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Acid-Base and Metal Complexation Chemistry of Phosphino-polycarboxylic Acid under High Ionic Strength and High Temperature Jianjun (Alan) Xiao Cognigen Corporation, 395 Youngs Road, Williamsville, New York 14221
A. T. Kan* and M. B. Tomson Department of Environmental Science and Engineering, MS-519, Rice University, Houston, Texas 77005 Received December 8, 2000. In Final Form: April 30, 2001 Phosphino-polycarboxylic acid (PPCA), a phosphorus-labeled linear polymer, has been widely used for controlling scale formation in the oil/gas field and water treatment processes because of its strong scaleinhibiting capability, high thermal stability, and environmental acceptability. PPCA is similar to poly(acrylic acid) with about 1% (w/w) phosphorus added to facilitate analytical measurement. In this study, the thermodynamic properties of PPCA and Ca-PPCA complexes in aqueous solution have been studied by combining electrostatic theory with potentiometric titrations. The acid-base and calcium complex solution chemistry of PPCA has been determined from 0.01 to 5 m ionic strength and from 25 to 90 °C. For simplicity, PPCA is reduced to a hypothetical, averaged monoacid, HA, with the same concentration of the PPCA monomer. Both proton and calcium dissociation reactions are defined as 1:1 type hypothetical reactions. That is, HA T H+ + A- with pKH ) pH - log(θu/θH) and Ca(A‚‚‚A) T Ca2+ + (A‚‚‚A)2- with pKCa ) pCa - log(θu/θCa) where θu stands for the dissociated fraction of HA, θH stands for the protonated fraction, θCa stands for the calcium-complexed fraction, (A‚‚‚A)2- is a unit of arbitrary combinations of two dissociated A- units, and pCa is the negative logarithm of the free calcium concentration. The corresponding constants for these two reactions, KH and KCa, are determined from the acid/base titrations. These constants are then fitted with a linear electrostatic model: pKH ) pKH,int + belecθu and pKCa ) pKCa,int + 2belecθu where KH,int and KCa,int are intrinsic constants at ionic strength (I), and refer to the condition of zero dissociation, and belec is an electrostatic factor determined from polyelectrolyte theory. Both pKH and pKCa are expressed as a function of ionic strength and temperature through pKH,int, pKCa,int, and belec: pKH,int ) 4.856 0.984I1/2+0.253I - 198.7/T for proton dissociation, pKCa, int ) 3.968 - 2.671I1/2 + 0.750I - 1102.3/T for Ca-PPCA dissociation, and belec ) 2.778 - 1.081I1/2 + 0.226I, where I stands for ionic strength in molality and T stands for temperature in Kelvin. The fitting results show that the electrostatic factor belec is not significantly influenced by temperature, which is reasonable according to electrostatic theory. These results can be used to analyze the equilibria of PPCA in a solution, given pH, ionic strength, temperature, and total metal (Ca2+ here) concentration.
Introduction Phosphino-polycarboxylic acid (PPCA) has applications in scale inhibition, corrosion inhibition, and sludge dispersion in oil/gas fields, water treatment, heat exchangers, boilers, and other areas. The formula of PPCA, H-{{CH2CH(COOH)}x-PO(OH)-{CH2CH(COOH)}y}zH, differs from poly(acrylic acid) (PAA) by only a few imbedded phosphino groups (P is about 1 wt % of PPCA). PPCA is representative of a flexible linear polymer scale inhibitor with an average molecular weight of around 3800. It is widely used in oil/gas fields and water treatment processes because of its strong scale-inhibiting capability,1-3 thermal stability, and environmental compatibility. Its concentration in solution can be much more easily and accurately monitored than PAA because of the imbedded phosphorus. In addition, PPCA or its derivatives might also be multifunctional inhibitors (e.g., for both scale and corrosion) in boilers, cooling water formulations, and oil wells, as has been observed for the corresponding molecular weight PAA.2,3
The application of polymeric inhibitors, including PPCA, is seriously limited by meager understanding of its solution chemistry, especially in quantitative description. Except for some experimental and field data,1-3 little systematic study on PPCA has been published. Even though PAA, analogue of PPCA, has been thoroughly studied as a model flexible linear polyelectrolyte, those PAAs studied are usually with high molecular weight, up to 100 000 and over. For lower molecular weight PAA, around 4000, which is in the molecular weight range of commercial scale inhibitors, its properties are not clear, especially under conditions such as high temperature, ionic strength, or both, as are often encountered in scale-forming environments. Therefore, it is necessary to quantitatively explore PPCA’s fundamental solution chemistry under a wide range of conditions. Generally, the proton and metal association/dissociation reaction of polyelectrolytes can be expressed in several ways, such as the aqueous/polymer phase equilibriaDonnan model,4 Manning theory,5 and 1:1 type site binding reaction.6 There are many similarities among these
* Corresponding author. E-mail:
[email protected]. (1) Chang, K. Y.; Patel, S. Mater. Perform. 1996, June, 48-53. (2) Grchev, T.; Cvetkovska, M.; Schultze, J. W. Corros. Sci. 1991, 32 (1), 103-112. (3) Patel, S.; Nicol, A. J. Mater. Perform. 1996, June, 41-47.
(4) Marinsky, J. A.; Elphralm, J. Environ. Sci. Technol. 1986, 20, 349. (5) Manning, G. S. J. Phys. Chem. 1981, 85, 870. (6) Lifson, S.; Katchalsky, A. J. Polym. Sci. 1954, 13, 43.
10.1021/la001720m CCC: $20.00 © 2001 American Chemical Society Published on Web 06/23/2001
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models. Among these models, the applicability of the aqueous/polymer phase equilibria-Donnan model to the multicomponent systems at high ionic strength and high temperature is not clear.7 The discontinuity of the predicted proton dissociation constants with the dissociated fraction makes the Manning theory not an ideal model for this study. The 1:1 type site binding model may not be appropriate for divalent and multivalent metal ion complexation with linear flexible polyelectrolytes.8 More complicated models of polymer acid-base and metal complexation reactions can be written than those used in this paper, but the simple semiempirical model developed in the next section has been found to be sufficient to represent the experimental data over wide ranges of composition and temperature. In this study, we take both proton and metal-polyelectrolyte association reactions as 1:1 type charge neutralization reactions, as will be discussed next. Equilibrium Reactions. To express the equilibrium reaction of PPCA with protons and metal ions, PPCA is reduced to a hypothetical, averaged monoacid, HA. HA and A- are used to represent its protonated and dissociated species, respectively. Since all functional groups are physically seated on the same flexible backbone, it is reasonable to define both proton and metal association/ dissociation reactions as 1:1 type charge neutralization reactions, similar to territorial binding. That is, the combination of one monovalent cation with PPCA neutralizes one charge, equivalent to one A- unit, while that of one divalent cation neutralizes two charges, equivalent to one (A‚‚‚A)2- unit; it is not necessary for the monomer units A to be physically adjacent to each other on the polymer backbone. In fact, using titration data it is not possible to determine where the individual A units are located on the polymer; for example, this would be equivalent to determining the four microconstants of the diprotic acid HBCH (B * C) from only a titration curve, which cannot be done. Molecular/electrostatic models can aid in this identification of microconstants when the backbone units are chemically quite different, such as phenolic versus carboxylic groups. Since the backbone units are nearly all the same and the calcium ion is not known to form highly specific complexes with carboxylic groups, there is no a priori reason to expect to need anything other than a reasonable polymer/electrostatic model, and this is born out by experiment, as will be shown. The speciation method used in this paper is similar to the method used by Morel and Hering8 to describe polyvalent ion sorption. Thus, the proton and the calcium dissociation reaction can be expressed by eqs 1-2 and 3-4, respectively:
HA T H+ + A-
pKH
(1)
with
pKH ) pH - log
() θu θH
(2)
and
Ca(A‚‚‚A) T Ca2+ + (A‚‚‚A)2-
pKCa
(3)
with (7) Miyajima, T.; Mori, M.; Ishiguro, S. J. Colloid Interface Sci. 1997, 187, 259. (8) Morel, F. M. M.; Hering, J. G. Principles and Applications of Aquatic Chemistry; John Wiley & Sons: New York, 1994.
pKCa ) pCa - log
( ) θu θCa
(4)
where θu ) [A-]/CA ) 2[(A‚‚‚A)2-]/CA stands for the dissociated fraction of the averaged monoacid of PPCA, and note that the second “)” is based on the assumption of charge neutralization reaction, that is, one Ca2+ neutralizes two charges, although not necessarily adjacent, as discussed above. θH ) [HA]/CA stands for the protonated fraction of the averaged monoacid of PPCA. θCa ) 2[Ca(A‚‚‚A)]/CA stands for the calcium-complexed fraction of the averaged monoacid of PPCA. [ ] stands for the free concentration of the corresponding species in the solution, and CA is the total concentration of PPCA as an averaged monoacid. Note that the pH in eq 2 is the negative logarithm of proton activity, which is usually measured by a pH meter, while pCa (in eq 4) is the negative logarithm of Ca2+ concentration, rather than its activity. In this study, all pKH values are obtained from the acid/ base titration of PPCA in the absence of Ca2+ and all pKCa values are obtained from the acid/base titration of PPCA in the presence of Ca2+. At each point of the titration, the protonated fraction θH can be calculated from the charge balance equation (in the absence of Ca2+):
[Na+]add + [H+] ) [A-] + [OH-]
(5)
which yields
θH ) 1 - ([Na+]add + 10-pH/γH - 10(pH-pKw)/γOH)/CA (6) where [Na+]add refers to the concentration of the strong base (NaOH) added; γH and γOH are the activity coefficients of H+ and OH-, respectively; pKw is the negative logarithm of the dissociation constant of water. The activity coefficients can be calculated with Pitzer theory.9 Note that the neutral salt for adjusting ionic strength is not shown in the charge balance equation, eq 5. In the absence of Ca2+ and other binding metal ions, θu ) 1 - θH and pKH can thus be calculated from eqs 2 and 6 at each titration point. For each titration curve, a plot of pKH versus θu can be derived (it is a horizontal line for a simple acid, such as acetic acid). This pKH versus θu curve is a straight line with a positive slope for PPCA; the slope and intercept can be interpreted with electrostatic theory for flexible linear polyelectrolytes.10 A set of pKH versus θu over a selected range of temperature (T) and ionic strength (I) were determined via potentiometric titrations. Acid/base titrations were also carried out for PPCA in the presence of Ca2+ and no other binding metal ions. If calcium is added as CaCl2, as was done herein, then the value of θH can still be calculated from eq 6. θu can then be calculated from eq 2 with the pKH determined separately in the absence of Ca2+. Thus, θCa ) 1 - θu - θH and [Ca2+] ) CCa - [Ca(A‚‚‚A)] ) CCa - θCaCA/2 where CCa refers to the total concentration of calcium in the solution, which is the sum of free and complexed calcium concentration. pKCa can be calculated from eq 4. Similarly to pKH, a curve of pKCa versus θu can also be derived from each titration curve of PPCA in the presence of Ca2+. A set of pKCa values versus θu over a selected range of ionic strength and temperature can then be determined via potentiometric titration. (9) Pitzer, K. S. Thermodynamics; McGraw-Hill: New York, 1995. (10) Tanford, C. Physical Chemistry of Macromolecules; John Wiley & Sons: New York, 1967.
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Electrostatic Model. The changes of pKH and pKCa with θu were treated with electrostatic theory. Since all functional groups are fixed on a backbone, the neighboring charged functional groups influence the association/ dissociation reactions on the given functional groups through electrostatic forces. Generally, the extent of this electrostatic effect depends on the polyelectrolyte structure, the properties of the binding ions, solution ionic strength, and temperature and can be quantitatively described with numerous molecular models.10-16 Among these models, the model of the spherical salt impenetrable macroions is the extension of the Debye-Huckel theory to polyelectrolytes. It is an appropriate approximation to express the solution properties of globular, compact proteins.10 The model of the spherical solvent-permeable macroions is a good approach to express the solution properties of partly packed proteins.10 The model of long rod shaped macroions is used to express the properties of highly charged polyelectrolyte ions at low ionic strength.10 The model of spherical, loosely coiled, and completely saltpenetrable macroions can be used to express the solution properties of flexible, linear polyelectrolyte ions. This last model, derived by Hermans and Overbeek,15 is employed in this paper because it fits PPCA’s structure well. According to this model, each polyelectrolyte molecule is spread over a relatively large volume so that the volume of the polyelectrolyte itself and that of the small binding ions are negligible. On the average, each polyelectrolyte molecule can be treated as a completely salt penetrable sphere with a uniform distribution of charge density in this spherical region. Thus, the relationship of pKH and pKCa with the dissociation fraction θu can be derived as15
pKH ) pKH,int + belecθu
(7)
pKCa ) pKCa,int + 2belecθu
(8)
and
with
belec ) 1.302N0ne2(RDT)-1R-1{(κR)-2 - 1.5(κR)-5 [(κR)2 - 1 + (1 + κR)2 exp(-2κR)]} (9) and
κ ) (8πF2/(DRT))1/2(I × 1000)1/2
(10)
where all variables are expressed in SI units: κ (m-1) is called the Debye-Huckel coefficient (1/κ is called the Debye length); N0 () 6.02 × 1023 mol-1) is the Avogadro constant; R () 8.314 J mol-1 K-1) is the gas constant; R (m) is the averaged equivalent radius of the polyelectrolyte molecules in solution; e () 1.6 × 10-19 C) is the elementary charge; n is the averaged number of functional groups on each molecule; D ) 4π0 with 0 () 8.854 × 10-12 C2 J-1 m-1) being the permittivity of vacuum and (dimensionless) being the dielectric constant of water. The dielectric constant of water is a function of temperature ( ) 78.3 at 25 °C); F () 96 485 C/mol) is the Faraday constant; I (mol/L) is the ionic strength; T (°K) is the temperature. (11) Debye, P.; Huckel, E. Phys. Z. 1923, 24, 185. (12) Harned, H. S.; Owen, B. B. Physical Chemistry of Electrolytes, 2nd ed.; Reinhold Publishing Corp.: New York, 1950; Chapters 2, 3. (13) Kirkwood, J. G.; Poirie, J. G. J. Phys. Chem. 1954, 58, 591. (14) Hill, T. L. Arch. Biochem. Biophys. 1955, 57, 229. (15) Hermans, J. J.; Overbeek, J. Th. G. Recl. Trav. Chim. 1948, 67, 761. (16) Tanford, C. J. Phys. Chem. 1955, 57, 229.
Note that in this specific model the electrostatic factor belec is independent of the properties of small binding ions. Furthermore, under a fixed ionic strength and temperature, the relationship of pKH or pKCa with the dissociated fraction θu depends on the relationship of the polyelectrolyte size R and θu. In other words, if R does not change significantly with θu during titration, belec will be a constant for a titration curve (eq 9). This means that pKH and pKCa are linearly proportional to θu, as explicitly shown by eqs 7 and 8. Therefore, eqs 7 and 8 were applied to determine the pKH,int, pKCa,int, and belec values from the experimentally determined pKH and pKCa, respectively, as will be illustrated below. Experimental Section Chemicals. Phosphino-polycarboxylic acid (PPCA) is the active component (50 wt %, average MW 3800) of a commercial chemical Bellasol S29 (FMC Co.). Bellasol S29 was used without further purification. NaCl, used for adjusting ionic strength, was from Fisher Scientific Co., 99.4%. CaCl2‚2H2O, used for titration, was from Fisher Scientific Co. NaOH titrant (1.600 ( 0.008 N and 0.1600 ( 0.0007 N) and EDTA titrant (0.800 ( 0.004 M and 0.0800 ( 0.0004 M) cartridges were from Hach Co. All stock solutions were made with deionized water. Analytical Techniques. PPCA was analyzed by the solidphase separation-UV digestion-ascorbic acid/phosphomolybdic acid blue method. Samples containing PPCA were acidified by HCl to pH 2. A C18 cartridge (Waters Corp.) was conditioned with methanol, water, and 0.01 N HCl in sequence. An acidified sample was then injected dropwise through the preconditioned C18 cartridge by a syringe, followed by about 5 mL of deionized water. The cartridge was then backward eluted dropwise with a 0.01 M Borax solution (pH 9.18). This eluate was collected and neutralized to pH 5. The phosphorus in the polymer was oxidized by potassium persulfate (0.1 g c.a.)/UV oxidation (HACH).17 The UV-digested solution was then cooled to room temperature, and phosphorus was analyzed by the ascorbic acid method.17 The total PPCA concentration in the product Bellasol S29 was determined by a COD method.17 Ca2+ concentration was analyzed either by EDTA titration or by ICP analysis.18 Apparatus. The experimental apparatus consisted of four 250 mL jacketed glass beakers connected in series to a temperature-controlled circulating bath ((0.5 °C) (VWR 1135). Two beakers were used for heating and degassing phthalate (pH 4.008) and Borax (pH 9.180) buffers for two-point calibration of the pH electrode. A third beaker containing electrode storage solution (phosphate buffer solution plus 5 g KCl/L) was used for preheating the pH electrode. The sample beaker and calibration beakers were each situated on an insulated stir plate and fitted with a Teflon lid. A small hole in the center of each lid allowed watermoistened nitrogen (ultrahigh purity) to be introduced into the sample solution or into the airspace above it via a small bore Teflon tubing. The thermometer, the pH electrode, and the titrant solution could be introduced through additional holes in each lid. These holes were covered with plastic or glass stoppers when not in use. Experiments were performed using a Hach digital titrator, a Ross combination pH electrode (Orion Inc.), and an Accumet model 15 pH meter (Fisher Scientific Co.), similar to previous work.19 Potentiometric Titration. Before each titration, the pH electrode was filled with new filling solution (Orion Inc.) and was kept in a storage solution at the same temperature as that of the sample for about 2 h to be stabilized. Immediately before and after each titration, the stabilized pH electrode was calibrated in preheated fresh buffer solutions (0.05 M potassium hydrogen phthalate and 0.01 M Borax). Around 100 mL solutions of different concentrations of PPCA with or without Ca2+ were (17) Water Analysis Handbook; Hach Co.: Loveland, CO, 1989. (18) Standard Methods for the Examination of Water and Wastewater, 18th ed.; Greenberg, A. E., Clesceri, L. S., Eaton, A. D., Eds.; American Public Health Association: Washington, DC, 1992. (19) Tomson, M. B.; Kan, A. T.; Oddo, J. E. Langmiur 1994, 10, 14421449.
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Table 1. pKH,int and belec Values Obtained by Fitting Equation 7 to Titration Data in the Absence of Ca2+ expt no.
PPCA (m monomer)
IS (m)
T (K)
pKH,int
belec
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
0.0250 0.0250 0.0025 0.0025 0.0250 0.0250 0.0250 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.2000 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025
1 1 1 1 1 1 1 1 1 0.2 0.05 3 5 1 2 1 2 3 5 1 0.01
297.6 297.7 297.7 297.9 343.2 343.2 343.2 343.2 343.2 343.2 343.2 343.2 343.2 343.2 343.2 323.2 343.2 343.2 343.2 363.2 343.2
3.473 3.473 3.436 3.457 3.493 3.519 3.549 3.546 3.559 3.832 4.084 3.365 3.331 3.541 3.371 3.511 3.446 3.393 3.304 3.591 4.223
1.839 1.845 2.018 1.987 1.898 1.921 1.906 1.929 1.914 2.322 2.438 1.643 1.445 1.858 1.782 1.984 1.740 1.589 1.481 1.866 2.803
Figure 1. Titration curves of PPCA (2.5 mm monomer) at 70 °C at different ionic strengths. The numbers in the legend are the ionic strengths (m) of the solution. See Table 1. transferred to a jacketed sample beaker. The ionic strength was adjusted by adding NaCl. After the solution was equilibrated at the fixed temperature, it was titrated by adding NaOH via digital titrator under N2 atmosphere. The potential of the pH electrode responding to each addition of NaOH was recorded. This potential was then converted to pH values. Thus, a titration curve of pH versus NaOHadd/CA ratio was obtained for each solution under a specific condition. The agreement between duplicate titration and back-titrations were generally excellent.
Results and Discussion Proton Dissociation. In Table 1 are listed PPCA concentrations and I and T values for the experiments that are used to determine pHH,int. Plots of acid/base titration curves (pH vs nonprotonated fraction, 1 - θH) at different ionic strengths and temperatures in the absence of Ca2+ are shown in Figures 1 and 2, respectively. As shown in Figure 1, pH decreases with an increase in ionic strength at a given dissociated fraction. This implies that the proton dissociation constant (KH) increases (eq 2) with ionic strength, as expected. In contrast to the effect of ionic strength, temperature has little significant effect on the proton association/dissociation between 25 and 90 °C (Figure 2). For all titration curves at high ionic strengths above 0.01 m, pKH is linearly related to the dissociated fraction, θu. This observation is consistent with that by Miyajima et al.20 for PAA and those by other researchers.21-24 Figure 3 shows one example of these linear relationships at ionic strength 1.0 m and 70 °C, expt 9 of Table 1. According to eq 9, this means that the polymer (20) Miyajima, T.; Mori, M.; Ishiguro, S.; Chung, K.; Moon, C. H. J. Colloid Interface Sci. 1996, 184, 279.
Figure 2. Titration curves of PPCA (2.5 mm monomer) at I ) 1.0 m at different temperatures. The numbers in the legend are temperature (°C). See Table 1.
Figure 3. The linear relationship of pKH of PPCA with dissociation fraction at I ) 1.0 m and 70 °C, expt 9.
size does not change significantly during titration in the range of dissociation fraction from 0.3 to 0.95. The variation of polymer size and configuration with the dissociation fraction may happen at very low ionic strengths or at the two extremes of the dissociation fraction. This is probably due to the direct binding of neutralizing cations (Na+) on the charged polymer backbone and has been thoroughly studied by Katchalsky,25 Gregor et al.,26 Nordmeier,27 and Huizenga et al.28 This Na+ binding would not change significantly with the dissociated fractions if the ionic strength is high enough (>0.05 M) and the dissociated fraction is in the range between 0.30 and 0.95. Therefore, in this paper, the effect from the direct binding of neutralizing cations (here Na+) is not treated separately, but combined with the ionic strength effect. The phosphino group (one per every 52 carboxyl groups) was not specifically treated, for the following reasons: (1) the proton on the phosphino group represents less than 2% of the total exchangeable protons, (2) the phosphino group on each PPCA molecule is always dissociated during the titration, and (3) the electrostatic effect due to the phosphino group is constant for all titrations. Furthermore, during titration, the concentra(21) Nagasawa, M.; Takashi, M.; Kondo, K. J. Phys. Chem. 1965, 69 (11), 4005. (22) Arnold, R.; Overbeek, J. Th. G. Recl. Trav. Chim. 1950, 69, 192. (23) Pals, D. T. F.; Hermans, J. J. Recl. Trav. Chim. 1952, 71, 433. (24) Kabo, V. Ya.; Itskovich, L. A.; Budtov, V. P. Polym. Sci. U.S.S.R. 1989, 31 (10), 2217. (25) Katchalsky, A. J. Polym. Sci. 1951, 7, 393. (26) Gregor, H. P.; Luttinger, L. B.; Loebl, E. M. J. Phys. Chem. 1955, 59, 34. (27) Nordmeier, E. Polym. J. 1994, 26 (5), 539. (28) Huizenga, J. R.; Grieger, P. F.; Wall, F. T. J. Phys. Chem. 1950, 72, 2636.
Chemistry of Phosphino-polycarboxylic Acid
tion of the polymer has a slight influence on the dissociation constant at low ionic strengths. At higher ionic strengths (g0.05 m), pKH does not change significantly with concentration. This is consistent with the observation of Nagasawa et al.,21 Arnold,29 and Samelson30 on PAA. In this study, the PPCA concentration is from 0.0025 to 0.2 m as monomer. To quantitatively describe effects of both temperature and ionic strength on the proton association/dissociation of PPCA, eq 7 was used to fit each titration datum in the absence of Ca2+ to obtain the corresponding pKH, int and belec values. Table 1 lists all pairs of pKH, int and belec for titrations under different ionic strengths and temperatures. These values were then analyzed with nonlinear regression via the SAS software package.31 The corresponding equations for pKH, int and belec as a function of ionic strength (I, m) and temperature (T, K) are as follows:
Langmuir, Vol. 17, No. 15, 2001 4665 Table 2. Relationship of belec with Temperature, Ionic Strength, and Ra I (m)
T (K)
DT (× 106) (C2 K/J/m)
κ (1/nm)
belec
Rcalb (nm)
Rgb (nm)
0.05 0.1 1 0.05 0.1
298.2 298.2 298.2 343.2 343.2
78.3 78.3 78.3 64.2 64.2
2.598 2.598 2.598 2.452 2.452
0.732 1.035 3.274 0.756 1.070
2.53 2.45 1.92 2.53 2.45
2.39 2.03 1.18 2.43 2.06
2.29 2.12 1.80
a R cal is calculated from belec (eq 12) through eq 9 for PPCA (MW ) 3800), while Rg is calculated from the equation derived by Rogan (ref 34) for PAA (MW ) 3800) at the same conditions. b Rcal and Rg are two radii of PAA using different definitions; see text.
pKH,int ) (4.856 ( 0.127) - (0.984 ( 0.048)I1/2 + (0.253 ( 0.019)I - (198.7 ( 43.0)/T r2 ) 0.999 (11) belec ) (2.778 ( 0.055) + (-1.081 ( 0.092)I1/2 + (0.226 ( 0.036)I
r2 ) 0.998 (12)
For each term, the number following “(” represents the standard deviation of the coefficient. These two equations were fitted under the following conditions: I of 0.01-5.0 m, T of 25-90 °C, and PPCA concentration of 0.0025-0.2 N. As shown in Figure 1, the model predictions (lines) by eqs 11 and 12 agree well with the experimental data (symbols). To test the reasonableness of these two equations, the results were compared to published data on PAA. From eq 11, pKH,int ) 4.19 for PPCA at zero ionic strength and 25 °C. This value is slightly smaller than pKH,int of 4.3 obtained by Miyajima et al.20 for PAA (with MW about 180 000 at 298 K and I of 0.01-0.5 M). However, it is close to the corresponding constant of PAA obtained by Arnold.29 Note that Arnold treated Samelson’s PAA data30 with a different electrostatic equation (pKH ) pKH,int + mθu1/3). This is because Samelson’s experiments were done with no added salt (I ) 0) so that both the ionic strength and the polymer configuration change significantly during the titration. Regardless, the pKH,int (4.25) of Arnold should be comparable to that of this study. In addition, a value of 4.2 for pKH,int of PAA (with MW 4000) at ionic strength of zero and 25 °C was used by Dupont et al.32 Therefore, the value of 4.19 for pKH,int at ionic strength of zero and 25 °C is reasonable. Although the extrapolation of eq 11, which was obtained from fitting experimental data at high ionic strength, down to ionic strength of zero seems applicable, the behavior of the polymer at very low ionic strength might actually be quite different (for example, the condensation of M+ on polymer might become important). The temperature dependence of pKH,int for PPCA at I ) 0 is ∂(pKH,int)/∂(1/T) ) -198.7 ( 32, which is close to that of propionic acid (pKH ) 4.87 at 25 °C33). No corresponding temperature-dependent data for PAA nor PPCA are available for comparison. (29) Arnold, R. J. Colloid Sci. 1957, 12, 540. (30) Samelson, H. Thesis, Columbia University; University Microfilms (Ann Arbor, MI), Publ. No. 9531, 1954. (31) SAS/STAT, version 6, SAS Institute Inc., 1991. (32) Dupont, L.; Foissy, A.; Mercier, R.; Motiet, B. J. Polym. Sci. 1995, 273, 364. (33) Handbook of chemistry and physics, 62nd ed.; CRC Press: Boca Raton, FL, 1981-1982.
Figure 4. Titration curves of PPCA in the absence and presence of 5 mm Ca2+. See expt 9 in Table 1 and expt 24 in Table 3.
According to eq 9, belec is the electrostatic effect of the charged neighboring groups on the given functional group. In this study, negligible temperature effect on belec is observed (eq 12). This is consistent with the electrostatic theory. The slight temperature dependence in D and T of eq 9 is partially canceled by a subtle change of both κ and the polymer size R at higher temperature. This is clearly demonstrated in Table 2. In Table 2, the ionic strength, temperature, , DT () 4π0T), and κ are listed in columns 1-5, respectively. The belec values calculated from eq 12 are listed in column 6. Rcal in column 7 is calculated from eq 9. Rg in column 8 is the gyration radius of the polyelectrolyte molecules derived from Rogan’s data on PAA.34 Interestingly, the absolute value of Rcal for PPCA obtained in this paper is reasonably consistent with the gyration radius (Rg) of a PAA34 of similar molecular weight. Contrary to temperature, ionic strength has a strong effect on belec (column 6). The effect of ionic strength on belec is related to both κ and R. An increase in ionic strength results in an increase in κ of eq 9 (column 5 in Table 2). This changes the charge distribution around the polymer backbone so that the configuration of the polymer changes. At high ionic strength, the charge centers on the polymer segments are effectively shielded by neutral salt cations so that the whole polymer is in a less expanded configuration. At low ionic strength, the polymer is in a highly expanded configuration due to strong intersegmental electrostatic repulsion. This trend of polymer size with ionic strength is demonstrated in Table 2. With an increase in ionic strength, both Rcal (column 7) and Rg (column 8) decrease, as expected. Calcium Complexation. The metal complex solution chemistry of PPCA is studied via acid/base titration of PPCA in the presence of Ca2+. An example of the effect of Ca2+ on the titration curve is shown in Figure 4. In the (34) Rogan, K. R. J. Colloid Interface Sci. 1993, 161, 455.
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Langmuir, Vol. 17, No. 15, 2001
Xiao et al. Table 3. pKCa,int Determined from Experimental Data and Equation 8, Where the Corresponding belec Value Was Determined in Table 1
Figure 5. Titration curves of PPCA (2.5 mm monomer) with 5 mm Ca2+ at 70 °C under different ionic strengths. The numbers in the legend are the ionic strengths of the solutions (m). See expts 22-26 in Table 3.
Figure 6. Titration curves of PPCA (2.5 mm monomer) with 5 mm Ca2+ at I ) 1.0 m and different temperatures. The numbers in the legend are the temperatures of the experiment (°C).
Ca2+
presence of ions, the solution pH decreases since protons are displaced by Ca2+ from the PPCA backbone. Ionic strength has a strong effect on the titration curves in the presence of Ca2+, as shown in Figure 5. In comparison with Figure 1, Ca2+ has a stronger effect on the titration curve at low ionic strength than at high ionic strength, which implies that the ionic strength not only affects the proton dissociation/ association but also affects Ca2+ binding with PPCA. Figure 6 shows the effect of temperature on the titration curves in the presence of Ca2+. The effect of temperature is not significant, which is similar to the titrations in the absence of Ca2+. For each titration in the presence of Ca2+, a corresponding titration in the absence of Ca2+ was performed. pKCa, int was then obtained by fitting eq 8 to the titration data in the presence of Ca2+ with the belec value obtained from the titration data in the absence of Ca2+ under the same conditions. A set of pKCa,int values under different ionic strengths and temperatures are listed in Table 3 and were analyzed by the SAS software package31 with the nonlinear regression equation as
pKCa, int ) (3.969 ( 0.415) + (-2.671 ( 0.149)I1/2 + (0.750 ( 0.057)I + (-1102.3 ( 139.7)/T r2 ) 0.998 (13) where the numbers following “(” are the parameter standard deviations. This equation was fitted with data over a wide range of conditions: I ) 0.05-5 m, T ) 298-
expt no.
PPCA (m monomer)
IS (m)
total Ca (m)
T (K)
pKCa,int
22 23 24 25 26 27 28 29 30 31 32
0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0250 0.2000
0.05 0.2 1 2 5 1 1 1 1 1 1
0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.0025 0.005 0.005
343.2 343.2 343.2 343.2 343.2 298.2 323.2 363.2 363.2 343.2 343.2
0.242 -0.346 -1.115 -1.483 -1.478 -1.647 -1.442 -0.971 -1.086 -1.175 -1.061
363 K, and PPCA concentration is from 0.0025 to 0.200 N. This equation shows that both ionic strength and temperature have stronger effects on pKCa,int than on pKH,int. In Figures 5 and 6, the lines are predictions from eq 13 and the symbols are experimental data. The model and experimental data match quite well. Extrapolation of eq 13 for pKCa,int gives a value of 0.27 at ionic strength of zero and room temperature. This value is smaller than the corresponding values of 0.68 for 1:1 type calcium-proprionic acid and 1.06 for calcium glutaric acid. Whereas the pKH,int values might be expected to be comparable to the value of the corresponding monomer carboxylic acid, the pKCa,int might not, because the interactions of Ca2+ ions with (A‚‚‚A)2- are not necessarily with adjacent A- units, as with glutaric acid; see modeling section for further discussion. Also, this difference between pKCa,int for PPCA and glutaric acid might also be due to the fact that the combination of Na+ with PPCA is treated in this paper as a part of the ionic strength effect. Dupont et al.32 reported pKCa,int ) 1.8 for PAA of an average MW 4000, but in their paper KCa is defined differently:
(COO‚‚‚Ca)+ T COO- + Ca2+ with
KCa ) [COO-][Ca2+]/[(COO‚‚‚Ca)+]
(14)
That is, the complex is treated as 1:1 type ion pairs with charge. Also, the condition used for Ca-PAA was at low neutral salt concentration (0.04 M KCl), high PAA concentration (2400 ppm), and high Ca2+ concentration (total 0.01 M). Under such conditions, network type PAACa-PAA interactions might form. At high PAA concentration and low ionic strength, PAA can even form a network with hydrogen bonds.35 Pochard et al.36,37 reported a Ba2+-PAA complexation constant as log Kc ) 6.5, where Kc is defined to be equal to [(-COO)2Ba]/[-COO-]2[Ba2+]. However, eq 13 represents the Ca-PPCA complexation well and is consistent with the value of log(β1)p ) 0.2 for the Ca2+-PAA complexation obtained at low ionic strength by Miyajima et al.7 with the definition of (β1)p ) [MA]p/[M]p[A]p via a Donnan based concept. Example Application. The equations above can be used to calculate the fractional distribution of protonated (θH), dissociated (θu), and complexed (θM) “species” of PPCA in solution at given I, T, pH, CCa, and CA values. The following conditions, which might exist in a cooling tower (35) Tanaka, N.; Kitano, K.; Ise, N. Macromolecules 1991, 24 (10), 3017. (36) Pochard, I.; Couchot, P.; Foissy, A. Colloid Polym. Sci. 1998, 276, 1088-1097. (37) Pochard, I.; Foissy, A.; Couchot, P. Colloid Polym. Sci. 1999, 277, 818-826.
Chemistry of Phosphino-polycarboxylic Acid
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or in oil/gas production, are used to illustrate the calculation procedure: I ) 1.0 m, T ) 343.2 °K, pH ) 5.5, CCa ) 0.1 m, and PPCA concentration equals 1.92 × 10-6 m. Note that for a polymer of 52 monomer units (e.g., PPCA), the total concentration of the monomer (CA) will be equal to 1.00 × 10-4 m. At the given I and T, the values of pKH,int ) 3.546, belec ) 1.957, and pKCa,int ) -1.165 can be calculated from eqs 11, 12, and 13, respectively. For a given set of pH, I, T, CCa, and CA values, the equilibrium speciation of the PPCA species can be calculated as follows. Equation 15 can be used to obtain a single equation with only θu as the unknown by combining eqs 2 and 7 to eliminate log KH and solving for θH and then combining eqs 4 and 8 to eliminate log KCa and solving for θCa. Finally, eq 16 can be used to eliminate free ionic Ca2+ concentration. The resultant equation, eq 17 below, is a function only of θu and known experimental values and can be readily solved by most root-finder routines, such as Microsoft Excel Goal Seek:
θu + θCa + θH ) 1
(15)
[Ca2+] ) CCa - θCaCA/2
(16)
2belecθu
θu + (CCaθu10
{ (
) KCa,int
)}
θu102belecθuCA 1+ 2KCa,int
-1
+
θu10-pH10belecθu ) 1 (17) KH,int
Once θu is calculated from eq 17, the values for θCa and θH can easily be calculated as the second and third terms, respectively, of the left-hand side of eq 17. Then, θCa can be substituted into eq 16 to calculate the concentration of Ca2+. When eq 17 was solved with the above set of conditions, θu ) 0.526, θH ) 0.063, θCa ) 0.411, and the concentration of free calcium Ca2+ = 0.100 m because CCa
. CA, as is typically the practical situation when PPCA is being used as a scale inhibitor. Conclusions This paper has systematically and quantitatively investigated the acid/base and metal complexation chemistry of the commercial scale inhibitor PPCA over a wide range of ionic strength and temperature. In this study, PPCA is reduced to a hypothetical, averaged simple monoacid and all association/dissociation reactions are simplified as hypothetical 1:1 type reactions. It has been found that both proton and calcium dissociation constants can be expressed by electrostatic theory and predicted at a given ionic strength and temperature. Both ionic strength and temperature affect pKH,int and pKCa,int. The effect of ionic strength and temperature on pKCa,int is much stronger than that on pKH,int. Ionic strength decreases polymer size and consequently decreases belec while temperature has no significant influence on the value of belec or on polymer size. The extrapolated values of both pKH,int and pKCa,int of PPCA at zero ionic strength and 25 °C are found to be consistently smaller than the corresponding values of model glutaric acid. pKH,int of PPCA at ionic strength of zero and 25 °C is in a good agreement with that of its analogue, PAA, while no appropriate reference is available for a direct comparison between pKCa,int of Ca-PPCA and that of Ca-PAA. The results of this study can be used to quantitatively evaluate the solution chemistry of PPCA and its performance under widely different conditions. Acknowledgment. The financial support of Rice University Energy and Environmental Systems Institute and Brine Chemistry Consortium of companiessAramco, B.J. Services, Baker-Petrolyte, Champion Technologies, Inc., Chevron Petroleum Technologies, Inc., Conoco, Inc., Texaco, Inc., British Petroleum, Nalco Exxon, Shell, and Solutiasto this research is greatly appreciated. LA001720M
