Energy versus Entropy in Cooperative Electrostatic Interactions

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J. Phys. Chem. B 2002, 106, 2175-2185

2175

Energy versus Entropy in Cooperative Electrostatic Interactions: Comparative Study of Binding of Sodium Poly(Styrenesulfonate), Dodecylbenzenesulfonate, and Methylbenzenesulfonate to Polycations J. Krˇ´ızˇ ,* J. Dybal, and D. Kurkova´ Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, HeyroVsky Square 2, 162 06 Prague 6, Czech Republic ReceiVed: July 27, 2001; In Final Form: December 2, 2001

Electrostatic coupling of sodium 4-dodecylbenzenesulfonate (DDBS), sodium 4-methylbenzene-sulfonate (MBS), and sodium poly(styrenesulfonate) (PSS) with the homopolymers of N-diallyldimethylammonium chloride (DADMAC) and chloride of 2-N-trimethylammonioethylmethacrylate (TMAEMA) or their copolymers with neutral hydrophilic groups were studied using 1H and 35Cl (single- and double-quantum) NMR, relaxations, and pulsed field-gradient (PFG) diffusivity measurements. Both DDBS and PSS were found to react cooperatively with the polycations, in contrast to MBS. However, PSS binds more strongly to the polycations and is thus able to substitute DDBS in its complex with an appropriate polycation. The binding isotherms of DDBS and MBS with DADMAC polymer agree well with the theoretical model suggested by Kuhn, Levin, and Barbosa, providing that the hydrophobic energy parameter χ has the value -3.5kBT for DDBS and 0.0 for MBS. Energy stabilization by hydrophobic interactions is thus suggested to be the determining factor in the cooperative binding of DDBS. This conclusion is supported by clusters of bound DDBS with a mean length of about 30 molecules in its complex with DADMAC polymer, which were indirectly found using 35 Cl T23 relaxation. In conclusion, the interaction of DDBS with polycations is suggested to be an example of cooperative electrostatic binding with unfavorable entropy but strong energy stabilization by, e.g., hydrophobic interactions. Entropy change due to liberation of bound counterions into a disordered state, found by us earlier1-3 to be operative in the couplings of complementary polyions, is thus not the only possible driving force of cooperative electrostatic interactions.

Introduction In our systematic study of cooperative electrostatic couplings,1-3 we explored the interaction of poly(sodium styrenesulfonate) (PSS) and polyphosphates (PP) with homo- or copolymers of chloride of 2-(N-trimethylammonioethylmethacrylate) (TMAEMA) or N-diallyldimethylammonium chloride (DADMAC), the latter having already been studied by other authors.4-8 Among factors influencing the effectiveness of binding, the most important appeared2 to be (i) entropy gain due to liberation of small counterions, (ii) energy stabilization due to long-range Coulomb forces, (iii) lateral correlation of the binding groups, and (iv) hydrophobic interaction. Among them, i depends on the polyelectrolyte effect (nonstatistical distribution of counterions in the parent polyions). All three factors i-iii are ascertained by the polymeric (or, to a lower degree, aggregate, e.g., micellar) structure of the reactants. Contrary to i, factor iv is probably anti-entropic and points to energy (or rather enthalpy) stabilization as the driving force of the process. It is quite difficult, both experimentally and theoretically, to establish the quantitative weights of the factors i-iv (or others). Some insight into the matter can be gained by comparing the binding of a polyion with that of analogous low-molecularweight compounds, which either do or do not possess the ability to gain additional stabilization by hydrophobic interactions in the electrostatic complex. Considering PSS as coupling partner to a polycation, two different low-molecular-weight compounds have a very similar structure in the vicinity of the same sulfonate * Corresponding author. E-mail: [email protected]. Fax: 420-2-35357981.

anionic group: sodium 4-methylbenzenesulfonate (MBS) and sodium 4-dodecylbenzenesulfonate (DDBS). MBS in water solution can be shown to remain in a monomeric form in the concentration range 2-30 mM/L, its hydrophobic interactions thus being too weak to overcome electrostatic repulsions and entropy demands of aggregation. In contrast to it, DDBS was shown9 to form various aggregates in this concentration range and was found by us to bind effectively to polycations even at very low concentrations below 2.0 mM/L. Interactions of surfactants with polyions were extensively explored.10-19 The binding of surfactant molecules (SM) to an oppositely charged polyelectrolyte (PE) usually occurs at surfactant concentrations much lower than the critical micellar concentration (CMC) and leads to electrostatic complexes, sometimes thought (but not evidenced) to contain micelle-like domains. Its cooperativity clearly can be viewed from two complementary perspectives:12 (i) the ionic binding of SM to PE is cooperative, being supported by the hydrophobic interactions of SM in the resulting complex; (ii) the hydrophobic selfassembly of SM micelles (unstable but possible below CMC) is cooperative, being facilitated by the neutralization of micellar charges by the bound PE. The first alternative can be further divided into two ways of binding, namely, (ia) gradual binding of individual SM, with a probability low for the first several SM but fast increasing as the collection of neighboring bound SM approximates a stable aggregate and (ib) almost-simultaneous binding of a micelle (or a smaller SM aggregate), which itself has to be assumed to be formed (and disintegrated again) even below CMC. If all the elementary steps are reversible and

10.1021/jp012916d CCC: $22.00 © 2002 American Chemical Society Published on Web 02/12/2002

2176 J. Phys. Chem. B, Vol. 106, No. 9, 2002 the system is at equilibrium, all these descriptions are equivalent in effect, providing that SM forms at least some aggregates at the given concentration. As shown by a recent study,9 DDBS does not form aggregates larger than a trimer at concentrations below 2.0 mM/L. Nevertheless, it binds effectively to a polycation even at 1.0 mM/L. This study explores this behavior in comparison to PSS and MBS.

Krˇ´ızˇ et al. SCHEME 1

Experimental Part Materials. DADMAC polymer (Mn ) 1.6 × 105, Z ) 898) and statistical copolymers with acrylamide, containing 8%mol (COP08, Mn ) 5.15 × 105, Z ) 597) and 21%mol DADMAC units (COP21, Mn ) 5.32 × 105, Z ) 1240), were prepared4 by feeded radical polymerization of the respective monomers. TMAEMA polymer (Mn ) 3.5 × 104, Z ) 167) and a statistical copolymer with 2-hydroxypropylmethacrylamide containing 25%mol TMAEMA units (Mn ) 1.6 × 104, Z ) 25) were prepared20 by radical polymerization of the respective monomers. Sodium polystyrenesulfonate (PSS, Mn ) 12900, Z ) 66) was purchased from Polymer Source, Inc. Sodium 4-dodecylbenzenesulfonate (DDBS) and sodium 4-methylbenzenesulfonate (MBS) were laboratory-grade commercial products purchased from Sigma-Aldrich. Sample Preparation. All measured samples were prepared by addition of a D2O solution of the actual anionic species (PSS, MBS, DDBS) to 2 mL of a vigorously stirred D2O solution of the given polycation at 298 K. The addition was very slow (0.5 mL/h); a Genie syringe pump and a polyethylene tube (inner diameter 0.2 mm) immersed into the polycation solution were used. NMR Measurements. 1H, 23Na, and 35Cl NMR spectra, relaxations, and pulsed-field-gradient stimulated spin-echo (PFG-SSE) measurements were obtained using a Bruker Avance DPX300 instrument with a broadband inverse-detection probehead (proton spectra, relaxations and PFG-SSE) or broadband direct-detection probehead (23Na and 35Cl spectra and relaxations). Most of the methods used were described earlier.1-3 1H PFG-SSE was done using the simple Tanner sequence with 3 ms pulses of field gradients incremented in the range 15-50 G/cm and constant diffusion delay (typically 80 ms). Its use for establishing the coupling degree R is described in Results and discussion. 35Cl T relaxations were measured using the ordinary inver1 sion-recovery sequence. 35Cl DQC (T 3) relaxations were measured using23-25 a π/22 τ0-π-τ0-π/2-nτ1-π/2-FID sequence, with τ0 ) τ1 ) 50 µs and n ) 1-32. T2 measurements were done with a simple π/2x-d2-πy-d2-FID sequence with a continuous-wave saturation of the nearest coupled proton signals (to remove possible echo effects due to spin coupling), d2 being incremented 10 ms in 16 points. T1F measurements were done using a pulsed (515 µs pulse delay) spin-lock and 16 points. In very dilute samples, WATERGATE suppression of the residual HOD signal was used. Results and Discussion Chemical structures of the reactants studied in this paper are given in Scheme 1. In the case of polycations, both DADMAC and TMAEMA homopolymers and copolymers were used (comonomers are included in the shown structures). Modes of Cation-Anion Binding and Structures of the Products. Like in our previous studies,1-3 most of the experiments were performed at 1.0 (exceptionally up to 4.0) mM/L

of the ionic groups. Such large dilution, needed for stability of products and reproducibility of results, sets rather close limits on the NMR techniques available. Except for 23Na and 35Cl in small counterions, the only nucleus available for NMR was 1H, providing that the spectra were measured with a highly sensitive probe and HOD signals were removed by a WATERGATE technique. Under these conditions, conventional 1D spectra took about 1 h, and any other experiment such as T2 or PFG SSE experiments needed 16 h or more. Special techniques such as high-resolution MAS or multiple-quantum spectra were virtually excluded by insufficient sensitivity. Even under such limitations, NMR offers valuable information about the systems as shown below. Aliphatic parts of 1H NMR spectra of the reaction mixtures of DADMAC or TMAEMA polycations (concentration of cationic groups 1.0 mM/L) with increasing quantities of DDBS and PSS (up to 0.5 equivalents) are compared in Figures 1 and 2, respectively, showing similar changes for PSS and DDBS. The aromatic parts of the spectra are not shown, the signals being either invisible (PSS) or very broad (DDBS). Broadening of the signals of aromatic protons shows that both DDBS and PSS bind to the polycations in the expected way, i.e., by electrostatic interaction of their sulfonate groups. In contrast, the aliphatic signals of both DDBS (1.5-0.5 ppm for the methyl and next nine CH2 groups) and PPS (around 1.5 ppm for skeletal CH2) are broadened but visible, showing thus motional freedom sufficient for removing static dipolar spin interactions. As for the polycations, mostly N-CH3 signals are visible (3.2-2.9 ppm) in both types (the splitting of signals in DADMAC units corresponds to cis configuration2,3). Their absolute intensity decreases with increasing coupling degree, which can be due only to extreme broadening of parts of the signals. Hindered motion of the methyl protons in coupled ammonium groups was already observed.1-3 In contrast to previous studies dealing with copolymers containing neutral hydrophilic units, the intensity

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J. Phys. Chem. B, Vol. 106, No. 9, 2002 2177

Figure 1. Aliphatic parts of 1H NMR spectra of DADMAC (left) and TMAEMA (right) polymers (1 mM/L of the cations) after addition of 5, 10, 20, 30, and 50 % mol of DDBS (D2O, 300 K).

loss in N-CH3 signals is unproportional to the fraction of groups already bound in the complex. The mobility of uncoupled groups thus must be affected by the binding state of their near and possibly farther neighbors. This could hardly happen if all coupled groups were distributed in one part of the polycation only, thus leaving the rest of the chain free. In principle, the reason for this phenomenon could be (i) interchain bridges with steric hindrances in the resulting complex structure or (ii) less ordered distribution of the bound anions (DDBS or PSS) along the polycation chain with a possible partial collapse of its conformation due to long-range electrostatic interactions. Apparent diffusion coefficients of the complexes of both polycation types with DDBS, obtained from an apparently monoexponential decay in PFG SSE, are given in Figure 3. Their increase with increasing amount of DDBS indicates that increased coupling of the polycation leads to a partially collapsed conformation rather than interchain bridging in the electrostatic

complexes. This holds for dilute solutions (cationic groups 1.0 mM/L) and agrees with the observations of analogous products with PSS.2,7,8 At this concentration, all results given below are reproducible within 5% of the respective experimental value. NMR spectra of these systems do not change appreciably after 1 month of storage. When the same experiments are done at concentration 4.0 mM/L and higher, the products with PSS flocculate and precipitate, whereas those with DDBS have diffusion coefficients almost 1 order of magnitude lower, thus indicating a super-molecular micelle-like structure. These systems are formed in a nonequilibrium state and change with time, according to NMR. The motional hindering of N-CH3 groups due to partially collapsed structure is generally stronger in DADMAC than in TMAEMA products. This also holds for the CH2 groups in PSS part of the corresponding products, as seen from the width of their signals.

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Krˇ´ızˇ et al.

Figure 2. Aliphatic parts of the 1H NMR spectra of DADMAC (left) and TMAEMA (right) polymers with indicated amounts (% mol) of added PSS (D2O, 300 K).

According to diffusivity, virtually no collapse of the structure due to electrostatic coupling is observed in polycations with their charges diluted by an excess of neutral hydrophilic monomer units. As shown in Figures 4 and 5, N-CH3 signal broadening is much slighter (no extreme broadening) in these cases. Again, weaker broadening effects are observed with the TMAEMA copolymer DO296. Analysis of the Binding Degree r of DDBS and MBS to the Polycations. The task to establish the coupling degree R of DDBS or MBS is rather difficult. In contrast to the analogous coupling of PSS2, neither relaxation nor diffusivity of 23Na ions can be used here owing to the almost full ion dissociation of DDBS at concentrations below 2 mM/L. As shown in the previous section, the absolute intensity loss of N-CH3 signal in the coupling product is not proportional to the conversion of corresponding groups in the present case and thus cannot be used for its quantitative measurement. Selective electrodes already used,21 e.g., for measuring the coupling of dodecyl(trimethyl)ammonium bromide to DNA are inappropriate for the parallel measurements with PSS and MBS. Considering the need of using the same measuring techniques for all reactants

in use, we used two NMR methods, namely, 1H PFG SSE and 35Cl T relaxation, combined with the measurement of the 1 activity of Cl- ions using a selective ion electrode. All these methods have their drawbacks, but their mutual agreement within experimental errors gives confidence that the results are meaningful. 1. PFG SSE. In this method, the analysis is complicated by chemical exchange between the free and bound diffusant (DDBS, MBS). Assume a chemical exchange between several sites characterized by the same chemical shift but different diffusion coefficients Di. The rate of transfer from a site i to site j is characterized by a rate constant kij ≡ τij-1. The over-all x-magnetization can be expressed as a column vector Mx ≡ (Mx1, Mx2, ... Mxn), which changes during the diffusion delay ∆ of the PFG SSE method according to the relation

d M ) AMx dt x

(1)

where A is a n × n matrix with the elements Aii ) ξi - ∑j*ikij, Aij ) kij (j * i), and ξi ) -T1i-1 - Diγ2g02δ2. Here Di and T1i

Cooperative Electrostatic Interactions

J. Phys. Chem. B, Vol. 106, No. 9, 2002 2179 length of the gradient pulse (we assume that δ is small compared to ∆). For ∆, the δ constant and g0 varied, and the expression for ξi can be simplified to ξi ) -ψDig02, with ψ ) γ2δ2. With the roots λi(t) (i ) 1, 2, ... n) of the matrix A, the over-all x-magnetization after the period ∆ is

Mx(∆) )

∑i Mxi(0) exp(λi∆)

(2)

For two sites

Mx(∆) ) Mx(0)[(1 - R) exp(λ1∆) + R exp(λ2∆)] (2a) with the roots λ1,2 of A:

λ1,2 )

(

ξ1 + k12 + ξ2 + k21 2

(ξ1 + k12 + ξ2 + k21)2 - (ξ1 + k12)(ξ2 + k21) + k12k21 4

Figure 3. Apparent self-diffusion coefficients of the complexes of DADMAC and TMAEMA polymers with DDBS at increasing relative concentration of DDBS (PFG SSE at 1.0 mM/L in D2O, 300 K).

are the true self-diffusion coefficient and relaxation time of the i-th species, respectively, and g0 and δ are the strength and

)

1/2

(3) where k21 ) τex-1(R-1 - 1) and k12 ) τex-1, τex being the exchange correlation time. The Mx(g02) dependence is approximately biexponential. With the known respective diffusion coefficients of the free diffusant (such as DDBS, MBS), D1, and of the complex (using signals independent of exchange),

Figure 4. 1H NMR spectra of COP08 and its complexes with PSS and DDBS (concentration of the ionic groups 1 mM/L in the original solutions, D2O, 300 K).

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Krˇ´ızˇ et al.

Figure 5. 1H NMR spectra of DO296 and its complexes with PSS and DDBS (concentration of the ionic groups 1 mM/L in the original solutions, D2O, 300 K).

Figure 6. PGF SSE decay curves of ω-CH3 DDBS 1H NMR signal in DDBS/PDADMAC systems (2 mmol/L DADMAC in D2O, 300 K). The curves correspond to fitting by eqs 2a and 3.

D2, the dependence has to be fitted for two independent parameters, namely, τex and R. For fast exchange (τex-1 . Di), the decay is virtually monoexponential, and the apparent coefficient Dap derived from

Figure 7. PGF SSE decay curves of CH3 (MBS) 1H NMR signal in MBS/PDADMAC systems (2 mmol/L DADMAC in D2O, 300 K). The curves correspond to fitting by eqs 2a and 3.

the Mx(g02) dependence is a weighted mean, Dap ) ∑iwiDi, leading to a simplified expression:

R)

D1 - Dap D1 - D2

(4)

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J. Phys. Chem. B, Vol. 106, No. 9, 2002 2181

TABLE 1: Apparent Self-diffusion Coefficients D1 and D2 of DDBS or MBS, Corresponding Correlation Times τEx, 35Cl Relaxation Rates REx, Cl- Activities aEx and the Corresponding Coupling Degrees rD, rR, rA, and the Mean r in the Systems of DADMAC with K (mol/mol) DDBS or MBS κ (mol/mol)

D1a (× 1010 m2 s-1)

D2a (× 1010 m2 s-1)

τex (× 104 s)

0.1 0.2 0.3 0.4 0.5 0.6

2.99 2.86 1.60 0.96 0.63 0.56

b b 1.56 0.89 0.35 0.29

b b 0.6 1.3 9.3 26.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7

4.56 4.42 4.23 4.05 3.82 3.68 3.30

b b b b b b b

b b b b b b b

Rex (s-1) DDBS 209.7 189.9 150.2 89.7 47.5 53.4 MBS 264.3 259.4 251.9 241.9 236.1 217.2 199.8

aexa

RD (mol/mol)

RR (mol/mol)

RA (mol/mol)

R (mol/mol)

0.031 0.074 0.168 0.308 0.450 0.558

0.33 0.36 0.57 0.76 0.93 0.95

0.28 0.32 0.52 0.73 0.90 0.91

0.31 0.37 0.56 0.77 0.90 0.93

0.30 0.35 0.55 0.75 0.91 0.93

0.005 0.012 0.033 0.064 0.110 0.150 0.238

0.06 0.10 0.14 0.17 0.23 0.25 0.32

0.04 0.08 0.11 0.15 0.18 0.22 0.30

0.05 0.06 0.11 0.16 0.22 0.25 0.34

0.05 0.08 0.12 0.16 0.21 0.24 0.32

a Normalized so that a ) 0 and a ) 1. b Monoexponential decay; D B F 1a and D2a are the apparent diffusion coefficients corresponding to the approximately biexponential PFG decay.

TABLE 2: Same Quantities as in Table 1 for the System TMAEMA with K (mol/mol) DDBS or MBS κ (mol/mol)

D1a (× 1010 m2 s-1)

D2a (× 1010 m2 s-1)

τex (× 104 s)

0.1 0.2 0.3 0.4 0.5 0.6

3.34 2.91 2.28 1.65 0.98 0.77

b b 2.24 1.59 0.85 0.55

b b 0.2 0.9 5.3 17.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7

4.65 4.56 4.32 4.13 3.90 3.76 3.43

b b b b b b b

b b b b b b b

Rex (s-1) DBBS 185.8 169.5 136.9 108.4 79.9 57.6 MBS 228.5 226.5 216.3 212.2 202.0 193.9 175.6

aexa

RD (mol/mol)

RR (mol/mol)

RA (mol/mol)

R (mol/mol)

0.023 0.074 0.150 0.268 0.435 0.534

0.28 0.36 0.52 0.66 0.83 0.91

0.24 0.32 0.48 0.62 0.76 0.87

0.23 0.37 0.50 0.67 0.87 0.89

0.25 0.35 0.50 0.65 0.82 0.89

0.002 0.008 0.024 0.056 0.090 0.126 0.182

0.05 0.07 0.13 0.17 0.23 0.25 0.32

0.03 0.04 0.09 0.11 0.16 0.20 0.29

0.02 0.04 0.08 0.14 0.18 0.21 0.26

0.03 0.05 0.10 0.14 0.19 0.22 0.29

a Normalized so that a ) 0 and a ) 1. b Monoexponential decay; D B F 1a and D2a are the apparent diffusion coefficients corresponding to the approximately biexponential PFG decay.

In this case, τex cannot be derived from the PFG SSE decay. As an example, experimental points of the decay of CH3 1H signals of DDBS and MBS in their mixtures with DADMAC polymer are given in Figures 6 and 7. The curves correspond to fitting by eq 2a (using eq 3). The fitting results of R and τex are given in Table 1. Quite analogous results for the systems DDBS and MBS with TMAEMA polymer are given in Table 2. 2. 35Cl Longitudinal Relaxation. In contrast to the relaxation of single and multiple quantum coherences, that of zero quantum coherence (longitudinal relaxation) can be treated in a relatively simple way. Under fast exchange of the bound and free ions, the relaxation rate is a weighted mean of the rates in a free and bound state, provided that extreme narrowing applies, i.e., the relaxation decay is monoexponential. In the solution of a strongly charged polycation, a substantial part of the chloride counterions is in a bound state, which dominates the relaxation rate. Under coupling of the polycation with an anionic species, part of the counterions (corresponding to the coupling degree R) is liberated into a free state. The value of R thus can be obtained1-3,22 using the approximation

R)

R1B - R1ex R1B - R1F

(5)

where R1i ≡ T1i-1, R1F and R1B being the relaxation rates of free and bound ions, respectively. The value R1B appropriate

for this approximation slightly depends on the ionic strength of the system,29 but its variation in the concentration range 0-2 mmol/L of NaCl (corresponding to R from 0 to 1 in the present case) is within experimental errors. R1F in all present systems is 31.1s-1, and R1B is 279.2s-1 and 234.6 s-1 for DADMAC and TMAEMA polymers, respectively. Experimental results for DDBS and MBS coupling with DADMAC polymer are given in Table 1, those for TMAEMA polymer in Table 2. In the case of weakly charged DADMAC-AA copolymers, where the polyelectrolyte effect on R1B is weaker, the values of R1B are 198.3, 133.6, and 93.4 s-1 for 57, 21, and 8 % mol of DADMAC in the copolymer, respectively. 3. Cl- ActiVity Measured with a SelectiVe Electrode. On the basis of the polyelectrolyte effect, the activity of Cl- ions (measured by a selective ion electrode) in a solution of DADMAC polymers and analogous polyelectrolytes is substantially lower than that in an equimolar solution of NaCl. Liberation of Cl- ions by coupling of the polycation with an anionic partner thus increases their activity. In analogy to eq 5, the coupling degree can be expressed by the relation

R)

aex - aB aF - aB

(6)

where aex, aB, and aF are the activities measured in the given experiment, in the parent solution of the polyelectrolyte and in

2182 J. Phys. Chem. B, Vol. 106, No. 9, 2002

Figure 8. Dependence of the coupling degree R of SO3- groups in PSS, DDBS, and MBS on the ratio κ to cationic groups in DADMAC polymer (n[can]0 ) 1.0 mmol/L, D2O, 300 K).

Figure 9. Dependence of the coupling degree R of SO3- groups in PSS, DDBS, and MBS on the ratio κ to cationic groups in TMAEMA polymer (n[can]0 ) 1.0 mmol/L, D2O, 300 K).

the equimolar solution of NaCl, respectively. The results for couplings of DDBS or MBS with DADMAC and TMAEMA polymers and DADMAC-AA copolymers are given in Tables 1 and 2, respectively. 4. Binding Isotherms of PSS, DDBS, and MBS with DADMAC and TMAEMA Polymers. Based on the results of the above three paragraphs, Figures 8 and 9 show the binding isotherms (or dependences of the coupling degree R on the ratio κ of cationic to anionic groups in the system) for PSS, DDBS, and MBS with DADMAC and TMAEMA polymers, respectively. Generally, the respective isotherms for DADMAC and TMAEMA are similar, the latter showing somewhat weaker binding. However, there is a striking difference between the isotherms for PSS, DDBS, and MBS bound to the same polycation. Both PSS and DDBS show clear cooperative behavior in contrast to MBS, which somewhat prefers the free state to that bound to the polycation. With both polycations, PSS binds more strongly than DDBS.

Krˇ´ızˇ et al.

Figure 10. Dependence of the coupling degree R of SO3- groups in DDBS on the ratio κ to cationic groups in DADMAC polymer and its copolymers with acrylamide (molar % of DADMAC in the copolymer indicated in the graph, n[can]0 ) 1.0 mmol/L, D2O, 300 K).

As expected, the binding of DDBS to the copolymers with regressive charge density (8-57 % mol of DADMAC) shows lower cooperativity. This is documented in Figure 11, where the binding degree R is generally lower for lower charge density but also the inflection point of its dependence on κ shifts to larger κ. In these cases, the binding of DDBS is markedly less effective than that of PSS (compare our previous results2). This can be demonstrated by a separate experiment. To a copolymer like COP21 (containing 21 % mol DADMAC and 79 % mol AA), an equivalent amount of DDBS (κ ) 1.0) could be added without risk of precipitation, thanks to a large content of neutral hydrophilic units. The value of R achieved in this case was 0.84 ( 0.04. When an equivalent amount of PSS was added to the resulting product, 92% of the bound DDBS was released and substituted by PSS. An analogous result was obtained with COP08. Comparison of Experimental Binding Isotherms with KLB Model. Kuhn, Levin, and Barbosa suggested18 an attractively simple model of interaction of a surfactant with a polyion (referred to as KLB). At the core of the model, electrostatic interactions between a polyion with the bound surfactant molecules and counterions (leading to the increment f pcs), between the free ions and free surfactant molecules (f ccs), and between polyions (f pp) are described by a linearized Poisson-Boltzmann equation. To the resulting free energy, entropy (f ent) and hydrophobic free energy (f hp, part of f ent in the original model) terms are added, and the total free energy

f ) f pcs + f ccs + f pp + f hp + f ent

(7)

is minimized with respect to the number of bound counterions (nB) and surfactant molecules (mB). Simulated dependences of β ≡ mB/(ZFp) (Fp being the volume density of the polyion and Z the number of charges per one molecule) on κ ≡ Fs/(ZFp) with the hydrophobic interaction parameter χ values -3.5kT and 0 are compared with our experimental results achieved for DADMAC (ZFp ) 2.0 mM/L) coupled with DDBS and MBS, respectively, in Figure 11. The agreement is remarkably good, considering the level of approximation. In particular, the

Cooperative Electrostatic Interactions

Figure 11. Comparison of experimental results (β ) mB/Z) for DDBS and MBS coupled with DADMAC (n[can]0 ) 1.0 mM/L) with simulations according to KLB model (position of the correctly predicted inflection point for DDBS indicated by dashed line).

inflection point in the DDBS dependence is predicted quite correctly. The main disagreement between the model and experiment is in the region of lowest κ and in the less sigmoidal shape of the predicted dependence. An analogous difference for the lowest κ was observed by KLB authors18 for the case of DNA coupled with dodecyl(trimethyl)ammonium bromide, where the stiff cylinder approximation of the polyion is much more grounded. However, the somewhat lower slope of the simulated curve in the inflection point compared with DDBS experimental results seems to be peculiar to our case. This slope (or sigmoidal shape of the curve) is a sign of cooperativity and has to be considered somewhat more closely. In the KLB model, f hp ) -Fpχ(Z - 1)y2, with y ) mB/Z. It thus reflects the probability of two surfactant molecules bound in adjacent positions; accordingly, the interaction energy given by χ was estimated in KLB to be roughly -3.5kT, i.e., a proportional part of about 20kT needed for a transfer of 12 carbon atoms from a bulk hydrocarbon into water solution. SCHEME 2

J. Phys. Chem. B, Vol. 106, No. 9, 2002 2183 In the case of more flexible polyions such as DADMAC or TMAEMA polymers, more than two adjacently bound DDBS molecules can have their hydrocarbon chains in contact gaining thus higher hydrophobic stabilization. To demonstrate that steric hindrances do not preclude such possibility, we performed quantum chemical MNDO calculations on model [DADMACDDBS]n and [TMAEMA-DDBS]n cyclic oligomers (n ) 4 and 5), in which the contacts between the DDBS chains were fixed by chemical bonds on the carbon second from the aromatic ring. Structures such as that shown in Scheme 2 are perfectly possible if less energetically preferable than those with a straight chain (quantum calculations cannot simulate the hydrophobic interactions resulting from the pressure of the surrounding water molecules). This accents the possibility of the aggregation of bound DDBS and a resulting higher cooperativity. Returning to KLB, the model could reflect the hydrophobic interactions of longer strings of surfactant molecules by including terms with higher order of y into f hp. Actually, we have found that already including y3 terms leads to a shape much closer to the experimental one. The difference of the two simulated curves in Figure 11 is much more apparent, however. It shows f hp to be a major factor both in the model (χ equal to -3.5kBT and 0, respectively) and experiment (DDBS and MBS). From this comparison, it is plausible to conclude that hydrophobic interaction of DDBS is the main factor in its cooperative coupling with DADMAC (and other polycations). Distribution of Bound DDBS along the Polycation Chain. In consequence of the conclusion of the preceding paragraph, the distribution of the bound DDBS along the polycation chain cannot be random. Considering the entropy requirements of any clustering, the mean length of DDBS sequences can serve as a kind of measure of hydrophobic stabilization. In principle, the clustering of DDBS should affect the transverse relaxation rate of its aliphatic protons, as it must lead to hindering of their motions. Unfortunately, this effect is distorted by overlapping of the signals of bound and free DDBS as well as chemical exchange between them. There is another indirect possibility to probe this phenomenon by 35Cl relaxation, however. This approach is based on the fact that in a polyion with part of its

2184 J. Phys. Chem. B, Vol. 106, No. 9, 2002

Figure 12. Simulated length distributions of uncoupled sequences in random couplings (R ) 0.5) of l-strings (l ) 1, 2, 3) to a 10000-mer.

charges neutralized by coupling, the degree of binding of the chloride counterions (and thus their quadrupolar relaxation rate) depends on the distribution of the remaining uncoupled charges. As explained below, longer sequences of charges bind counterions tighter, leading thus to faster relaxation of their nuclei. Our analysis is based on the statistical connection between the distribution of uncoupled charges and that of the couplings. Let us have a polycation with Z cationic groups in a chain and couple with it in a random way n strings of anionic groups with a length l so that n × l ) Z/2. This can be simulated in the following way. A matrix of Z ) 10000 elements a1, a2, ... aZ with their values originally equal to zero (vacant) is formed. Using a random number generator, a random number r in the interval from 1 to Z is produced. For a string length l, elements ar, ar+1, ... ar+l-1 are examined: if all these elements are zero, they are changed to 1, and the number l is added to the number of successful hits; if any of them is already nonzero, this trial is dropped. The same procedure is repeated until n strings are successfully placed on the matrix. After that, the distribution of sequences of remaining zeroes is examined. The whole procedure is repeated at 100 times to give the result statistical significance. As partly shown in Figure 12, the distribution of length of uncoupled cation sequences is shifted to short ones for l ) 1 (the alternation of cation and neutral pair being most probable) and gradually evens out as l increases; for l ) Z/2, it becomes quite flat. Any clustering of anionic groups coupled to a polycation thus makes longer sequences of remaining uncoupled cations more probable. From the early simulations of polyelectrolyte behavior using the Poisson-Boltzmann approximation,26-28 it has been known that the relative population of counterions in the vicinity of the polyion increases with its increasing charge density and, up to some limit, with its length.29 Owing to the latter, polyions with different charge distribution give rise to different populations of counterions in the “bound” area, i.e., near to the polyion. This can be illustrated by two extreme cases of a polyion with one-half of its units charged and the other neutral: (i) with a block of charged units spanning one-half of the chain, most of the counterions will be distributed near its densely charged surface; (ii) with a random distribution of charged units, their alternation with neutral units can be shown to be most probable and, due to the consequent charge dilution, the counterions will be much more randomly distributed in the surrounding medium.

Krˇ´ızˇ et al.

Figure 13. Relaxation rates of the 35Cl T23 coherence in a DADMAC polymer with 50% of its ionic groups coupled to a phosphate glass (PG) with n phosphate groups compared with those of the same polymer coupled to the same degree to PSS (P h n ≈ 66) and DDBS and MBS (n[can]0 ) 1.0 mM/L, D2O, 300 K).

As shown by Halle, Wennerstro¨m and Picullel29 for the quadrupole relaxation of I ) 3/2 nuclei such as 23Na+ or 35Clin the counterions, an important increment ∆R2 of their transverse relaxation rate, defined as

∆R2 ≡ R2+ - R2-

(8)

R2+ ) (2π2/3)[J(0) + J(ω0)]

(9a)

R2- ) (2π2/3)[J(ω0) + J(2ω0)]

(9b)

is approximately proportional to the squared probability P of the counterion being in the “bound” area (i.e., in the vicinity) of the polyion:

∆R2 ) (π2/20)P2(χjb)2τrad/(1 - S2)

(10)

(χb is the quadrupole coupling constant, τrad is the correlation time of radial diffusion in the given area, and S2 is the order parameter). ∆R2 is normally extracted29 from the two components of the biexponential decay of the single-quantum coherence. This is rather difficult under the conditions of chemical exchange with free ions (produced in our case by a partial coupling of the polyion with its counterpart such as DDBS). Incidentally, the term ∆R2 as defined by eqs 8 and 9a,b is equal to the relaxation rate23-25 of the third-rank tensor of the second-order coherence T23, which can be indirectly measured as a pure state (at the given dilution and 35Cl receptivity, the relaxation experiment takes at least 16 h, however). Figure 13 shows the relaxation rate R23 of T23 of “bound” 35Cl- ions vicinal to the DADMAC polymer, in which 50% of its groups were coupled with phosphate groups, dependent on the polyphosphate length. Polyphosphates were used in this study, as they are commercially available in different and rather narrowly distributed lengths. As expected, longer polyphosphates leave statistically longer sequences of uncoupled cations with more tightly bound and thus faster relaxing 35Cl- counterions. The dependence expectedly converges for longer polyphosphates (and consequently longer cation sequences). In Figure 13, there are added points corresponding to the relaxation rates obtained with 50% of DADMAC groups saturated by PSS, DDBS and MBS. The rather good agreement between the length of PSS

Cooperative Electrostatic Interactions (P h n ) 66) and that expected from the curve (about 71) shows that the effect is not peculiar to a polyphosphate but depends on the length of the polyanion. From this point of view, the correspondence of DDBS to a mean sequence about 30 anions long (contrasting to that of MBS, corresponding to about 3) can be considered to be a very strong argument for its clustering in the complex with DADMAC. Conclusions In the present study, we have shown that DDBS, in contrast to MBS, binds cooperatively to polycations such as DADMAC and TMAEMA polymer or their statistical copolymers with neutral hydrophilic monomers. The binding of DDBS has some similar features with that of PSS regarding both cooperativity and relative collapse of the polycation conformation with higher coupling degrees. However, it is somewhat weaker, thus allowing DDBS to be substituted by PSS. The experimental binding isotherms of DDBS and MBS to DADMAC polymer agree well with the theoretical model proposed by Kuhn, Levin, and Barbosa (KLB) provided that the value of the hydrophobic interaction parameter χ is put to -3.5kT for DDBS and 0.0 for MBS. According to this, the binding is expected to be cooperative and nonrandom for DDBS, forming larger clusters of the bound molecules, whereas almost random for MBS. The clustering of bound DDBS molecules was supported by an indirect evidence using the relaxation rate R23 of the T23 tensor state of 35Cl- ions bound to a complex of DADMAC polymer and 0.5 equiv of DDBS. The mean length of a DDBS cluster in the complex was found to be near 30. Clustering of DDBS bound to DADMAC polymer offers stabilization energy (due to hydrophobic interactions) but lowers entropy in two different ways: (i) by nonrandom distribution of DDBS itself along the chain and (ii) by stronger binding (and thus narrower distribution) of the counterions by the consequently longer strings of the still uncoupled DADMAC groups. The only plausible conclusion from it thus is that the driving force of the cooperative interaction of DDBS with DADMAC (or analogous) polycation is energy stabilization by hydrophobic interactions of its aliphatic chains. In our previous studies,2,3 entropy gained mostly by liberating the small counterions (and presumably also hydrating water molecules) bound by the polyions into a disordered state was suggested to be the driving force of the cooperative electrostatic interactions of complementary polyelectrolytes. Considering the conclusions of the present study, we have to admit that the validity of this hypothesis is not quite general. Clearly, electrostatic couplings with unfavorable entropy change are possible providing that there is a sufficient source of stabilization energy (or rather enthalpy) to counterbalance it. Hydrophobic interactions of longer aliphatic chains attached to at least one of the reacting species clearly can offer such an effect.

J. Phys. Chem. B, Vol. 106, No. 9, 2002 2185 Both entropy and energy factors in the overall free energy balance thus have to be considered in cooperative electrostatic interactions. Acknowledgment. The authors thank the Grant Agency of the Academy of Sciences of the Czech Republic for financial support given under Grants A4050206 and K4050111. They also thank Dr. H. Dautzenberg of the Max-Planck-Institute for Colloid and Interface Research, Berlin, for the DADMAC polymer and copolymers, Dr. D. Oupicky´ and Dr. T. Reschel from the Institute of Macromolecular Chemistry, Prague, for the TMAEMA polymer and copolymer, and Mrs. D. Kanˇkova´ for technical assistance. References and Notes (1) Krˇ´ızˇ, J.; Kurkova´, D.; Dybal, J.; Oupicky´, D. J. Phys. Chem. A 2000, 104, 10972. (2) Krˇ´ızˇ, J.; Dautzenberg, H. J. Phys. Chem. A 2001, 105, 3846. (3) Krˇ´ızˇ, J.; Dybal, J.; Dautzenberg, H. J. Phys. Chem. A 2001, 105, 7486. (4) Brand, F.; Dautzenberg, H.; Jaeger, W.; Hahn, M. Angew. Makromol. Chem. 1997, 248, 41. (5) Brand, F.; Dautzenberg, H. Langmuir 1997, 13, 2905. (6) Dautzenberg, H.; Hartmann, J.; Grunewald, S.; Brand, F. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 1024. (7) Dautzenberg, H. Macromolecules 1997, 30, 7810. (8) Dautzenberg, H.; Karibyants, N. Macromol. Chem. Phys. 1999, 200, 118. (9) Krˇ´ızˇ, J. Langmuir, submitted for publication. (10) Goddard, E. D. In Interactions of Surfactants with Polymers and Proteins; Goddard, E. D., Ananthapadmanabhan, K. P., Eds.; CRC Press: Boca Raton, FL, 1992; p 123. (11) Lindman, B.; Thalberg, K. In Interactions of Surfactants with Polymers and Proteins; Goddard, E. D., Ananthapadmanabhan, K. P., Eds.; CRC Press: Boca Raton, FL, 1992; p 203. (12) Wallin, T.; Linse, P. J. Phys. Chem. 1996, 100, 17873. (13) Wallin, T.; Linse, P. Langmuir 1996, 12, 305. (14) Fundin, J.; Brown, W.; Vethamuthu, M. S. Macromolecules 1996, 29 (4), 1195. (15) Liu, J.; Takisawa, N.; Shirahama, K.; Abe, H.; Sakamoto, K. J. Phys. Chem. B 1997, 101, 7520. (16) Yoshida, K.; Morishima, Y.; Dubin, P. L.; Mizusaki, M. Macromolecules 1997, 30, 6208. (17) Kosmella, S.; Kotz, J.; Shirahama, K.; Liu, J. J. Phys. Chem. B 1998, 102, 6459. (18) Kuhn, P. S.; Levin, Y.; Barbosa, M. C. Chem. Phys. Lett. 1998, 298, 51. (19) Konop, A. J.; Colby, R. H. Langmuir 1999, 15, 58. (20) Oupicky´, D.; Konˇa´k, C ˇ .; Ulbrich, K. Biomed. Sci., Polym. Educ. 1999, 10, 573. (21) Gorelov, A. V.; Kudryashov, E. D.; Jacquier, J.-C.; McLoughlin, D.; Dawson, K. A. Phys. A 1998, 249, 216. (22) Bull, T. E. J. Magn. Reson. 1972, 8, 344. (23) Jung, K. J.; Tauskela, J. S.; Katz, J. J. Magn. Reson. B 1996, 112, 103. (24) Eliav, U., Navon, G. J. Magn. Reson. A 1996, 123, 32. (25) Jung, K. J.; Katz, J. J. Magn. Reson. B 1996, 112, 214. (26) Alfrey, T.; Berg, P. W.; Morawetz, H. J. Polym. Sci. 1951, 7, 543. (27) Lifson, S.; Katchalsky, A. J. Polym. Sci. 1954, 13, 43. (28) Selegny, E., Ed. In Polyelectrolytes and Their Applications; Reidel: Dordrecht, The Netherlands, 1975. (29) Halle, B.; Wennerstro¨m, H.; Picullel, L. J. Phys. Chem. 1984, 88, 2482.