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Polyelectrolyte Titration: Theory and Experiment
Itamar Borukhov(a)∗, David Andelman(b)†, Regis Borrega(a), Michel Cloitre(a) Ludwik Leibler(a)^ and Henri Orland(c)‡ (a) (^) Unit´e Mixte de Recherche CNRS/Elf Atochem (UMR 167) 95 rue Danton, B.P. 108, 92303 Levallois-Perret Cedex, France (b) (^) School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv 69978, Israel (c) (^) Service de Physique Th´eorique, CE-Saclay, 91191 Gif-sur-Yvette, Cedex, France (May 6, 2000) Titration of methacrylic acid / ethyl-acrylate copolymers is studied experimentally and theoret- ically. At low salt concentrations, this polyacid exhibits a plateau in the titration curve below the neutralization point. The plateau has been often attributed to a first-order phase transition associ- ated with polymer conformational changes. We argue that the specific shape of titration curves of hydrophobic polyelectrolytes is due to electrostatics and does not necessarily require a conformation change of the polyelectrolyte chains. We calculate the free energy at the mean-field level and its first-order (one loop) correction using a loop expansion. The latter is dominated by Debye-H¨uckel– like charge-charge correlations as well as by correlations between dissociation sites along the polymer chain. We show that the one-loop corrections to the free energy lead to titration curves that agree with experiments. In particular, the model explains the decrease of the pH at the plateau when the polymer concentration is increased or when salt is added to the solution.
I. INTRODUCTION
In recent years the search for environment-friendly materials has promoted the development of numerous water- soluble polymer applications. Polymers can be made water soluble by making them compatible with the strong polar environment of the aqueous media, e.g., by introducing either charges or strong dipoles on the chains [1–3]. Most charged polymers (polyelectrolytes) have a hydrophobic backbone. This hydrophobicity induces an effective attraction between monomers which competes with the Coulomb repulsion between charges. As a result, hydrophobic polyelectrolytes exhibit complex behavior including conformation changes, macro- and meso-phase separation, self- association and aggregation [4–6]. In spite of considerable theoretical and experimental effort, many questions remain open. In particular, the role of the coupling between hydrophobic attractions and long range Coulomb interactions on the physico-chemical properties of polyelectrolyte solutions is not fully understood. In this paper, we consider the role of such coupling in solutions of weak polyacids. Weak acid monomers (denoted HA) can undergo dissociation of the type
HA ⇀↽ H+^ + A−
where A−^ is the charged monomer attached to the backbone, while the dissociated H+^ charge dissolves into the solution. The dissociation/association is an equilibrium process satisfying detailed balance. The ionization degree determines the effective amount of charge on the polyelectrolyte chain, and it depends on the pH of the solution. At low pH the polymer is weakly charged while at high pH a larger fraction of monomers is dissociated and the polymer charge saturates to its maximal value. The most visible consequence is the solubility in water: hydrophobic polyacids can become water soluble at high enough pH, where the polymer charge is strong enough to overcome the chain hydrophobicity. In contrast to low molecular weight acids, the charged groups of polyacids are correlated as they are linked together along the chain. Indeed, the dissociation of one acid group is correlated in a complex way to the position and number of other charged groups on the chain. As a result, when the amount of charge on the chain
∗present address: Department of Chemistry and Biochemistry, UCLA, 607 C. Young Dr. East, Los Angeles, CA 90095-1569,
USA. e-mail: itamar@chem.ucla.edu †e-mail: andelman@post.tau.ac.il ‡e-mail: orland@spht.saclay.cea.fr
varies, the chain conformation is affected, and in turn, influences the dissociation of other groups [4,5,7,8]. Studies of the interplay of these competing effects has attracted a large amount of experimental [9–11] and theoretical [12–14] interest. Additional motivation for these efforts is related to the use of water-soluble polyelectrolytes and in particular alkali-swellable polymers in many industrial applications, such as coatings, food and cosmetic industries [15,16]. An elementary and wide spread experimental tool to characterize polyacids is to perform titration experiments. In these experiments, a strong base like NaOH is added to a solution of weakly charged polyelectrolytes. The pH of the solution and the equilibrium dissociation depends not only the amount of added base (like for ordinary acids [19]), but also on the polyelectrolyte concentration and presence of salt [17,18]. The overall shape of the titration curve depends on the nature of the polyelectrolyte which is titrated [17]. In the case of poly(acrylic) acid, the pH increases steadily with the dissociation degree. In contrast, the titration curve of poly(methacrylic) acid shows a maximum at low dissociation. Such a non-monotonous dependence has been related to a conformational transition. The case of hydrophobic polyelectrolytes is still more intriguing: the titration curve shows a large plateau where the pH is almost constant before the neutralization point is obtained. It has been argued in the literature that this plateau could be associated with a first-order phase transition between collapsed and swollen states of the polyacid chains [7,20]. In this paper we show that this peculiar shape of the titration curves can be explained without the need to rely on conformational phase transition of the polyacid chains. By including the effect of correlations of charges along the chains in the free energy, titration curves are calculated and show a behavior similar to the curves obtained in experiments. In particular, we explain how the pH depends on the polymer concentration and the amount of added salt. This dependence is special to polyacid solutions and is much weaker in monomeric acid solutions. The paper is structured in the following way: in the next section the mean-field free energy, its one-loop correction and the resulting titration equations and curves are presented. Then, the experimental measurements of the titration of methacrylic acid / ethyl-acrylate copolymers (MAA-EA) are discussed in Sec. III, and the comparison between theory and experiment is presented in Sec. IV. The full formalism relying on a field theoretical approach will be presented in a forthcoming publication.
II. THEORY
We present first the free energy leading to titration curves of weak polyacids in presence of added salt. Our system consists of four dissociating species: water, methacrylic acid monomers (denoted HA), NaOH titrating base and NaCl salt. The dissociation reactions are written as:
H 2 O ⇀↽ H+^ + OH−, HA ⇀↽ H+^ + A−, NaOH → Na+^ + OH−, NaCl → Na+^ + Cl−.
The partial dissociation of water and acid monomers is accompanied by an energy cost of breaking the molecular bond. They are denoted as ∆ 1 and ∆ 2 , respectively (in units of the thermal energy kBT ), and they are related to the mass action law [19] as will be detailed below. Since NaOH is a strong base it is fully dissociated. Similarly, the salt is fully dissociated. The polymer used in the experiment is a statistical copolymer composed randomly of two monomers: methacrylic acid and ethyl-acrylate. The total polymerization index is denoted as N. A fraction f = 1/3 of the monomers are composed of the methacrylic acid, i.e. they are ionizable. Since the charged monomers are distributed uniformly along the chain, we assign a partial charge f e to each monomer (“smearing” the charges along the chain). Note that f e is the nominal charge of each monomer, while the actual charge is related to the partial dissociation of the acid monomers. The concentration of monomers in the solution is denoted cm while the concentrations of the small ions are denoted cH, cOH, cNa, cCl. The base concentration added to the solution is denoted cB and its ratio to the MAA monomer concentration is defined as the degree of neutralization γ
γ ≡ cB f cm
where γ varies between zero (no added base) and infinity (large base excess). It is also useful to define the degree of ionization α of the acid
- One-loop correction to the free energy
The one-loop correction to the mean-field free energy ∆F will be presented in detail in a forthcoming publication. It is obtained by integrating over the quadratic fluctuations of the concentration fields. The correction term to the free energy is given (up to an additive constant) by
β∆F =
d^3 q (2π)^3 log Σ(q) (6)
Σ(q) =
[
1 + (wcm − v)cmN D(η)
](
q^2 + κ^2
η =
a^2 q^2 N (8)
where the Debye-H¨uckel screening length κ−^1 depends on the total concentration of small ions cI and the effective charge of the chain f A(1 − A)cm
κ^2 = 4πlB
cI + f A(1 − A)cm
≡ 4 πlBceff (9) cI = cH + cOH + cB + 2cCl (10)
lB = e^2 /εkBT ≃ 7 A is the Bjerrum length. The dissociation fraction of the acid monomers is given by
A =
eβeϕ^0 1 + eβeϕ^0
The Debye function D(η) [2] entering eq. (7) is given by
D(η) =
η
e−η^ − 1 η
In addition to the small ion contribution to the electrostatic screening, eq. (9) includes a term proportional to A(1 − A), where A is defined in eq. (11) above. This term accounts for changes in the dissociation degree of monomers depending on the local electrostatic potential. At small A values, A(1 − A) ≈ A, and the polymer contribution is the same as that of disconnected monomers. However, as A → 1 there is a substantial reduction (by the factor 1 − A) of the effective polymer charge that contributes to screening [8,21]. This reduction can be understood in terms of charge correlation along the chain. Dissociated monomers prevent further dissociation of other monomers. In our model, the concentrations cNa, cCl and cm are fixed by the amount of NaCl salt, NaOH base and polymer in the solution while the concentrations of dissociated H+^ and OH−^ ions depend on the degree of dissociation of water molecules and acid monomers (through the electrostatic potential ϕ 0 ). Thus cH, cOH and ϕ 0 are variational parameters, determined by the requirement that the free energy F = F 0 + ∆F is an extremum. In titration experiments cH can be monitored directly through the pH of the solution.
- Simplified free energy
In order to better understand the main contributions to the correction ∆F let us return to the expression for Σ(q), eq. (7). First, we note that the Debye function is bound between 0 and 1 (see also eq. (27)). For typical values of the physical parameters (see also Sec. IV),
v ≃ 25A^3 w ≃ 100A^6 N ≃ 104 cm ≃ 1mM (13)
it is easily seen that for long chains (N ≫ 1), the Debye function can be approximated by N D(η) ≃ 12 /a^2 q^2 , and
|wcm − v|cmN D(η) ≪ 1 (14)
Thus, the effect of the solvent is small and will be neglected in this section
∆F ≃
d^3 q (2π)^3
log
[
q^2 + κ^2 + z
]
where
z = 4πlBf 2 A^2 N D(η)cm (16)
For the same range of parameters κ^2 ≫ z and ∆F can be expanded to first order in z, ∆F = ∆F 0 + ∆F 1 :
∆F 0 =
d^3 q (2π)^3 log
[
q^2 + κ^2
]
The integral can be calculated by including a cut-off at large q, which cancels out the Coulomb self-energy, yielding
∆F 0 = −
κ^3 12 π
This result is the well known Debye-H¨uckel correlation energy of an electrolyte, because the polymer contribution is negligible and does not appear in this leading term. The correction ∆F 1 is given by
∆F 1 ≃
d^3 q (2π)^3
z q^2 + κ^2
≃ 6 f 2 A^2
lBκ−^1 a^2
cm (19)
This correction is due to correlations along the chain between dissociated monomers as can be seen from the following simple argument. Consider the screened electrostatic interaction between charged monomers on a single infinite chain. For a specific chain configuration the Coulomb energy per monomer is
Uel =
j 6 =
f 2 A^2 e^2 ε|rj |
e−κrj^ (20)
where rj is the spatial distance between the j = 0 monomer and another j = ± 1 , ± 2 , ... monomer. For a Gaussian random walk, the typical (most probable) distance between monomers is given by
〈rj 〉 =
2 j 3
a (21)
Assuming that rj can be approximated by 〈rj 〉 in eq. (20), and κa ≪ 1, the sum over j can be replaced by a continuous integral leading to eq. (19).
B. Titration equations
The free energy depends on the species concentrations: cOH, cNa, cH, cm, cCl. However, the concentration of monomers, Na+^ and Cl−, is fixed by the amount of polymer, base and salt added to the solution. On the other hand, the concentration of H+^ and OH−^ ions is determined self-consistently by minimizing the full one-loop free energy F = F 0 + ∆F , eqs. (5), (6), with respect to cH, cOH and ϕ 0. The resulting equations of state determine the dependence of the pH of the solution on the other system parameters.
log(cHωH) + βeϕ 0 − β∆ 2 + 2πlBI 1 = 0 (22) log(cOHωOH) − βeϕ 0 − β∆ 1 + β∆ 2 + 2πlBI 1 = 0 (23)
cH + cB − cOH − f cmA
1 − 2 πlB(1 − A)
(1 − 2 A)I 1 + 2AN f I 2
where the quantities I 1 and I 2 are defined as:
I 1 =
0
k^2 dk 2 π^2
1 + N cm(wcm − v)D(η) Σ(k)
I 2 =
0
k^2 dk 2 π^2
D(η) Σ(k)
and Λ is a short distance (large q) cut-off.
III. EXPERIMENT
A. Preparation of samples
The copolymer polyelectrolyte chains used in this study are prepared by standard emulsion polymerization tech- niques using neutral ethyl-acrylate (EA) monomers and methacrylic acid (MAA) monomers. The methacrylic acid is a weak acid with pKA=4.5. The weight fraction of MAA in the copolymer is equal to 0.35 taken to be f = 1/3 in the theoretical section. The emulsion polymerization is performed under starved monomer conditions. Under these conditions, the MAA and EA monomers are evenly distributed along the polymer chain. From size exclusion chro- matography, we have estimated the molecular weight of the chains to be of the order of 106 daltons, corresponding to a polymerization index N ≃ 104. The MAA-EA copolymers precipitate from the aqueous solution, as they are insoluble in water. The precipitate is carefully washed by ultrafiltration in order to remove surfactants, unreacted monomers and initiators. This cleaning procedure is stopped when the resistivity of the water flushed through the separation membrane is that of pure water (18.2 MΩ/m). The solid content of the stock solution is then determined accurately by drying and weighing.
B. Titration measurements
Polymer solutions at different weight concentrations are prepared by mixing weighted amounts of the stock solutions with de-ionized water. The dissolution of CO 2 is prevented by carefully de-gassing the solution with nitrogen. Each of the polyelectrolyte solutions is neutralized by an NaOH solution with a molar concentration ranging from 0.1M to 2M depending on the polymer concentration. Upon neutralization, the methacrylic acid is neutralized and repulsive forces due to the negative charges cause the chain to expand, resulting in the progressive solubilization of the polymer chains. As a result the solution becomes transparent and its viscosity increases. In the following, we shall characterize the neutralization of the polymer chains by the degree of neutralization, γ = cB/(f cm), which is the ratio of the amount of added base to the amount of available acids groups. Titration experiments are performed using a pH-meter (Metrohm 691) with a combined glass electrode. The measurements are made at 20oC with constant stirring under a nitrogen atmosphere. In parallel, the conductivity of the solution is measured (Metrohm 712). The measurements reveal that all small ions present in solution are free and contribute to the conductivity. This was checked at different γ values by changing the polymer concentration. The measured conductivity is the exact sum of the conductivities coming from the Na+^ ions (whose concentration is known from the amount of added NaOH base), and H+, OH−^ (known from the pH), while the small contribution of the polymer is negligible. These conductivity measurements demonstrate that the distribution of small ions is homogeneous in the solution, and no evidence for the Donnan effect and counter-ion condensation is observed.
IV. RESULTS: COMPARISON OF EXPERIMENTS WITH THEORY
In this section we present the experimental results for the titration curves of MAA-EA copolymers and compare them with the theory of Sec. II. Fig. 1(a) shows the titration curves measured for different polymer concentrations. They differ substantially from the titration curves of monomeric methacrylic acid (MAA). Methacrylic acid in aqueous solution has the typical behavior of a weak monomeric acid. The pH increases monotonously with γ, takes the value pKA = 4.5 for γ=0.5 and then jumps near γ=1. By contrast, the titration curves of MAA-EA copolymers exhibit the following behavior: in the range 0 < ∼ γ < ∼ 0 .2, the pH increases sharply to a plateau value which remains nearly constant up to γ ≈ 1 where the amount of added base equals that of the monomer (neutralization point). The jump of the pH for γ < ∼ 0 .2 is associated with a swelling phase transition of the latex particles i.e. a change of conformation of the chains. A similar phenomenon has been observed during the titration of pure poly(methacrylic acid) [23]. Let us mention that it is possible to calculate the swelling transition of the polymer as function of monomer concentration, cm. This can be done by minimizing the free energy with respect to cm (in addition to the other annealed degrees of freedom discussed above). Any non-convexity of the free energy signals the existence of a polymer precipitate in excess water. On Fig. 1 several titration curves are plotted for different concentrations of the same MAA-EA copolymer. As the concentration cm increases, the jump in the pH at γ ≈ 1 increases. Note that the deviation between the pH of the MAA monomeric acid and the MAA-EA copolymers (cm = 0.1 M in Fig. 1), is large in the plateau region (γ < 1).
On the other hand, the deviation is quite small for γ > 1 when the acid is almost completely dissociated. The value of the pH at the plateau depends strongly on the polymer concentration: the larger the cm, the lower the plateau. It is surprising that even though MAA-EA copolymers contain carboxylic groups, the plateau value of the pH below γ = 1 may be neutral or even greater than 7. This difference in behavior between the MAA monomers and MAA-EA copolymers might be associated with a complex structure of the MAA-EA copolymer for γ < ∼ 1. Collapsed microdomains may exist on the chains due to the competition between the hydrophobic attraction and Coulomb repulsion. The plateau region suggests the existence of such collapsed microdomains along the chains. For γ > ∼ 1 the chains are in a swollen state and their behavior resembles that of monomeric weak acid. In Fig. 1(b) we plot for comparison the titration curves as calculated from the theory. The titration equations presented in Sec. II.B are solved numerically by an iteration procedure starting from the mean field values as the first iteration and including the corrections of the second iteration. Since there are a couple of unknown physical parameters (like the second and third virial coefficients, v, w and the monomer size a), we do not try to fit the experimental titration curve. Rather, we note that the corresponding theoretical curves look very similar to the experimental ones, demonstrating the same type of plateau for low γ and the same trend with the monomer concentration. Figure 2 shows the effect of adding a monovalent salt (NaCl). The overall shape of the titration curves remain unchanged. However, the value of the pH at the plateau increases upon the addition of salt, while the pH for γ > ∼ 1 is essentially independent on the salt concentration. This indicates that neutralization of the carboxylic groups carried by the chains becomes easier as the ionic strength is increased. As the Coulomb interaction is screened by the salt, there is less correlation between charged groups along the chain when their distance is larger than the Debye length. Therefore, monomers separated by distances larger than the Debye length can dissociate independently from each other. For γ > ∼ 1, almost all the charged groups on the chains are dissociated; thus, the salt has no effect on the pH. The comparison between experiment and theory is presented in part (a) and (b), where a good agreement can be observed. Note the larger deviation from the experimental results for higher salt concentration (cNaCl = 0.1 M). In Fig. 3, the plateau value of the pH is plotted on a semi-log plot as function of the ionic strength cI = cNa + cH + cCl + cOH. The plateau value of the pH is taken at γ = 0.5. It is interesting to note that the different points taken at various salt and base concentrations collapse on a single curve. This result indicates that the pH is dominated by electrostatic effects and not by conformation changes of the chains. When the ionic strength increases, the fixed charges carried by the polymer are screened and the energy associated with the dissociation of a carboxylic group decreases. This behavior can be characterized semi-quantitatively by noting that when γ ≃ 1 /2 the pH is much higher than the pKA and, therefore, γ(0)^ → 1 and A → 1. Consequently, at the plateau ∆γ ≃ 1 /2. To first order in 10pKA−pH^ the plateau value can be obtained from the simplified free energy, Sec. II.A.3 and reads:
pHplateau ≃ pKA + log 10
[
κlB + 24πf 2 cm l^2 Bκ−^3 a^2
]
Recall that κ depends on the total amount of ions in the system through eq. (9). The above expression is used for the plot of the solid curve of Fig. 3b, where the pH of the plateau is shown as a function of the effective ionic strength ceff defined in eq. 9. In this regime, the difference between ceff and cI is a small correction of order 10pKA−pH. Equation 39 exhibits three different regimes depending on the relative importance of the various contributions to the logarithmic term. In each regime, cH has a different power law dependence on the total ionic strength ceff = cI +f A(1 −A)cm leading to different slopes in Fig. 3. At low ionic strength the second term dominates and cH ∝ c−eff^3 /^2
(long dashed line in Fig. 3b). At intermediate ionic strength the last term dominates and cH ∝ c−eff^1 /^2 (short dashed
line). Finally, at high ionic strength (beyond the experimentally accessible values) cH ∝ c^1 eff/^2. In Figure 4 the effect of polymer concentration and ionic strength on S(q) is shown. The structure function is calculated at the plateau regime were A ≃ 1. The dependence on the controlled system parameters: the pH, the salt concentration and the degree of neutralization γ is taken implicitly into account in ceff. At low values of ceff , the structure function exhibits a peak at finite q corresponding to the most favorable density–density fluctuations. As ceff increases, S(q → 0) increases until the peak disappears. This increase is accompanied by an increase in the osmotic compressibility of the solution. The effect of varying the polymer concentration cm at a fixed ceff is depicted in the inset. The peak is stronger at lower polymer concentrations indicating that fluctuations become considerably stronger at lower concentrations. This effect agrees well with the titration curves shown in Fig. 1. Indeed, at low polymer concentrations the shift in the titration curve is stronger than at high concentrations. The peak position shifts to higher wavenumbers (smaller length
[16] Jenkins, R.D.; Lelong, L.M.; Bassett, D.R. in “Hydrophilic Polymers : Performance with Environmental Acceptability”, ed. by Glass, J.E., ACS Symposium Series N 248, American Chemical Society 1996 , 23, 425. [17] Mandel, M. in “Encyclopedia of Polymer Science and Engineering” 1988 , 11, 739. [18] Strauss, U.P.; Schlesinger, M.S. J. Phys. Chem. 1978 , 82, 571. [19] Atkins, P.W. Physical Chemistry, Oxford University: Oxford, 1985. [20] See, for example, Adv. Polym. Sci. 1993 , 109; 1993 , 110. [21] Borukhov, I.; Andelman, D.; Orland, H. Eur. Phys. J. B 1998 , 5, 869. [22] Auvray, X.; Anthore, R.; Petitpas, C. J. Phys. (France) 1986 , 47, 893. [23] Heitz, C.; Raviso, M.; Fran¸cois, J. Polymer 1999 , 40, 1637.
Fig. 1 pH plotted as a function of the degree of neutralization γ with no added salt. In (a) the experimental titration curves are shown for different polymer concentration: cm = 1 mM (10−^4 g/g) full circles; cm = 10 mM (10−^3 g/g) full triangles; cm = 0.1 M (10−^2 g/g) full squares. The open circles denote the titration curve of the monomeric MAA at cm = 0.1 M (10−^2 g/g). In (b) the theoretical results are plotted for the same polymer concentrations: cm = 1 mM (dotted curve); cm = 10 mM (short dashes); cm = 0.1 M (long dashes). The solid line corresponds to the mean field solution (no correlations and no connectivity) for cm = 0.1 M. It reproduces nicely the conventional monomeric titration curve. The polymer parameters we take are: a = 5 A, f = 1/3 and pKA = 4.5.
Fig. 2 pH plotted as a function of the degree of neutralization γ for different amount of added salt. The experimental values in (a) are compared with the theoretical ones in (b). The polymer concentration is cm = 1 mM (10−^4 g/g). The concentration of NaCl in the solution is: cNaCl = 0 (open circles/solid line), 1 mM (full circles/dotted curve), 10 mM (full diamonds/short dashes), 0.1 M (full squares/long dashes). The theoretical curves are calculated using the same parameters as in Fig. 1.
Fig. 3 Semi-log plot of the pH at the plateau as a function of the ionic strength. Comparison between experimental results (a) and theoretical calculation (b). The experimental measurements are plotted in (a) as function of the total amount of small ions cI. The open circles give the pH of salt-free solutions at different monomer concentration extracted from Fig. 1. The full symbols give the pH of solutions with increasing amount of added salt: cm = 1 mM (10−^4 g/g) full circles, and cm = 0.1 M (10−^2 g/g) full triangles. The data collapse indicates that the pH depends only on the total ionic strength. The theoretical results are plotted in (b) as a function of ceff = cI + f A(1 − A)cm using the same parameters as in Fig. 1. See text for details of the approximations made in the calculation.
Fig. 4 The structure function S(q) as function of the wavenumber q. The structure function is calculated at the plateau of the titration curve where A ≃ 1. The physical parameters used in the calculation are a = 5 A, v = 25 A^3 , w = 100 A^6 , N = 10^4 , f = 1/3 and pKA = 4.5. The monomer concentration is cm = 1mM while the effective ionic strength is ceff = 0.01 mM (solid line), ceff = 1 mM (dotted curve) and ceff = 0.1 M (dashed curve). In the inset the effective ionic strength is fixed at ceff = 1 mM while the polymer concentration is varied: cm = 1 mM (solid line), cm = 10 mM (dotted curve) and cm = 0.1 M (dashed curve).
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