Transport Studies of Divalent Ions through Nafion-117 Ion Exchange

Article pubs.acs.org/IECR

Transport Studies of Divalent Ions through Nafion-117 Ion Exchange Membrane in the Presence of Polyacrylate Chhavi Agarwal, Amol Mhatre, and A. Goswami* Radiochemistry Division, Bhabha Atomic Research Centre, Trombay, Mumbai, 400085 Maharashtra, India ABSTRACT: The effect of binding of divalent ions (Mg2+ and Ba2+) with a polyacrylate anion on the Donnan membrane equilibrium distribution and the transport kinetics have been studied using Nafion-117 membrane. The extent of association of the divalent counterions with polyacrylate has been ascertained from the deviation from Donnan membrane equilibrium condition. The equilibrium concentrations of ions in polyacrylate compartment have been found to be in excess of expected equilibrium concentrations obtained from Donnan membrane equilibrium condition. This has been attributed to higher electrostatic interactions of divalent ions with the polyacrylate backbone compared to the monovalent ions. The strength of binding has been found to depend on the nature of divalent ion. Precipitation has been observed for Ba2+, thereby changing the total salt concentration of the polyacrylate compartment. Also, the strength of counterion binding has been found to depend on salt concentrations of the two compartments. The rate of transport of the ions with polyacrylate in the receiver solution has been calculated based on modified Nernst−Planck (NP) approach. The calculated transport kinetics have been found to agree with the experimental transport kinetics.

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condensation.15,18,19 For multivalent ions, stronger binding may lead to site binding of counterion on the polymer backbone.20 As shown by Rubinstein et al., such binding may be opposed by the loss of translational entropy of the counterions due to their localization on the backbone.21−23 However, condensation may also lead to positive entropic effect owing to liberation of water molecules from the hydration shells of the counterions.19,24,25 The counterion binding has been studied by several methods e.g. for example, pH titration, conductivity,26 viscosity measurements, small-angle X-ray and neutron scattering,27 isothermal titration calorimetry,25 counterion diffusion measurement and partition of counterion between polyelectrolyte solutions and ion exchange resins. These studies are of interest for understanding the nature of binding between the counterion and polyelectrolyte backbone. Such studies are also useful for application of these polyelectrolytes as scale inhibitors in both detergents28 and seawater desalination29 and as masking agents.30 Due to the high charge on the polyelectrolytes, these ions are expected to be completely excluded from the ion-exchange membrane. Also, the presence of polyelectrolytes in any compartment can reduce the effective concentrations of the cations depending upon the extent of binding of the cations to the polyelectrolyte backbone. This may appreciably affect the Donnan equilibrium distribution of the cations between the two compartments. In our earlier work,31 the extent of binding of different monovalent ions (Li+, Na+, and Cs+) with polyacrylate anion was studied using Donnan membrane equilibrium. The binding of Cs+ ions with the polyacrylate anion was found to be stronger relative to Li+ and Na+ ions.

INTRODUCTION Ion-exchange membranes are permselective, allowing the passage of counterions while excluding co-ions. This phenomenon is called Donnan exclusion. It depends on the charge on the co-ion and concentration of the ions in the solution as well as in the membrane. Ion-exchange membranes are widely used in Donnan dialysis, electrodialysis and as polymer electrolyte membranes in fuel cells.1 The Donnan dialysis process is ecofriendly where the separation of metal ions and charged organic species is achieved based on the differences in solution concentrations and volume of the feed and receiver solutions.2−11 In Donnan dialysis, selective preconcentration and recovery can be achieved only for ions of different valences.12 This poor selectivity is a major disadvantage of the process. In order to overcome this problem, a chemical reagent selective to analyte ion is introduced in the feed or the receiving solution. This may alter the free metal ion concentrations, thereby altering the Donnan membrane equilibrium condition as well as the kinetics of transport of a metal ion. This is called as chemically facilitated Donnan dialysis (CFDD).7,12 The effectiveness of this process depends on the nature and the strength of binding of the metal ion with the complexing agent. Polyelectrolytes are water-soluble polymers with ionizable functional groups.13,14 These polymers find immense applications in the field of biology, medicine, and cosmetics and the food industry.15 Virtually all physicochemical properties of polyelectrolyte systems such as electrophoretic mobility, diffusion, and viscosity depend on the interactions between the polyanion chains and the counterions present in the solution. These interactions mainly depend on the nature of the functional groups on the polymer backbone, pH, ionic strength of the solution, and the nature of the counterions (size and valence), co-ion and temperature.16,17 An important area of polyelectrolyte research is the counterion binding whereby the counterion tends to neutralize the charge of the poly ion backbone by way of what is called Manning counterion © 2015 American Chemical Society

Received: Revised: Accepted: Published: 3445

December March 17, March 18, March 18,

22, 2014 2015 2015 2015 DOI: 10.1021/ie504957p Ind. Eng. Chem. Res. 2015, 54, 3445−3450

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Industrial & Engineering Chemistry Research

ing anions (X and Y). In order to prevent the film diffusion, the solutions were continuously stirred at 500 rpm. With time, the cations in the two compartments undergo ion-exchange until Donnan equilibrium is established. In order to obtain the transport kinetics, the concentration of the cations in each compartment has been monitored as a function of time by tagging one of the cations with a corresponding radiotracer and monitoring its activity. The distribution of the radiotracer between the two compartments at time, t → ∞ gives the experimental equilibrium distribution of the cations. Compartment I (feed compartment) was spiked with the carrier free 22Na and 133Ba radiotracers, giving approximately 6000 counts per minute per 100 μL solution. The radiotracers were obtained from the Board of Radiation and Isotope Technology, Mumbai, India. In order to monitor the transport of cations (Na+ and Ba2+), the radioactivity in each compartment was measured by pipetting out 100 μL of solution at regular time intervals until the equilibrium was reached. The samples were counted in a NaI (Tl) detector coupled to a 4k MCA. To ensure a constant counting efficiency, the sample− detector geometry was fixed. The radioactivity loss in the membrane was accounted for while calculating the percent transport.

The transport rate for the system studied was obtained by modifying Nernst−Planck approach32 to account for the fraction of metal ion bound to the polyacrylate. The bivalent ions due to their higher charge are expected to be strongly bound to the polyelectrolyte backbone compared to monovalent ion, thereby significantly altering the equilibrium distribution of the cations in the two compartments. In the present work, the effect of binding of divalent ions (Mg2+ and Ba2+) on the Donnan membrane equilibrium distribution and the transport kinetics have been studied using Nafion-117 membrane using a two compartment cell. These ions were chosen based on their difference in hydration characteristics. The deviation from Donnan equilibrium has been used to obtain the fraction of the counterion bound to the polyacrylate. A modified Nernst−Planck approach has been used to calculate the transport rate of counterions in the presence of polyacrylate and results have been compared with experimental transport kinetics.31 The dependence of interaction between the polyacrylate and counterion on their concentrations has also been studied based on Donnan membrane equilibria. Results have been qualitatively explained based on the difference in hydration characteristics of the ions.

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EXPERIMENTAL SECTION In the study, all the reagents used (NaCl, BaCl2, and MgSO4) were analytical grade. Deionized water (Milli-Q, U.S.A.) was used throughout the experiments. Nafion-117 ion-exchange membrane with an equivalent weight of 1100 g, and thickness of 200 μm was used. Removal of organic impurities and conditioning of the membrane with relevant salt solution were done as described in ref 33. Aldrich make poly(acrylic acid) (Mw = 250 000) (PAA) was procured as aqueous solution (35 wt % solution in water). Standardization of PAA solution and subsequent preparation of 0.1 N sodium polyacrylate (NaPAA) stock solution were done as described in ref 31. A typical arrangement of a two-compartment cell separated by the Nafion membrane, used for the present experiment, is shown in Figure 1. The feed (I) and the receiver (II) compartments initially contain AX and BY salt solutions, respectively, where A and B refers to the two counterions and X and Y are corresponding anions. Due to Donnan exclusion, the Nafion membrane allows only the exchange of cations (A and B) between the compartments, leaving behind the correspond-

THEORETICAL EQUATION +

The transport rate of the ions AzA in the ion-exchange process +

+

AzA ⇆ BzB has been calculated theoretically using Nernst− Planck equation. A modified Nernst−Planck approach was used for systems with polyacrylate, taking into consideration the counterion binding. The method has been described in our + earlier publication.31,32 In this method, the flux of the ions (AzA) across the thickness in the membrane is obtained by solving Nernst−Planck equation under electroneutrality and zero + electric current condition. The flux of the ions AzA is then integrated over the thickness of the membrane (L) and is given by JA =

1 (X1 + X 2) L

(1)

where X1, X2 are given by X1 =

⎛ z B y1 ⎜ y zA( CAI − CAII) − DA zA zA y22 ⎜⎝ 3 ⎡ (y + zADA )Q − y zA CAII ⎤⎞ 3 ⎥⎟ Q ln⎢ 3 ⎢⎣ (y3 + zADA )Q − y3 zA CAI ⎥⎦⎟⎠

X2 =

⎛ zA2 y1 ⎜ y ( CAII − CAI) − DBz B z B2 y42 ⎜⎝ 4 ⎡ DBz BQ + y CAII ⎤⎞ 4 ⎥⎟ Q ln⎢ ⎢⎣ DBz BQ + y4 CAI ⎥⎦⎟⎠

+

of AzA and BzB in the two compartments. CAI , C BI , CAII and C BII are z+A

(3)

where y1, y2, y3, and y4 are given by

Figure 1. Schematic representation of a two compartment cell experiment at time, t = 0. CAI, CBI, CAII, and CBII are the concentrations +

(2)

z+B

the membrane concentrations of A and B at the two membrane solution interfaces. 3446

y1 = DA DBz B2

(4)

y2 = DBz B2 − DA zAz B

(5) DOI: 10.1021/ie504957p Ind. Eng. Chem. Res. 2015, 54, 3445−3450

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Industrial & Engineering Chemistry Research

Table 1. Initial Composition of the Two Compartments for Different Sets of Experiments, Corresponding Expected and Observed Equilibrium Concentrations, %, and Fraction of Counter Ions, f, Bound to the Polyacrylate Donnan equilibrium concentration(%) of AzA+ in I

composition experiment no.

I(AX)

II(BY)

expected

observed

f

1. 2. 3. 4. 5. 6. 7.

MgSO4(0.05M) MgSO4(0.05M) BaCl2(0.1M) BaCl2(0.1M) BaCl2(0.05M) BaCl2(0.0025M) BaCl2(0.0005M)

NaCl(0.1M) NaPAA(0.1M) NaCl(0.1M) NaPAA(0.1M) NaPAA(0.1M) NaPAA(0.005M) NaPAA(0.001M)

50 50 70 70 50 50 50

53 41 69 63 37 39 52

NA 0.66 NA 0.79 0.80 0.57 NA

y2

y3 =

zB

was solved to obtain CA as a function of time (t) using fourth order Runge−Kutta method.

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(6)

y4 = DA zA2 − DBzAz B

RESULTS AND DISCUSSION Donnan Membrane Equilibrium Studies. Table 1 gives the initial composition of the feed (I) and the receiver (II) compartments for the experiments carried out. The table also gives the expected and the observed equilibrium distribution (%) of the ions. The condition of Donnan membrane equilibrium (eq 12) has been used to calculate the expected equilibrium concentrations

(7)

Here, Ji and Ci are the flux and the concentration of the ith ion in the ion-exchange membrane, Di the self-diffusion coefficient (SDC), zi the ionic charge, and F, R, and T the Faraday and gas constants and the temperature, respectively. Q is the ionexchange capacity of the membrane, obtained using literature value of equivalent weight of the dry Nafion membrane and its measured density. It is assumed that the swollen membrane has approximately same Q as that of dry membrane. The + concentration of ion AzA in the membrane/solution interface is calculated using Donnan relation34 ⎛ CA ⎞ ⎜ ⎟ ⎝ CA ⎠

zB

⎛C ⎞ = ⎜ B⎟ ⎝ CB ⎠

⎛ CAI ⎞ z B ⎛ C BI ⎞ zA ⎟ ⎟ =⎜ ⎜ ⎝ CAII ⎠ ⎝ C BII ⎠

When the experiment is carried out with 0.05 M MgSO4 and 0.1 M NaCl in the two compartments, the Donnan equilibrium principle has been found to be valid. When NaCl is replaced with NaPAA, favored transport of Mg2+ to the second compartment has been observed. The result is similar to CsCl-NaPAA system studied earlier.31 This indicates strong interaction of Mg2+ ion with the polyacrylate anion. Though divalent, the fraction of Mg2+ ions bound to the polyacrylate ion (0.66) is comparable to Cs+ ions.31 This is due to higher hydration radii of Mg2+ ions. For the 0.1 M NaCl−0.1 M BaCl2 system also, in absence of polyacrylate, Donnan equilibrium condition is obeyed. However, preferential transport of Ba2+ is observed when NaCl is replaced with NaPAA. In this experiment (0.1 M NaPAA−0.1 M BaCl2), BaPAA precipitate appears at the end in the second compartment, thus reducing the effective salt concentration. However, complete transport of Ba2+ is not observed in spite of precipitation. The fraction of bound Ba2+ ions to the polyacrylate backbone ( f), as obtained from the measured equilibrium concentrations, is found to be 0.79 (Table 1), indicating 79% of the Ba2+ ions are bound to the polyacrylate anion. This is due to the lower hydration radius of Ba2+ compared to Mg2+ ions, leading to stronger electrostatic interactions between Ba2+ ions and the polyacrylate anions. Divalent Ba2+ ions are capable of binding two COO− groups simultaneously, both intramolecularly and intermolecularly causing them to collapse, leading to precipitation of Ba2+ polyacrylate.35 Absence of precipitation in Mg2+−Na+ system at same concentration shows that as the hydration radii of the multivalent ion increases, the threshold concentration required for precipitation increases. In order to see the effect of Ba2+ concentration on the counterion binding, experiments have been performed at lower Ba2+ concentration (0.05 M), keeping NaPAA concentration fixed (0.1 M). The deviation from Donnan membrane equilibrium is again observed (Table 1), followed by

zA

(8)

For systems with polyacrylate in compartment II, a part of the counterion is not available for ion-exchange and the above equation is replaced by ⎞ z B ⎛ C B ⎞ zA ⎛ CA =⎜ ⎟ ⎟ ⎜ ⎝ (1 − f )CA ⎠II ⎝ C B ⎠II

(9)

Here, f represents the fraction of the counterions in solution that are bound to the polyacrylate. It is defined as

f=

CAII(b) CAII

(10) z+A

where CAII(b) is the equilibrium concentration of A bound to polyacrylate in compartment II and CAII(eq) represent the total + equilibrium concentration of AzA in compartment II. The factor “f ” is obtained from the deviation of measured and predicted equilibrium concentrations. The method of its calculation is given in detail in our previous paper.31 In the present calculations, f has been assumed to be constant throughout the transport experiment. The rate of change of concentration + + of AzA in the solution (CA) is obtained from the flux of ions AzA through the membrane, using law of conservation of mass, as given by dCA S = − JA dt V

(12)

(11)

where V and S are the volume of the compartment and the effective membrane area, respectively. This differential equation 3447

DOI: 10.1021/ie504957p Ind. Eng. Chem. Res. 2015, 54, 3445−3450

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Industrial & Engineering Chemistry Research precipitation at the end of the experiment. The fraction of bound Ba2+ ions has also been calculated for this system, and found to be same as 0.1 M NaPAA−0.1 M BaCl2 system. Since the counterion condensation of an ion may depend upon the concentration of the polyelectrolyte in solution, experiments have also been carried out at low polyacrylate concentrations for NaPAA−BaCl2 system. A significant deviation from the Donnan equilibrium has been observed for 0.005 M NaPAA concentrations (Table 1). Here also, favored transport of Ba2+ ions to polyacrylate side followed by precipitation in polyacrylate side at the end has been observed. For still lower concentration of NaPAA (0.001 M), Donnan equilibrium has been found to follow and no precipitate has been observed. At this low NaPAA concentration, a given divalent metal ion is more likely to become associated with two carboxylate ions in the same polymer molecule. Also, as explained from Rubinstein point of view, for dilute polyelectrolyte solution, the entropic penalty for counterion localization is very high and the counterions prefer to remain dissociated.21−23 The absence of precipitation in 0.001 M NaPAA−0.0005 M BaCl2 system, whereas its presence in all the other NaPAA-BaCl2 systems studied can also be explained using the phase boundaries calculated by Schweins et al.36 for NaPAA−Ca2+/Sr2+/Ba2+ systems in the presence of 0.01 M NaCl. For our systems, the critical concentrations of Ba2+ ions required for precipitation of NaPAA (CBa2+(c)) has been calculated based on the linear relationships between the critical concentration of Ba2+ ions and NaPAA concentration as given by36 C Ba 2+(c) = 0.622 + 0.160C NaPAA

Figure 2. Kinetics of the exchange Mg2+(I) ⇆ *Na+(II), for the equilibrating solution containing 0.05 M of MgSO4 and 0.1 M of NaCl/NaPAA. The symbols represent the experimentally obtained curves and the solid lines represent the theoretical curves obtained from NP equation.

and experimental time profile during the initial stage, while the two profiles match close to equilibrium. The mismatch in the experiment and calculations in the initial time profile of NaPAA−MgSO4 system is due to the assumption of constant fraction bound to the polyacrylate ( f) even when the concentration of Mg2+ ions is low in compartment II. Figure 3 shows the plot of variation of ratio of Mg2+ ion concentration in the receiver (Cr) to feed (Cf) compartment as

(13)

For our system, the equilibrium concentration of Ba2+ in the receiver is found to be less than the critical Ba2+ concentration required for precipitation of 0.001 M NaPAA, whereas at all the other higher concentrations of NaPAA, the equilibrium concentration of Ba2+ in the receiver is well above the corresponding critical Ba2+ concentrations. This explains the polyacrylate precipitation in systems with 0.1 and 0.005 M NaPAA, but with no precipitation for 0.001 M NaPAA. It has also been shown by Schweins et al. that the critical concentration of metal ion required for precipitation increases in the order Ca2+ > Sr2+ > Ba2+. The Mg2+ equilibrium concentration in our study is well below the expected critical concentration required for polyacrylate precipitation, explaining the absence of precipitation in Mg2+−NaPAA system. Transport Kinetics Studies. Figure 2 shows the time profile of concentration of Mg2+ ion in compartment I for Mg2+−Na+ systems. The initial time profile for the system with and without polyacrylate has been found to be comparable. This shows that at low Mg2+ ion concentration in compartment II, the Mg2+ ions does not get bound to the polyacrylate and almost behaves as NaCl−MgSO4 system. But as the transport progresses, due to counterion binding, a part of Mg2+ ions present in polyacrylate compartment (II) is not available for ion-exchange, thereby driving the equilibrium toward higher Mg2+ percent transport. The kinetics of exchange of Mg2+ ⇆ Na+ with and without polyacrylate has been calculated based on modified Nernst− Planck approach, with the value of f obtained experimentally from Donnan membrane equilibrium studies. For MgSO4− NaCl system, the experimental and the calculated time profile agrees reasonably well. This shows the validity of the theoretical approach for monovalent−divalent system. For MgSO4− NaPAA system, there is a mismatch between the calculated

Figure 3. Ratio of Mg2+ ion concentration in receiver compartment (Cr) to its concentration in the feed compartment (Cf) as a function of time for systems with and without polyacrylate.

a function of time for theoretical as well as experimental profiles, for both the systems. The permeability coefficient can be obtained from the slope of the curves using the equation34,37 ⎡ VL ⎛ d(C /C ) ⎞⎤ r f t P(exp) = ⎢ ⎜ ⎟⎥ ⎝ ⎠⎦ dt ⎣ S

(14)

where S and L are the area and thickness of the membrane, respectively, and V is the volume of each compartment. For MgSO4−NaCl system, the permeability coefficients from the 3448

DOI: 10.1021/ie504957p Ind. Eng. Chem. Res. 2015, 54, 3445−3450

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Industrial & Engineering Chemistry Research experimental and theoretical profiles have been found to be 4.4 × 10−10 m2/s and 4.2 × 10−10 m2/s, respectively. However, for MgSO4−NaPAA system, the experimental (2.3 × 10−10 m2/s) and theoretical (8.9 × 10−10 m2/s) permeability coefficients do not match. As explained earlier, the discrepancy may be due to use of constant “f ” in theoretical calculations. Figure 4 shows the time profile of Ba2+ concentration in compartment I for the Na+−Ba2+ system. In this case, the

Figure 5. Ratio of Ba2+ ion concentration in receiver compartment (Cr) to its concentration in the feed compartment (Cf) as a function of time for systems with and without polyacrylate.

for BaCl2−NaPAA system, the permeability coefficient values are 3.0 × 10−10 m2/s and 2.7 × 10−10 m2/s, respectively. The experimental and theoretical permeability coefficients have been found to agree well for both the systems. The higher permeability coefficients in the presence of polyacrylate show faster transport kinetics.

Figure 4. Kinetics of the exchange Ba2+(I) ⇆ * Na+(II), for the initial concentrations of 0.1 M of BaCl2 and 0.1 M of NaCl/NaPAA in the respective compartments. The symbols represent the experimentally obtained curves and the solid lines represent the calculated values obtained from NP equation.

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CONCLUSION In this study, Donnan equilibrium principle has been used to obtain information about the nature of interaction of monovalent and divalent ions with the polyacrylate ion. Both the divalent ions (Ba2+ and Mg2+) are found to bind strongly with the polyacrylate, with polyacrylate getting precipitated in the presence of Ba2+ ions. The fraction of ions bound to the polyacrylate has been obtained from the deviation from Donnan membrane equilibrium condition. The modified Nernst−Planck approach has been used successfully to predict the rate of ion exchange in the Donnan dialysis process even in the presence of a complexing agent. The difference in affinity of polyacrylate with Mg2+ and Ba2+ ions can be exploited to separate divalent ions from each other.

transport rate has been found to be faster in the presence of polyacrylate. This shows that Ba2+ binds to the polyacrylate strongly throughout the transport process, thereby leading to higher concentration gradient across membrane. The transport profiles of the 0.1 M NaCl−0.1 M BaCl2 system and 0.1 M NaPAA−0.1 M BaCl2 system have also been calculated using the input parameters given in Table 2. The value of f obtained Table 2. Input Parameters for the Calculations input parameters ion-exchange capacity, Q (kmol m−3) membrane thickness (m) membrane area (m2) DNa+ (m2 s−1) DBa2+ (m2 s−1) DMg2+ (m2 s−1) a

1.6 2.0 9.6 1.0 1.5 4.0

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× 10−4 × 10−4 x10−10 a × 10−11 a × 10−11 b

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +91-22-2559-3688. Fax: +91-22-2559-5151. Author Contributions

Ref 33. bRef 32.

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

experimentally has been used to calculate the kinetic profile for Ba2+−NaPAA system. The calculated profiles are shown in Figure 4. For both systems (with and without polyacrylate), the calculated and the experimental profiles match very well, indicating the generality of modified NP approach. Figure 5 shows the plot of ratio of Ba2+ ion concentration in the receiver (Cr) and feed (Cf) compartment as a function of time for theoretical as well as experimental profiles, for both the systems. For BaCl2−NaCl system, the permeability coefficients from the experimental and theoretical profiles have been found to be 1.3 × 10−10 m2/s and 1.4 × 10−10 m2/s respectively and

Notes

The authors declare no competing financial interest.

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ABBREVIATIONS Ci = Concentration of the ith ion in solution Ci = Concentration of the ith ion in ion-exchange membrane Ji = Flux of the ith ion in ion-exchange membrane Di = Self-diffusion coefficient in of the ith ion in ionexchange membrane zi = Ionic charge

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(19) Molnar, F.; Rieger, J. Like-charge attraction between anionic polyelectrolytes: Molecular dynamics simulations. Langmuir 2005, 21, 786. (20) Boisvert, J.-P.; Malgat, A.; Pochard, I.; Daneault, C. Influence of the counter-ion on the effective charge of polyacrylic acid in dilute condition. Polymer 2002, 43, 141. (21) Dobrynin, A. V.; Rubinstein, M. Counterion Condensation and Phase Separation in Solutions of Hydrophobic Polyelectrolytes. Macromolecules 2001, 34, 1964. (22) Deshkovski, A.; Obukhov, S.; Rubinstein, M. Counterion Phase Transitions in Dilute Polyelectrolyte Solutions. Phys. Rev. Lett. 2001, 86, 2341. (23) Liao, Q.; Dobrynin, A. V.; Rubinstein, M. Molecular Dynamics Simulations of Polyelectrolyte Solutions: Nonuniform Stretching of Chains and Scaling Behavior. Macromolecules 2003, 36, 3386. (24) Bulo, R. E.; Donadio, D.; Laio, A.; Molnar, F.; Rieger, J.; Parrinello, M. Site Binding of Ca2+ ions to polyacrylates in water: A molecular dynamics study of coiling and aggregation. Macromolecules 2007, 40, 3437. (25) Sinn, G.; Dimova, R.; Antonietti, M. Isothermal titration calorimetry of the polelectrolyte/water interaction and binding of Ca2+: Effects determining the quality of polymeric scale inhibitors. Macromolecules 2004, 37, 3444. (26) Rieger, J. A new approach towards an understanding of scaling in the presence of polycarboxylates. Tenside, Surfactants, Deterg. 2002, 39, 221. (27) Rieger, J.; Hädicke, E.; Büchner, K.-H. In Proceedings of the IDA World Congress on Desalination and Reuse; Int. Desalination Association: Topsfield, MA, 2002 (CD-ROM). (28) Volchek, K.; Krenstel, E.; Zhilin, Y.; Shtereva, G.; Dytnersky, Y. Polymer binding/ultrafiltration as a method for concentration and separation of metals. J. Membr. Sci. 1993, 79, 253. (29) Leeuwen, H. P. V.; Cleven, R. F. M. J.; Valenta, P. Conductometric analysis of polyelectrolytes in solution. Pure Appl. Chem. 1991, 63, 1251. (30) Schweins, R.; Huber, K. Collapse of sodium polyacrylate chains in calcium salt solutions. Eur. Phys. J. E 2001, 5, 117. (31) Agarwal, C.; Mhatre, A.; Goswami, A. Transport studies of monovalent ions through Nafion-117 ion-exchange membrane in presence of polyacrylate. Sep. Sci. Technol. 2014, 49, 2650. (32) Agarwal, C.; Chaudhury, S.; Pandey, A. K.; Goswami, A. Kinetic aspects of Donnan dialysis through Nafion-117 membrane. J. Membr. Sci. 2012, 415−416, 681. (33) Goswami, A.; Acharya, A.; Pandey, A. K. Study of self-diffusion of monovalent and divalent cations in Nafion-117 ion-exchange membrane. J. Phys. Chem. B 2001, 105, 9196 and references therein.. (34) Helfferich, F. G. Ion Exchange; McGraw-Hill Book Company: New York, 1962. (35) Huber, K. Calcium induced shrinking of polyacrylate chains in aqueous solution. J. Phys. Chem. 1993, 97, 9825. (36) Schweins, R.; Goerigk, G.; Huber, K. Shrinking of anionic polyacrylate coils induced by Ca2+, Sr2+, and Ba2+: A combined light scattering and ASAXS study. Eur. Phys. J. E 2006, 21, 99. (37) Suresh, G.; Pandey, A. K.; Goswami, A. Permeability of water in poly(perfluorosulfonic) acid membrane with different counter ions. J. Membr. Sci. 2007, 295, 21.

x = Distance in the membrane F = Faraday constant R = Gas constant T = Temperature φ = Electrical potential Q = Ion-exchange capacity of the membrane f = Fraction of counterions bound to the polyacrylate backbone Cio = Initial concentration of ith ion in solution V = Volume of the compartment S = Effective membrane area L = Thickness of the membrane Ci(c) = Critical concentration of ith ion required for precipitation of polyelectrolyte

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REFERENCES

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DOI: 10.1021/ie504957p Ind. Eng. Chem. Res. 2015, 54, 3445−3450