SAXS and ASAXS on Dilute Sodium Polyacrylate Chains Decorated

Apr 29, 2013 - ... Anomalous Small-Angle X-Ray Scattering. Guenter Johannes Goerigk. Advances in Linear Algebra & Matrix Theory 2013 03 (04), 59-68 ...
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SAXS and ASAXS on Dilute Sodium Polyacrylate Chains Decorated with Lead Ions Sebastian Lages,† Günter Goerigk,‡ and Klaus Huber* Department of Chemistry, Physical Chemistry, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany S Supporting Information *

ABSTRACT: Bivalent lead ions as representative main group heavy metal cations form specific interactions with the negatively charged COO− residues of sodium polyacrylate chains in dilute aqueous solution. The interactions eventually lead to aggregation and precipitation of sodium polyacrylate chains partially neutralized with Pb2+ cations. The present work outlines a small-angle X-ray (SAXS) and light scattering study of the polyacrylate chains undergoing changes in coil conformation and successive aggregation while approaching and crossing the Pb2+induced precipitation threshold. The study reveals a coil shrinking while approaching the precipitation threshold. Anomalous SAXS (ASAXS) complemented this information with a first insight into the spatial distribution of the Pb2+ cations captured by the polyacrylate chains together with a semiquantitative estimation of the amount of Pb2+ cations located within the collapsed domains of the shrinking chains. Conformational aspects of the shrinking coils could be established by means of model form factors of hybrid chains formed by a freely jointed chain of rods with spheres located on all or part of the joints. Development and application of the form factors of the pearl-necklace-like hybrid model chains and the use of quantitative analysis of ASAXS data are described and discussed in detail.

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INTRODUCTION Organic polyelectrolytes are highly charged water-soluble polymer chains. The subtle interplay of their electric charges and the hydrophobic chain backbone make these polyelectrolytes extremely sensitive to variations of solvent conditions and to the addition of salts. Alkaline salts exhibit the most simple interaction pattern in aqueous solution, which can be described by counterion condensation based on electrostatic interactions between counterions and backbone charges.1,2 Unlike to such chemically inert salts, specifically interacting cations (SIC) bind more strongly to the oppositely charged residues of the polyelectrolyte chains. The most prominent example for such an interaction is the neutralization of COO− residues in sodium polyacrylates (NaPA) or proteins with protons. Sodium polyacrylate as one of the most frequently investigated polyelectrolyte chains exhibits specific interactions also with alkaline earth cations and with a variety of transition metal cations and heavy metal cations.3 This property renders NaPA and related polymers interesting candidates to be used as antiscalent agents, as additives in water desalination processes, or as efficient dispersants of minerals.4 They may also serve as a model compound to study modulated crystallization as it occurs in biomineralization. Since the interactions underlying all these processes and applications impose drastic changes in the conformation of the NaPA coils,5,6 the phenomenon is also highly relevant for the development of new responsive materials © 2013 American Chemical Society

or for a better understanding of the complex interaction patterns of proteins containing COO− residues in saline media of living systems. Whereas scattering techniques are highly suitable to analyze such conformational changes, anomalous small-angle X-ray scattering (ASAXS) turned into a powerful complement to conventional scattering analysis of polyelectrolytes7−9 since ASAXS may quantify the amount of condensing or specifically interacting counterions. Along this route, a considerable extent of information could be established for the interaction of alkaline earth cations with NaPA coils in dilute solution.9−11 NaPA coils precipitate with alkaline earth cations whereby the precipitation does not obey the law of mass action but requires stoichiometric amounts of both components to hit the precipitation threshold.3,12−14 Upon approaching this threshold by stepwise adjusting the ratio of [COO−]/[M2+] to the proper value (with M2+ denoting the respective SIC), intermediates of the coils can be trapped at varying degree of shrinking. This shrinking becomes largest at the precipitation threshold, where the chains adopt the shape of a collapsed sphere.5 While light5 and small-angle neutron scattering15,16 indicated a pearlnecklace-like shape of the shrinking polyacrylate coils with Received: February 27, 2013 Revised: April 15, 2013 Published: April 29, 2013 3570

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Table 1. Light Scattering Characterization of All Samples Used for ASAXS Experiments at the JUSIFAa sample

[NaPA]/mM

[Pb2+]/mM

Mw/g/mol

Rg/nm

Rh/nm

αg

ρ

H-029 H-030 H-031 H-034 H-037 H-038

7.20 6.45 3.33 6.18 4.84 5.80

1.65 1.65 1.00 1.65 1.25 1.65

449 000 571 000 7 676 000 620 000 655 000 1 093 000

28.0 26.7 80.7 23.8 30.3 29.4

15.7 14.5 47.5 13.9 15.6 15.7

0.89 0.85 2.56 0.75 0.96 0.93

1.78 1.84 1.70 1.71 1.95 1.87

a

Characterization includes the apparent weight averaged molar mass Mw, the square root of the z-averaged squared radius of gyration Rg, the hydrodynamically effective radius Rh, the expansion factor αg, and the structure sensitive parameter ρ. The radius of gyration of the unperturbed dimensions required for he estimation of the αg is derived from the respective scaling law18 at Mw = 457 000 g/mol, resulting in Rg(Θ) = 31.5 nm.

nodules of COO−−M2+ complexes, ASAXS experiments with Sr2+ ions made available for the first time the quantitative extent of decoration of the NaPA coils with Sr2+, being predominantly located in the nodules or pearls.9 Experiments with Cu2+ ions17 similar to those carried out with alkaline earth cations revealed a pattern which partly differs from the one established for the alkaline earth cations. The threshold where precipitation occurs also follows a stoichiometric relation, but the extent of shrinking accessible before precipitation occurs is considerably smaller than observed with alkaline earth cations and the aggregates, which precede the actual precipitate are less dense than in the case of the alkaline earth cations.17 The bivalent lead ion is a typical heavy metal ion from the main groups and thus may serve as a representative candidate to investigate its SIC interaction pattern with NaPA coils in dilute aqueous solution. The fact that SAXS experiments are available at the LIII absorption edge of Pb enables ASAXS with Pb2+ and hence makes this cation particularly interesting for a study of SIC−NaPA interactions. The present study will be performed with a NaPA sample with a molecular weight of Mw = 457 kDa in aqueous solution, which is 0.1 M in NaNO3 and which has a pH between 8 and 9. The pH makes sure that the polyelectrolyte is fully dissociated and the excess level of NaNO3 serves as a background electrolyte, which suppresses the impact of a structure factor from the highly charged polyelectrolytes on the scattering pattern. The present SAXS and ASAXS study analyses the extent of shrinking of NaPA coils upon addition of Pb2+ cations and reveals structural details of the shrinking coils. Beyond this, quantitative evaluation of the ASAXS data provides information on the extent of decoration of the NaPA chains with the Pb2+ cations close to the precipitation threshold. Analogous investigations on alkaline earth ions in preceding work10,11 enables comparison of the impact of the Pb2+ ions with the impact of Sr2+ ions in the same system.

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Combined static (SLS) and dynamic light scattering (DLS) experiments were carried out in cylindrical scattering cells with an outer diameter of 20 mm from Hellma (Müllheim, Germany) using an ALV 5000E goniometer from ALV-Laser Vertriebsgesellschaft (Langen, Germany). The goniometer was equipped with a 200 mW Nd:YAG laser operating at a wavelength of 532 nm, and the scattering intensities were recorded in an angular regime of 30° ≤ Θ ≤ 150° at a temperature of 25 °C. A dilution series of four different concentrations of the NaPA standard P585 in the regime of 0.072 g/L ≤ c ≤ 0.289 g/ L in 0.01 M NaCl was analyzed by combined SLS and DLS experiments in order to characterize the NaPA sample. The samples used for the SAXS and ASAXS experiments were denoted as H-029, H-030, H-031, H-034, H-037, and H-038. As PbCl2 precipitates in the presence of 0.1 M Cl− ions, NaNO3 and Pb(NO3)2 are used for the sample preparation. A stock solution of the NaPA standard was prepared in 0.1 M aqueous NaNO3 and gently mixed at room temperature for 3 days. The pH of the 0.1 M aqueous NaNO3 solution was set to 9 using a 0.1 M aqueous NaOH solution prior to dissolving the NaPA standard. Another salt solution containing 5 mM Pb(NO3)2 and 90 mM NaNO3 in water was used to prepare the SAXS samples by mixing appropriate amounts of the 0.1 M NaNO3 solution, the NaPA stock solution, and the solution containing the Pb2+ ions. Details on the composition of the six samples can be taken from Table 1. In order to scrutinize temporal stability of the samples, the samples denoted as H-029, H-030, H-031, and H-034 were analyzed with combined DLS and SLS in Paderborn before and after the SAXS experiments. None of the four samples revealed significant changes. All samples could thus be considered to keep their solution state within the 2 weeks of the experiments. The samples denoted as H-037 and H038 were characterized by combined SLS and DLS experiments only after the SAXS experiments. Results of the characterization are summarized in Table 1. Quartz capillaries with an inner diameter of 2.00 mm and a wall thickness of 0.05 mm were purchased from Hilgenberg (Malsfeld, Germany) and used as scattering cells for the SAXS experiments, which were performed at the JUSIFA beamline (DESY, Hamburg, Germany). Processing of Light Scattering Data. The SLS experiments yield the excess Rayleigh ratio of the solute as a function of the scattering vector and were evaluated in a q regime of 0.67 × 10−4 nm−2 ≤ q2 ≤ 9.3 × 10−4 nm−2 according to Zimm’s approximation19 by extrapolating to q2 = 0:

EXPERIMENTAL SECTION

Materials. NaCl, NaNO3, Pb(NO3)2, and NaOH of analytical grade were purchased from Fluka (Buchs, Switzerland), the sodium polyacrylate standard P585 was purchased from Polymer Standards Service (Mainz, Germany), and bidistilled water with a conductivity 2, corresponding to M > 1, the relationship for a freely jointed chain32 Rg2 =

(29)

and eq 22 via

M(M + 2) 2 A 6(M + 1)

(32)

has been applied. 3574

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For fits with md Model 2 the number of rodlike segments has been adjusted beforehand to make sure that the radius of gyration of the model hybrid chains gets close to the value measured by static light scattering. Adjustment of M has been performed by eq 32 using a rod length of A = 4.2 nm. This value corresponds to the Kuhn length of sodium polyacrylate chains in aqueous NaCl established in a range of 0.1 M < [NaCl] < 1.5 M, independent of [NaCl].18 Unlike to Model 1, the number of pearls can be varied independently of M. The fit routine only accepted integer numbers for g, N, and M. As in the case of fits with md Model 1, successful adjustment of the coil size is demonstrated by a comparison of theoretical and experimental Rg values. Individual fits were again carried out with three variable parameters (μs, μr, R) at fixed numbers of pearls and repeated at increasing number of pearls in the range of 2 ≤ N ≤ 12. In order to take into account polydispersity, pd Model 2 had been applied with (Ps, μr, Vw) as fit parameters replacing the set of (μs, μr, R) of the md models. An averaged pearl radius Rw was calculated from Vw by means of eq 26. A procedure analogous to the one applied for md Model 2 was used, however now with additional individual fits at six different polydispersities for each pearl number N. The polydispersities defined by eq 25 adopted values of z = 1, z = 2, z = 3, z = 5, z = 10, and z = 24, corresponding to relative standard deviations σR of the Schulz−Zimm distribution (cf. eq 24) of σR = 0.71, σR = 0.57, σR = 0.50, σR = 0.41, σR = 0.30 and σR = 0.20. Fit parameters established for all samples at the respective χ2min are summarized in Tables 3 and 4 and in additional tables in the Supporting Information. The model fitting was performed using Maple 9.5. Evolution of Quantitative Information from ASAXS Data. Data treatment of the energy dependent recordings of SAXS scattering curves make accessible the so-called pure resonant scattering contribution of the resonant scattering lead ions. In order to achieve this goal, the present work applies a procedure introduced in refs 9−11. The procedure, which is based on the recording of three scattering curves at three different energies close to the LIII absorption edge of Pb2+, shall be briefly summarized by the following equations. The recorded differential scattering cross section of the SAXS curves result from an interplay of the two components, resonant scattering Pb2+ ions and nonresonant scattering NaPA, which can be subdivided into three contributions.

ΔfC = ((f0, C − ρS VC) + fC′ (E) + ifC″ (E))

Equation 34 represents the difference of the electron number of a monomer including the fraction of a Na+ cation condensed per monomer to NaPA in order to neutralize the chain, and eq 35 corresponds to the difference of the electron number of the Pb2+ cations, with both differences referring to the electron number of the respective solvent background. The parameters in eqs 34 and 35 are as follows: VM and f M are the volume and the electron number of a monomer including the fraction of a Na+ cation condensed per monomer to NaPA in order to neutralize the chain, respectively, f 0,C is the electron number of a Pb2+ cation, f′C and f ″C are the energy-dependent anomalous dispersion corrections of the Pb2+ cations, ρs is the electron density of the solvent, and VC is the volume of a Pb2+ cation, respectively. A subtraction of the scattering curves at energy E3 and E2 and at the energies E3 and E1 respectively leads to dΣ dΣ dΣ = (q , E3) − (q , E 2 ) dΩ dΩ dΩ dΣ dΣ dΣ Δ3,1 = (q , E3) − (q , E1) dΩ dΩ dΩ

Δ3,2

1 dΣ 1 dΣ Δ3,1 − Δ3,2 (fC′ (E3) − fC′ (E1)) dΩ (fC′ (E3) − fC′ (E2)) dΩ = F(E1 , E2 , E3)SCC(q)

(37)

The prefactor of the pure scattering contribution from the resonant species reads ⎛ |Δf (E )|2 − |Δf (E )|2 3 1 C F(E1 , E2 , E3) = ⎜⎜ C ⎝ (fC′ (E3) − fC′ (E1)) −

(33)

|ΔfC (E3)|2 − |ΔfC (E2)|2 ⎞ ⎟ (fC′ (E3) − fC′ (E2)) ⎟⎠

(38)

Since all contributions to Δf C(E) which are independent of the energy cancel, the term simplifies to

The three contributions to eq 33 are the so-called partial scattering functions indicating the contribution from the nonresonant species SPP(q), the cross-term SPC(q), and the term from the resonant species SCC(q). All three terms SPP(q), SPC(q), and SCC(q) are given in units of cm−1 and thus have incorporated already the differential scattering cross section of an electron r02 as a constant factor. The first term refers to the monomers which include Na+ ions to the extent incorporated in the polyelectrolyte species to neutralize it, and the third term corresponds to the Pb2+ ions. The prefactors of the three terms are based on the following differences in electron numbers ΔfM = (fM − ρS VM )

(36)

Expressions like those in eq 36 are denoted as separated scattering curves. If SAXS curves are available at three different energies, three such differences can be performed. All separated scattering curves are stripped off the nonresonant contribution SPP(q) but still include the cross-term SPC(q) being influenced by both components. Isolation of the pure contribution from the resonant component SCC(q) is achieved by forming one more difference for which again three alternatives exist. One of the alternatives is explicitly given with the following relationships.

dΣ (q , E) = ΔfM 2 SPP(q) + 2ΔfM (f0, C − ρS VC + fC′ (E)) dΩ SPC(q) + |ΔfC (E)|2 SCC(q)

(35)

F(E1 , E2 , E3) = fC′ (E1) − fC′ (E2) + −

fC″ (E3)2 − fC″ (E1)2 (fC′ (E3) − fC′ (E1))

fC″ (E3)2 − fC″ (E2)2 (fC′ (E3) − fC′ (E2))

(39)

All subtraction steps in eqs 36 and 37 have to be performed with all contributing components being given at identical qvalues, respectively. This could be achieved as follows. The twodimensional scattering patterns have been processed by the JUSIFA20 software, thereby calculating the scattering curves for all energies upon the same q-grid by use of a Jacobi determinant

(34) 3575

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larger apparent molar mass value, and one of them also has a much larger size value indicating the onset of aggregation. Yet, we have deliberately included these two samples in order to see whether such samples would indicate significant differences from stable single chains with respect to their internal morphology and extent of decoration with Pb2+ cations. Details of the light scattering experiments are given in the Supporting Information. The information can be used to generate a diagram shown in Figure 2, which reflects the phase behavior of the system NaPA

transforming from Cartesian space coordinates to spherical coordinates in the q-space. Extraction of the scattering curve stemming from the resonant scatterersin the present case the Pb2+ ions makes accessible a quantification of the Pb2+ cations being bound and/or condensed to the NaPA chains.11 This quantification is provided by the averaged number density of the resonant scattering Pb2+ counterions in solution ⟨ν⟩, which is related to the experimentally accessible invariant [I] of the pure resonant curves according to11,34 ⟨v⟩ =

VC 2π 2r0 2

V

∫q q2SCC(q) dq = 2π 2Cr 2 [I] 0

(40)

However, it has to be stressed that in an experiment the integral in eq 40 is limited to the accessible q-space, and it has to be scrutinized in any system under consideration, whether the experimentally accessible q-regime is sufficient to separate the Pb2+ correlated to the NaPA chains from the isotropically scattering free Pb2+ cations.11,34 The only constant required in eq 40 is the radius of the Pb2+ ion, which enables calculation of the volume VC of the Pb2+ ion. A value of 0.112 nm had been used for this radius,35 which leads to VC = 5.885 × 10−3 nm3. Together with the absolute scattering length of an electron of r0 = 2.82 × 10−13 cm, the front factor of the integrals amounts to 3.75 cm.

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Figure 2. Phase diagram for P585 in 0.1 M NaNO3 solutions in the presence of Pb2+ cations. The NaPA concentration is expressed as [COO−]. The hollow squares denote states which exhibit single chain behavior, and the filled squares indicate states where aggregation takes place.

RESULTS AND DISCUSSION In a first step, solutions with suitable concentrations of NaPA and Pb2+ cations had to be selected for ASAXS experiments which are capable of giving insight into the morphology of shrinking polyacrylate coils and of revealing the spatial distribution of the Pb2+ cations inducing this shrinking process. In this respect, knowledge of the precipitation behavior of PbPA in 0.1 M NaNO3 is of utmost importance. Six samples differing in their Pb2+ and/or NaPA concentration had been selected and carefully characterized in order to accomplish this step. Characterization was based on a detailed light scattering analysis of the respective solutions. Three criteria have been used in order to establish an overview on the phase behavior and in order to ensure suitability of the samples for SAXS and ASAXS. These criteria are as follows. The molecular weight must not deviate significantly from the specified value of 457 kDa, DLS has to result in a diffusional mode which corresponds to single coils and the coil dimensions should be shrunken considerably with respect to the state in the absence of Pb2+ cations. As can be seen from Table 1, this holds for four out of the six samples. Those four samples exhibit apparent molar mass values, which are acceptable within uncertainty, particularly in the light of the fact that partial decoration of the coils with Pb2+ ions may increase the apparent Mw value compared to the value specified by our light scattering analysis. They all have particle size values, which are smaller than Rg(Θ) = 31.5 nm corresponding to the value expected for the respective unperturbed dimensions18 of sample P585. This means, that the dimensions are considerably smaller than the average size the coils would adopt in the absence of lead ions but at otherwise the same conditions, because an aqueous solution with 0.1 M NaNO3 is considered to act as a good solvent for NaPA. If the scaling law established at 0.1 M NaCl is used to estimate a radius of gyration of the sample,18 a value of Rg = 53.2 nm is predicted. The remaining two samples out of the six have a considerably

+ Pb(NO3)2 in water. A straight line is resulting when separating the states where precipitation occurs from the states which exhibit single chain behavior. It can be considered as a precipitation threshold governed by stoichiometry. The threshold has a slope of [Pb2+]/[COO−] = 0.28, indicating a degree of neutralization of the carboxylate residuals by lead ions of 56%. However, since this slope is based on two points only, further data are required to confirm its value. SAXS and ASAXS experiments have been performed at three energies close to the LIII absorption edge of Pb2+. The SAXS experiments performed at an energy of 13 035 eV shall first be used for a conventional SAXS study. As is shown in Figure 3, the scattering curves of all six samples exhibit the same characteristic trend. An initial steep decay turns into a shoulder which is followed by a second steep decay with an exponent close to −4. Obviously, the overall size of the entities in all six samples was too large to make accessible a radius of gyration of the polymer particles.19 This is compatible with the particle size values inferred from static light scattering. The size values observed with the samples which show single chain behavior cover a regime of 25 nm < Rg < 30 nm for the radius of gyration Rg, which is beyond the q-regime of the SAXS experiments. The intermediate shoulders indicate substructures with a size in the range of a few nanometers. It has to be emphasized that the two samples, which aggregate, exhibit scattering curves close to those of the four samples where single chain behavior dominates. Apparently, aggregation does not significantly influence the substructures of the interconnecting chains, at least not in the two intermediates captured by sample H-031 and H-038. 3576

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Figure 4. Representation of the SAXS scattering curve of sample H034 recorded at an energy of 13 035 eV in comparison to optimized fits with model form factors. The symbols have the following meaning: (△) H-034; optimized least-squares fit based on (black line) md Model 1;25 (red line) md Model 2, and (blue line) pd Model 2.

improvement more obvious. The upturn observed at relatively high q values in the case of md Model 1 is due to the scattering contributions of the rodlike linkers. The q−1 decay due to those rods eventually outweighs the q−4 decay of the pearls. This upturn sets in at a higher q value once the linker chains are formed with FJC segments. Thus, the intermediate states exhibit small collapsed nodules connected with semiflexible chain segments. Though only marginally, adoption of a polydispersity for the size distribution of the spheres further improves the quality of the data fitting in the regime of 1.1 nm−1 < q < 1.4 nm−1. This is most suitably demonstrated by means of a Kratky plot (Figure 5) of the same data shown in Figure 4.

Figure 3. Representation of the SAXS scattering curves recorded at an energy of 12 400 eV (A) and of the pure-resonant scattering contribution from the Pb2+ cations (B) for all six solutions investigated by ASAXS. SCC(q) in (B) is established by means of eq 37. All measurements are performed in aqueous 0.1 M NaNO3. The symbols have the following meaning: (■) H-029; (□) H-030; (▲) H-031; (Δ) H-034; (●) H-037; (○) H-038.

In order to extract further information from the SAXS curves, we applied two closely related theoretical form factors for leastsquares fits. Both model form factors are based on a chainsphere hybrid model suitable to describe pearl-necklace chains. They are composed of freely jointed chains (FJC) and regular spheres, where the spheres are located on the joints formed by two neighboring rigid-rod-like monomers, respectively. Model 1, which had been designed earlier to analyze SANS curves of shrinking NaPA chains, where the shrinking had been induced by Ca2+ cations,15 is a simple FJC with a spherelike pearl sitting on every joint and on both chain ends.25 Model 2 is a generalized version of Model 1 and is presented for the first time in the present contribution. It separates two neighboring pearls by FJC-like subchains all with the same degree of polymerization. It thus differs from Model 1 by the fact that two neighboring pearls are separated by a variable distance. The distribution of the distance is given by the distribution of the end-to-end distance distribution of the FJCs forming the subchains. An illustrative example is given in Figure 4, which compares best fits of md Model 1, md Mode1 2, and pd Model 2 to the experimental data recorded with sample H-034. Only pd Model 2 accounts for a polydispersity of the spheres outlined in detail in the theoretical section on the model form factors. Although all three models succeed to reproduce a significant part of the scattering curve, qualitative agreement increases in the order of md Model 1 < md Model 2 < pd Model 2. The most significant improvement could be achieved by replacing the rodlike link between two neighboring pearls with a FJC-segment as link via md Model 2. The following considerations make this

Figure 5. Representation of Kratky plots of the SAXS scattering curve of sample H-034 recorded at an energy of 13 035 eV in comparison to optimized fits with model form factors. The symbols have the following meaning: (△) H-034; optimized least-squares fit based on (black line) md Model 1;25 (red line) md Model 2 and (blue line) pd Model 2.

A more quantitative overview on the fit quality is received with a comparison of all fit results in Tables 3 and 4 and in a summary of all fit parameters in the Supporting Information. While the number of pearls fluctuates, the χ2min values stemming from the best fits with md Model 2 are without exception lower than with md Model 1, confirming the feature of flexible links between the connected pearls. As is indicated by 3577

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the χ2min values summarized in Table 3, consideration of a polydispersity of pearls with pd Model 2 further improves the fit for all six samples. All fits performed with Model 2 revealed radii of 2 nm < R < 3 nm as size values for the pearls. They are also in fair agreement with the respective values retrieved by md Model 1, which also kept below 4 nm. Values for R have been summarized in Table 3. The optimal numbers of pearls are in most cases smaller for fits with Model 2 than for fits with md Model 1. Yet, this is due to the fact that the number of pearls are predetermined for md Model 1 as an inevitable consequence of the necessary adjustment of the number of rods M = N − 1 to the proper size of the coils. Fits with the md Model 2 reveal N > 2 only for one sample. In order to further illustrate that the number of rods M has been preselected to adjust the overall size of the hybrid particle to the respective radius of gyration established by SLS, the values from SLS are compared in Table 3 with the respective values used for the best fit. Because of the fact that each sample had been characterized at three different energies (wavelengths) close to the LIII absorption edge of the Pb2+ cations, information on the shrinking NaPA coils could be provided by ASAXS in addition to the conventional SAXS analysis just presented. As will be pointed out in the following paragraphs, ASAXS gives insight into the distribution of the lead ions within the shrinking NaPA coils. By using eqs 37−39, pure resonant scattering curves have been established for the Pb2+ cations, which represent the isolated scattering pattern emanating from the ensemble of the inhomogeneously distributed Pb2+ cations in the solution. A collection of all six curves is represented in Figure 3B. The curves exhibit Guinier shoulders, which in all cases appear at a regime of 0.5 nm−1 < q < 2 nm−1. Since this corresponds to the regime where model interpretation with all three pearl-necklace chain models revealed domains with a size between 2 and 3 nm, we can now identify these domains as nodules filled with condensed Pb2+ cations and NaPA segments. A quantitative estimation of the amount of Pb2+ ions condensed onto the PA chains can be retrieved from the invariant of the pure resonant curve. As is indicated by eq 40, the invariant corresponds to the area under the respective Kratky curves. However, as has been outlined in a preceding work on quantitative ASAXS experiments with the system Sr2+ cations/NaPA chains in dilute solution,11 nonbound and noncondensed Pb2+ cations do not contribute significantly to the invariant in the q-regime probed by small angle experiments like the one in the present work. A collection of Kratky representations from the pure resonant curves of all six samples is given in Figure 6. Although the plots seem to indicate a maximum, the experimental uncertainty of the data points beyond q = 2 nm−1 gets too large to admit quantitative determination of the area under the curve. Plots of experimental uncertainties are indicated in Figure 7. The areas established in Figure 6 and indicated as [I] in Table 5 are close to the true values only if the decay on the right-hand side of the maxima would obey a power law of P ∼ q−4. An interference with this q−4 decay could for instance be caused by additional Pb2+ ions nonbound but condensed to the domain of the polyacrylate coils. These loosely captured ions would lead to an increase of the area under the curves of Figure 6 beyond q ∼ 2 nm−1. Hence, the values for the areas determined in Figure 6 are considered to represent lower limits of the true values.

Figure 6. Representation of Kratky plots of all six pure resonant scattering curves Scc(q) extracted via eq 37. The gray areas indicate the invariants extracted from all six graphs and successively used as [I] in eq 40 to calculate the averaged number density ⟨ν⟩ of Pb2+ cations bound to the NaPA coils.

Figure 7. Representation of Kratky plots of all six pure resonant scattering curves Scc(q) extracted via eq 37 including error bars. The red lines indicate the contribution of the sphere self-terms q2Sss(q) extracted from the optimized form factor fits with pd Model 2 by setting the rod self-term and the rod−sphere cross-term equal to zero. Adjustment to the experiment has been achieved by the application of eq 41.

Table 5. Lower Limits for the Concentration of Pb2+ Cations Bound to NaPA Chains [Pb2+]ba sample

[I]/10−12 nm−4

[Pb2+]/mM

[Pb2+]b/mM

[Pb2+]b/[NaPA]

H-029 H-030 H-031 H-034 H-037 H-038

3.22 3.50 3.23 5.10 2.78 3.83

1.65 1.65 1.00 1.65 1.25 1.65

0.196 0.204 0.150 0.317 0.162 0.231

0.027 0.032 0.045 0.051 0.033 0.040

a

These lower limits are calculated from the lower limit of the corresponding averaged number density ⟨ν⟩ of the Pb2+ counterions bound to the NaPA chains in solution via [Pb2+]b = ⟨ν⟩/NA. Values for ⟨ν⟩ are extracted from eq 40 for all six samples based on [I] illustrated in Figure 6.

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description of the SAXS curves and the pure resonant curves with pd Model 2. As can be inferred from this comparison, the model form factors confirm the small coverage of the NaPA chains with Pb2+ cations. Values for the percentage of neutralization by Pb2+ cations would in fact only deviate significantly from the lower limits if the decay of the scattering curves beyond q = 2 nm−1 would obey a power law smaller than q−4. The interpretation of the SAXS and ASAXS data reveals crucial aspects of the morphological changes induced by the addition of Pb2+ cations within the NaPA chains in dilute solution. As the ratio of [Pb2+]/[COO−] increases, the NaPA coil dimensions start to shrink via formation of condensed Pb2+−COO− nodules. Apparently, Pb2+ cations accumulate along finite chain segments, thereby neutralizing these segments and tightening them to nodules. More than one such nodule is expected to grow on many of the chains. Beyond a certain threshold ratio of [Pb2+]/[COO−] the chains start to aggregate without further noticeable change of the nodules. Pb2+ cations simply start to interconnect chains once the threshold is reached. The nodules formed for Pb-PA are smaller than those found in shrinking Sr-PA or CaPA as is the amount of condensed Pb2+ cations compared to the corresponding amount of condensed Sr2+.9,11,15,16

As is indicated by the numbers shown in Table 5, the lower limits indicate a fraction of 5%−10% of COO− residues being neutralized by complex bound Pb2+ cations. This is a fairly small fraction and in fact smaller than the values observed earlier for the system Sr2+−PA.11 Although a definite conclusion based on true values is not possible, because we can only present estimates of a lower limit for the Pb2+ system, the differences between the Sr2+−NaPA system and the Pb2+ system are significant. In line with the lower limiting values for the fraction of acid residues neutralized by M2+, also the pearls identified as domains with condensed, specifically interacting M2+ cations is significantly smaller for the Pb2+ system compared to the Sr2+−NaPA. This feature is nicely illustrated by Figure 8.

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SUMMARY Bivalent lead ions as representative main group heavy metal cations form specific interactions with the negatively charged COO− residues of sodium polyacrylate chains in dilute aqueous solution. The interactions lead to a precipitation of NaPA coils partially decorated with Pb2+ cations. Light scattering analysis of the global dimensions of NaPA coils in 0.1 M NaNO3 in the presence of various amounts of Pb2+ cations enabled selection of six suitable solutions close to the precipitation threshold. Four samples exhibited single chain behavior of the NaPA coils with the coil dimensions significantly smaller than those observed in the absence of Pb2+ cations. The remaining two samples showed aggregate formation and were thus beyond the Pb2+-induced precipitation threshold. SAXS curves from all six samples could be successfully interpreted with a new model of pearl-necklace-like hybrid chains. The hybrid chains are composed of rodlike segments of length A, linked together to form a freely jointed chain and of sphere-like pearls which are placed on the two ends and on joints separated by a distinct number of rodlike segments. Polydispersity is taken into account only via a distribution of the sphere size, which varies while going from one chain to another. The model is an extension of a simple hybrid chain, published earlier25 were a sphere was placed on both chain ends and on every joint of the freely jointed chain. Optimized model fits to the SAXS curves from all six samples revealed feely jointed chains with a small number of pearls. The pearls have an average radius of 2 nm < R < 3 nm. Recording of the SAXS curves at three different energies close to the LIII absorption edge of the Pb2+ cations enabled extraction of the pure resonant scattering curve of the Pb2+ cations. The resulting ASAXS analysis identified the pearls extracted already by conventional SAXS as nodules with a high concentration of Pb2+ cations. Evaluation of the invariant of the pure resonant scattering enabled an estimation of a lower limit of Pb2+ cations per COO− groups collected in the NaPA coil domains.11,34 These lower limits covered a regime between 2.7% and 5.1%. Noteworthy, the size of the collapsed Pb2+−

Figure 8. Representation of the SAXS scattering curve recorded at an energy of 12 400 eV (A) and of the pure-resonant scattering contribution from the Pb2+ cations (B) for solution H-034 (▲) in comparison to an experiment performed with Sr2+−NaPA in 0.01 M NaCl at an energy of 15 507 eV with [Sr2+] = 1.5 mM and [NaPA] = 3.23 mM (■).11

As a test for consistency of our model interpretations, we took the respective best fit with form factors based on pd Model 2 to each experiment and used the pearl contribution Sss(q) thereof expressed by eq 27; i.e., we simply set mr = 0 in eq 30. The resulting model curves Sss(Vw,q) were rescaled with a factor according to33 SCC(q) =

NVw⋅[I] 2

2π Sss(q = 0)

SSS(V w , q)

(41)

and compared with the resulting curves in the corresponding Kratky representations of the experimental pure resonant curves in Figure 7. [I] in eq 41 corresponds to the invariant of the pure resonant scattering established in Figure 6 and summarized in Table 5. The theoretical curves nicely overlay with the respective experiments, thus confirming a consistent 3579

dx.doi.org/10.1021/ma400427d | Macromolecules 2013, 46, 3570−3580

Macromolecules

Article

COO− nodules are significantly smaller than the respective features established from NaPA coils shrinking and precipitating at a threshold induced by specifically interacting Sr2+ cations.11 In line with these findings the extent of decoration of the shrinking NaPA coils with Sr2+ cations at the respective threshold is also significantly larger than observed with the Pb2+ cations. The results presented in this work support a shrinking mechanism induced by the Pb2+ cations, which generates intermediate structures with compact nodules formed by collapsed segments of Pb-PA and interconnected by wormlike polymer segments. Two out of the six samples under consideration exhibited weight-averaged molecular weights significantly higher than the values expected for single chains and thus indicate aggregation of the shrinking coils. The internal morphology of the aggregates as well as the extent of decoration by specifically interacting Pb2+ cations in the nodules did not show significant changes compared to the single coils existing at the aggregation threshold, i.e., in the stable solution state of single shrinking coils. Hence, the onset of aggregation does not seem to change the morphology of the partially collapsed coils but merely interconnect these coils to a loose network of pearl-necklace-like Na+ + Pb2+−PA coils.

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ASSOCIATED CONTENT

S Supporting Information *

Summary of all fit-parameters received from fits with model form factors; combined DLS/SLS analysis of all samples prior to and after SAXS/ASAXS experiments. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Ph (+49) 5251 602125; Fax (+49) 5251 604208. Present Addresses †

S.L.: Lund University, P.O. Box 124, SE-22100 Lund, Sweden. G.G.: Institute of Soft Matter and Functional Materials, Helmholtz-Zentrum Berlin, Hahn-Meitner-Platz 1, D-14109 Berlin, Federal Republic of Germany. ‡

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by the Deutsche Forschungsgemeinschaft (grant HU807/7) is gratefully acknowledged.

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