agreement with a report by Benitezet al.[23].Depending on the values ofLpandrp,we can derive an effective membrane thickness(Δx/Ak)of 2.23 μm.

Fig.3.Permeate volume flux of pure water vs.trans membrane pressure.

4.2.Effect of concentration

Fig.4 depicts the changes in rejection for different concentrations of KCl as a function of the permeate volume flux.As shown,the rejection decreases significantly as the concentrations increase.Table 1 shows the values ofXdandci,w/Xdat different concentrations of KCl.TheseXdvalues are slightly lower than the values reported by Maet al.[24],mainly because the experimental method makes several assumptions(such as Gouy-Chapman's electrical double layer model)and does not take the relocation of charges in and on the membrane into account.The effect of a support layer is also ignored,so the values of|Xd|from this method should be higher than the real situation.The values ofXdandci,w/Xdgradually decrease as the concentration of KCl increases.This phenomenon can be explained as follows:

Fig.4.Real rejection of KCl at various concentrations(VVM)vs.the permeate volume flux.

Table 1Relationships between the wall concentration of KCl and the charge density

In Donnan theory for ion exclusion,the increase in the fixed charge and the electrolyte concentration have opposite effects on the retention phenomenon because the former makes the Donnan effect stronger,whereas the latter screens the electrostatic interaction between the ions and the fixed membrane charges[25].The results show that the driving force,which was generated by the increasing concentration,cannot balance the resistance generated by the increasingXd.

4.3.Discussion of influential factors

Other influential factors that might cause a decrease in electrolyte rejection need to be discussed in this work.Some potential factors are listed below:

(1)The viscosities of mixed and pure solutions are almost the same,as shown in Table 2.In this work,the viscosity does not significantly change,which suggests that this influencing factor can be neglected.

(2)Fig.5 shows the rejection data for KClin various mixed solutions.The concentration of KCl is 7.5 mol·m-3,and the concentration of the organics is 3.7 mol·m-3.The values of the ionization constant(pKa)of benzyl alcohol,methanol,water,ethanol,and isopropanol are 15.4,15.5,15.7,15.9 and 16.5,respectively.In previous studies[6,7],researchers have concluded that the presence of neutral solutes does not affect the retention of ions,which is dominated by electrostatic effects(i.e.,Donnan exclusion and/or dielectric exclusion).The addition of organics without large π bonds in KCl solutions does not change the rejection of KCl,which is in agreement with the previous studies[6,7].However,upon addition of benzyl alcohol,the rejection of KCl decreases noticeably.Because the value of the dipole moment of benzyl alcohol is 1.7,which is the same as ethanol,its ionization constant is very close to the dipole moment of methanol.Therefore,we can conclude that the decreased rejection of KCl is not caused by ionization or the dipole moment.

Table 2Dynamic viscosities of pure KCl and mixed KCl/benzyl alcohol solutions

Fig.5.Realrejection of KCl(7.5 mol·m-3)in various mixed solutions(3.7 mol·m-3)vs.the permeate volume flux.

4.4.The rejection of salts decreases in mixed solutions

Fig.6(a-d)shows the rejection data for KCl in single-solute and mixed-solute solutions as functions of the permeate volume flux at concentrations of 5.5 mol·m-3,7.5 mol·m-3,10.5 mol·m-3and 12.5 mol·m-3,and the concentration of benzyl alcohol is 3.7 mol·m-3.As shown,the rejections of KCl decrease in the presence of benzyl alcohol.This decrease is less obvious as the salt concentration increases.Fig.7(a-c)presents the rejection data for NaCl in single and mixed solutions as a function of the permeate volume flux at concentrations of 3.7 mol·m-3,5.5 mol·m-3,and 7.5 mol·m-3,and the concentration of benzyl alcohol is 3.7 mol·m-3.As shown,the rejections of NaCl decrease at lower electrolyte concentrations(3.7 mol·m-3),and this decrease disappeared as the salt concentration increased.Table 3 shows the values ofXdand the Stokes radius of K+at different concentrations of KCl in the mixed solutions.Compared to Table 1,the values ofXdand the Stokes radius change simultaneously.We explain these phenomena below.

4.4.1.Cation-πinteraction

In biological systems,cation-π interactions are a type of interaction between cations and aromatic systems,and the interaction energy between K+and benzene is 15 kcal·mol-1(1 cal=4.1840 J)[15].However,in aqueous environments,the water molecules that directly coordinate with cations neutralize the charge of the cations and suppress the interactions of the cations with the π cluster,leading to a reordering of the strengths of cation-π interactions.Due to the lower solvation energy of alkali cations,the screening effect of the hydration shell is weaker than the screening effect of other multivalent cations,which have solvated energies approximately three to four times lower than direct interaction[26].In this case,the methylol and-CH2-are electron donating groups,leading to the formation of a delocalized π bond on the benzene ring and a higher electron density,causing the electrostatic interaction between the π-bond and cation(or H2O)to be stronger than with benzene,but because this mechanism cannot change the bond strength significantly,the higher electron density can be neglected.

Figs.6 and 7 provide strong evidence for this theory.A difference in the rejection of 5.5 mol·m-3NaCl was observed only at low pressure(0.4 MPa),and at low concentration,the difference can easily be observed.However,for KCl,a difference in rejection is observed in the nanofiltration system at 5.5 mol·m-3.In previous studies[13,16],researchers have concluded that the cation-pi interaction of Na+is weaker than the cation-π interaction of K+,and our experiment is in agreement with this conclusion.In addition,the experiment could even become a type of proof for this conclusion.

4.4.2.Molecular dynamics theory

Since cation-π interactions are formed to create larger materials,the steric effect should increase and lead to higher retention,but the experiments gave a completely opposite phenomenon.In membrane theories,this phenomenon can only have one hypothesis:cations are dehydration at pore entrance.But this hypothesis cannot find appropriate explanation under membrane theories.However,molecular simulation can explain these two questions perfectly.

Previous investigations[27,28]have reported that a phenomenon exists called the activation energy that is defined as the potential required by an individual molecule to overcome the potential barrier created by its neighbors and squeeze past them into the next equilibrium position.The classical molecular simulation conclusion in carbon nanotubes(CNTs)indicates that friction at the inlet of the CNT is far larger than the friction in the pores;hence,the activation energy that the solution needs to break also satisfies this requirement.In other words, fluid transport across the pores is controlled by the pore entrance effect[27].

Organic membranes are different from CNT,but these investigations can still validly explain our phenomenon.The flow of a fluid through a pore is in essence a process in which fluid molecules overcome two types of flow resistance(i.e.,the friction between the pore wall and the fluid molecules and the interactions among the fluid molecules themselves)under a certain driving force[29].So different materials mean different frictions between pore walls and fluid molecules(as to aqueous systems,the materials' problem is mainly laid under different hydrophilic)[29].If we take rigid wall and chemical reaction between fluid molecules and materials into consideration,it doesn't matter if the materials are made of organic or not,which should not affect our explanation.Since solutes and solvent used in this experiment can also be depicted by expanded Nernst-Planck equations as mentioned above,special chemical reaction should not exist.

In this case we focused on cation hydrated radii's change.It's essence is some interactions between water molecule and pore wall.So it's reasonable to use CNT to compare.

Fig.6.Permeate volume flux vs.real rejection of KCl and KCl in KCl/benzylalcohol(3.7 mol·m-3)nanofiltration systems(VVMs);symbols:round-shaped points,real rejection in mixed solutions;square-shaped point,real rejection in pure solutions(VVM);the concentrations of KCl were(a)5.5 mol·m-3;(b)7.5 mol·m-3;(c)10.5 mol·m-3;and(d)12.5 mol·m-3(the higher line in each figure is the simulation rejection curve of KCl in pure solution,and the lower line of each figure is the simulation rejection curve of KCl in mixed solution which computed using the DSPM-DE model).

The range of bond strengths in cation-π interaction is approximately 3 to 4 times lower than direct interaction[26],which means that the interaction energy is between 3.75 and 5 kcal·mol-1.Because the DK Desal membrane is made of a type of hydrophilic material and the pore radius is 0.53 nm,the activation energy should be much higher than reported by Jeetu Set al.[28].Cation-π interaction should be the only type of bond that interacts directly with the cation in this system because in a previous report,Zhuet al.[29]have proved that the friction between the pore wall and the water molecules in the nanopores with higher levels of hydrophilic material accounts for 50%of all friction.Because CNT is a type of hydrophobic material,the activation energy in this case should be much higher than reported previously[28].In addition,the pore radius decrease leads to an increase in the water-water interaction,which is rather strong,leading to an increase in the activation energy and hence a subsequent increase in friction.The diameter of CNT is between 1 and 1.5 nm,so in this case,we can use the results of a previous report[28].This part should be a direct explanation for the K+Stokes radius decrease in the nanofiltration system.

4.4.3.Nanofiltration theory

Due to their smaller hydration radii compared to cations,anions adsorb to a greater extent onto the hydrophobic surface of the membrane than cations[30].However,at low concentrations,the adsorption of chloride ions on the surface is negligible[31],and cations may play an important role in the Donnan effect.Therefore,the Stokes radius of a cation drops constantly,allowing the cations to penetrate into the membrane more easily,which would cause an increase in the value ofXd.From Section 4.1,we can see that the pore radius is approximately 0.53 nm,so the ratio of the ion radius to the pore radius is close to 0.5,and thus,the steric hindrance effect will be very important[32].Based on Eq.(3),we can see that decreases in the Stokes radius of the ion definitely cause an increase in diffusion if the dynamic viscosity of the solution does not change.The decreased Stokes radius of the cation would allow the ions to penetrate into the membrane more easily.Therefore,in extremely dilute solutions,the adsorption of chloride ions on the surface is not very significant[31],and the cation may play an important role in the Donnan effect.From Eq.(4),in a negatively charged membrane,a smaller Stokes radius for the cation increases the value ofXd,which is another reason for the decreased rejection.Fig.6 also shows that in mixed solutions with lower volumetric fluxes,certain deviations were observed.These phenomena might occur because,with a lower volumetric flux,the coordination compounds formed by cations and benzyl alcohol abound in the bulk solution.When the volumetric flux increases,experiment time also increases,and more interactions between the cation and benzyl alcohol would disappear.Therefore,we observed that in systems with lower volumetric fluxes,the rejection decreases more obviously than in systems with higher volumetric fluxes.At the same time,in mixed solutions with higher electrolyte concentrations,the model fitted the experimental data perfectly,but in systems with lower concentrations,certain deviations were observed.As the concentration of the electrolyte increased,the fitting line indicated that the dominant substance was electrolyte and that the effect of the electrolyte-benzyl alcohol decreased as the concentration of electrolyte increased.

Fig.7.Permeate volume flux vs.real rejection of NaCl and NaCl in NaCl/benzyl alcohol(3.7 mol·m-3)nanofiltration systems(VVM);the concentrations of NaCl were(a)3.7 mol·m-3;(b)5.5 mol·m-3;and(c)7.5 mol·m-3.

Table 3Relationships between the wall concentration of KCl,charge density and the Stokes radius of K+.

5.Conclusions

The rejection rate of electrolytes by an NF organic membrane was found to decrease in an obvious manner when benzyl alcohol was added to the solution.The rejection drop depends on the salt and decreases with the salt concentration.The decrease in the electrolyte rejection rates was interpreted in terms of both the decrease in the Stokes size of the cation(due to partial dehydration of the cationviacation-π interactions)and the decrease in the effective charge density(as a result of the better penetration of the smaller cations into the inside of the pores,which results in a decrease in the membrane density).The cation-π interaction should be responsible for these phenomena.That is not only the cation-π interaction actually existing in the bulk solutions,and other factors cannot explain the retention of the salt decreasing in this case.Nanofiltration and molecular simulation theories can explain these phenomena perfectly.

Cation-π interaction was quantified using a DSPM-DE modelapplied to the KCl rejection rate.The variation in the rejection rates is higher for lower volumetric fluxes,and as the experiment time increases,the variations decrease because many cation-π bonds on the membrane broke.Additionally,as the concentration of electrolyte increases,the fitting line approaches perfection,which indicates that the dominant substance was the electrolyte and that the effect of the electrolytebenzyl alcohol decrease as the concentration of electrolyte increase.These phenomena finally proved that cation-π interactions can be proposed as a new reason for retention change in the electrolyte/organic nanofiltration system.

Nomenclature

ciconcentration of ioniwithin pore,mol·m-3

ci,fsolute feed concentration,mol·m-3

ci,psolute permeate concentration,mol·m-3

ci,wwall concentration of ioni,mol·m-3

Dimass diffusivity,m2·s-1

dthickness of oriented solvent layer(0.28/nm),m

dHcharacteristic linear dimension,m

Jvvolumetric permeation flux,m3·m-2·s-1

KKarman constant(dimensionless)

Lppermeability of membrane,m·s-1·Pa-1

mthe number of ions

NAAvogadro constant(6.022×1023mol-1)

nthe number of experimental points

Runiversal gas constant(8.314 J·mol-1·K-1)

Ri,expreal rejection of ioni,obtained experimentally

Ri,modreal rejection of ioni,predicted by the model

Ri,obsobserved rejection of ioni(dimensionless)

Ri,realreal rejection of ioni(dimensionless)

ri,sStokes radius of ioni,m

rpeffective pore radius,m

rsStokes radius of K+,m

ReReynolds number

ScSchmidt number

ShSherwood number

Tabsolute temperature,K

Utaverage tangential velocity,m·s-1

Xdeffective charge density,mol·m-3

Δx/Akeffective membrane thickness,m

zivalence of ioni

ε* dielectric constant of oriented water layer(dimensionless)

εppore dielectric constant(dimensionless)

η solvent viscosity within pores,N·s·m-2

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