Quenched polyelectrolytes with hydrophobicity independent from chemical charge fraction: A SANS and SAXS study

2.1 Polymer synthesis

The tri-co-polyelectrolyte noted (AMx-STy-AMPSz), the chemical structure of which is shown in Fig. 1, is a random copolymer made out of three monomers: acrylamide whose rate is x/(x + y + z), styrene whose rate is y/(x + y + z) and 2-Acrylamido-2-methyl-1-propane sulfonic acid sodium salt whose rate is z/(x + y + z). This polyelectrolyte is characterized by decorrelated electrostatic charge and hydrophobic fractions, i.e. the hydrophobicity rate “y/x + y + z” can be varied, within a certain range, without changing the electrostatic charge fraction “z/x + y + z” and vice versa.

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Figure 1

Chemical Structure of poly(AMx-STy-AMPSz), the chemical charge fraction is f_chem = z/x + y + z.

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Acrylamide, styrene and N,N dimethylformamide (DMF) were purchased from Sigma Aldrich, the 2-acrylamido-2-methylpropane sulfonic acid and the benzoyl peroxide were purchased from Scharlau and sodium hydroxide was from Carlo Erba. All the reagents were used as received.

The poly(AMx-STy-AMPSz) was prepared by direct polymerization of the three monomers with a free-radical initiator in the N,N dimethylformamide or cyclohexane in case of high styrene content (y > 40%). This method is inspired from that of Peiffer et al. (1988) used to prepare copolymers of styrene and 2-acrylamido-2-methylpropane sodium sulfonate.

The experimental data of the resulting tri-co-polyelectrolyte are summarized in Table 1. An example of a polymerization of the tri-co-polyelectrolyte AM26-ST23-AMPS51, is described as follows. A three-necked 100 ml contains a solution of AMPS (7.45 g) in DMF 12 ml, then a solution of acrylamide (0.42 g in 1 ml DMF) was added to the mixture and finally styrene (2.4 ml) was added. The mixture was stirred for approximately 2 h under nitrogen gas at room temperature. The used initiator, benzoyl peroxide, was subsequently added (0.11 g in 1 ml DMF) and the solution temperature adjusted to 70 °C. The temperature was controlled continually throughout the polymerization time depending on the chemical charge (i.e. 4–12 h). Also, the solution was stirred continually during the reaction and kept flooded with dried nitrogen.

Depending on the polymer composition, the tri-co-polyelectrolyte precipitation occurred during the polymerization when the % of styrene is low (<40%), which rendered it easy to recover. But when styrene % was high, the polymer remained in cyclohexane solution; in this case it had to be recovered by precipitation in ethanol.

The resulting tri-copolymer was then dissolved in aqueous solution and neutralized with sodium hydroxide, purified by dialysis, concentrated and freeze dried.

2.2 Polymer chemical characterization

2.2.1

2.2.1 NMR

The ¹H NMR, ¹³C NMR spectroscopies and Elemental Analysis were used to determine the chemical composition of the synthesized tri-co-polyelectrolyte.

The NMR measurements were made using Ultra shield 500 plus at the Institut National de Recherche et d’Analyse Physico-Chimique (INRAP)-Sidi Thabet. The frequency is equal to 500 MHz for ¹H NMR and 125 MHz for ¹³C NMR. Two solvents were used for NMR analysis: D₂O for hydrophilic polyelectrolyte and DMSO-d6 for hydrophobic polyelectrolyte.

The NMR was mainly used to check the presence of the three monomers on the resulting polyelectrolyte but cannot be quantitative due to the overlapping of some resonance peaks.

Fig. 2 shows a typical ¹H NMR spectrum of a tri-co-polyelectrolyte in DMSO-d6. It is composed of a broad band (δ = 1.25–1.55 ppm) corresponding to the aliphatic protons of the polymer backbone (that of styrene, acrylamide and 2-acrylamido-2-methylpropane sodium sulfonate). The protons of CH₃ groups in AMPS are detected at around 2.0 ppm and the signal of the CH₂ group bonded to SO₃Na is detected at 3.2 ppm (Jamshidi and Rabiee, 2014).

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Figure 2

¹H NMR of AM17-ST44-AMPS39 in DMSO-d6.

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The sharp peak at δ = 2.56 ppm corresponds to the residual protons from the imperfect deuteration of the solvent DMSO and the second one at δ = 3.42 ppm is attributed to water either in the solvent or as the residual water in the polyelectrolyte powder (Baigl et al., 2002). The signals at 6 ppm < δ < 7 ppm correspond to the presence of the aromatic zone hydrogen of the styrene monomer. Finally, the peak at δ = 8.31 ppm corresponds to the protons of the amide function of acrylamide and 2-acrylamido-2-methylpropane sodium sulfonate.

In the ¹³C NMR spectrum of the same tri-co-polyelectrolyte of AM17-ST44-AMPS39 in DMSO-d6, the presence of C = O function in AM and AMPS gives a peak around 176.6 ppm (Jamshidi and Rabiee, 2014). The CH and CH₂ carbons of the main chain are detected around 39 ppm. The two CH₃ groups of AMPS are detected around 25.92 ppm; the quaternary carbon of the AMPS group was detected at 42.9 ppm, and the CH₂ of the same AMPS being near the SO₃Na group is detected at 67 ppm. The peak at 128 ppm is assigned to aromatic carbons of the styrene monomer (Baigl et al., 2002; www.unige.ch). The figure is not reproduced here due to the low quality of signal.

In summary, both ¹H NMR and ¹³C NMR confirmed the presence of the three monomers in the polyelectrolyte.

2.2.3

2.2.3 Size exclusion chromatography measurements

For the AM54-ST04-AMPS42 tri-copolymer, the molecular weight distributions and average molecular weights were determined by multidetection Size Exclusion Chromatography (SEC) using one SEC line of the Institut Charles Sadron (ICS) - Strasbourg. This line involves a usual HPLC part (Dionex), a refractometer and a multi-angle light scattering apparatus (Wyatt technology) completed by a dynamic light scattering detector. The fractionation was carried out through four Shodex columns based on poly(vinyl alcohol) gel arranged in series. The eluent was a mixture of water and acetonitrile with a volume composition 60–40, respectively, in the presence of 0.1 M NaNO₃. Measurements were performed at room temperature using a constant flow rate of 0.5 ml mn⁻¹. Tri-copolymer solution of concentration 1.2 g L⁻¹ was filtered on a 0.45 μm membrane prior to be injected.

The differential refractometer detector enables determining the concentration of each fraction, while the light scattering detector enables determining the molecular weight of each fraction, the radius of gyration of those associated with large enough molecular weight, as well as the hydrodynamic radius (Fig. 3a).

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Figure 3

Chromatograms of AM54-ST04-AMPS42 (a) and the corresponding elution volume dependence of the molecular weight measured from the multi-angle light scattering detector (b).

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By considering a refractive index increment value dn/dc = 0.134 cm³ g⁻¹ (Read et al., 2014; http://www.chemicalbook.com), the analysis of the chromatograms (Fig. 3b) gives the number average molecular weight M_N = 347.000 g mol⁻¹ and the weight average molecular weight M_W = 561.000 g mol⁻¹. The polydispersity index is therefore I = M_W/M_N = 1.62.

Moreover, it allows determining the various averages of the gyration and hydrodynamic radii 〈R_G〉 and 〈R_H〉, respectively. So, for AM54-ST04-AMPS42 they are as follows:

• 〈R_G〉_N = 27.6 nm; 〈R_G〉_W = 38.8 nm; 〈R_G〉_Z = 63.2 nm,

• 〈R_H〉_N = 13.6 nm; 〈R_H〉_W = 17.8 nm; 〈R_H〉_Z = 21.8 nm,

where the indexes N, W and Z refers to the number, weight and z averages.

We do not have values for all samples. However measurements for samples such as di-copolymers and homopolymers (not shown here) show that our copolymerization conditions (solvent, catalyst) lead to quite long chains (> several 100,000 g mol⁻¹). We thus expect to be always in semidilute regime c > c^∗(∼ 0.00381 g cm⁻³ for AM54-ST04-AMPS42, knowing that c^∗ =  $\frac{M_w}{\frac{4}{3}\pi N_A\langle R_G{\rangle }_w^3}$ ), for which chains interpenetrate so that the scattering does not depend on the molecular mass.

2.3 Preparation of solutions for scattering measurements

Solutions were prepared by dissolving the dry polyelectrolyte in the solvent. Solvent is D₂O for SANS measurements, in order to get the best contrast, and D₂O as well for SAXS, in order to be exactly in the same solvent. Solutions were let at rest 2 days before scattering measurements. Concentrations c_p are expressed in moles per liter of the corresponding monomers and range between 0.04 mol L⁻¹ and 0.68 mol L⁻¹, the most explored being 0.34 mol L⁻¹.

2.4 Small angle scattering measurements

We use both Small Angle Neutron Scattering and Small Angle X-Ray Scattering to determine the structure of the PE solutions, which will enable useful comparisons.

3.1 A methodology point: comparison of SAXS/SANS

Let us now compare the SANS and SAXS profiles of the same tri-co-polyelectrolyte. Fig. 4 shows the comparison for AM17-ST44-AMPS39. For that we divide SANS as well as SAXS intensities by the square contrast values, here K_N² = 6.83 10⁻²³ cm² for SANS and K_X² = 1.99 10⁻²³ cm² for X-rays. The first is the value chosen in Table 2 for SANS for this tricopolymer composition ignoring the neutron scattering from sodium counterions (Dubois and Boué, 2001). The second is also the contrast value with a condensed counterions fraction equal to zero. This is not far from the fraction of condensed Na ions (f_chim − f_eff) = 39–36% = 3%, calculated assuming Manning condensation, for which f_eff = a/l_B = 36% (a is the length of AMPS repeat unit).

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Figure 4

Comparison of the SANS and SAXS profiles for a tri-co-polyelectrolyte AM17-ST44-AMPS39, c_p = 0.34 mol L⁻¹. Both scattered intensities are divided by the contrasts, equal to 6.83 10⁻²³ cm² for SANS and chosen equal to 1.99 10⁻²³ cm² for X-rays, in the absence of condensation. The units of S_T(q)/c_p are in Å⁻³ mol⁻¹ L.

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The difference found above between SAXS and SANS profiles is within the error limits due to normalization, evaluation of the scattering volumes and partial molar volume. Superposing the maxima for neutron and X_rays is possible for K_X² = 2.3 · 10⁻²³ cm². It corresponds to a quite different value of f_eff = 36%, whereas we remain within the error limits. In summary, it is not possible to evaluate further the counterion binding from a SANS/SAXS comparison. For the sake of simplicity, we will use in the following the values of K²_X without condensed counterions.

Another interesting point is that the heterogeneity of the chemical composition along the chain, has no influence. Indeed the 3 different types of statistical units, AM, ST, and AMPS have different scattering length, and hence different individual contrast but if their distribution among all different chains is random, the system is equivalent to all chains with all monomers having the same average contrast. This is consistent with the shape of all curves and with the fact that the scattering of neutrons and that of X-rays are quite close when both have been measured. Moreover, this supports the assumption of really random statistical copolymerization.

Lastly, the low q upturn is more pronounced for SANS. This can be due to the angular width of the neutron incoming beam, which in this setting is larger than for X-rays. This produces the enlargement of the upturn commonly observed for most of the polyelectrolyte solutions which is still an open question in the field.

3.2 Effect of the hydrophobic styrene content at constant chemical charge fraction

Let us first follow the effect of the hydrophobicity, controlled by the amount of polystyrene repeat unit, ST, hydrophobic, in the backbone of the polyelectrolyte and of AM (hydrophilic), at a constant chemical charge fraction (AMPS content) on the polyelectrolyte structure properties. That can be monitored by both SANS and SAXS. We use the values of K_X² (K_N²) calculated by neglecting the condensed counterions.

3.2.1

3.2.1 For a constant chemical charge fraction f_chem ≈ 40%

Fig. 5a shows S_T(q) obtained from SAXS at a polymer concentration c_p = 0.34 mol L⁻¹, for three tri-co-polyelectrolytes of chemical charge fractions (AMPS content f_chem = z close to 40%) and of two quite distinct hydrophobic contents. Compared to AM17-ST44-AMPS39 (already shown in Fig. 4), the most hydrophilic AM54-ST04-AMPS42 copolymer shows an increase in the peak position q^∗, with a broader peak and a very strong decrease in the peak intensity at q^∗, I(q^∗). On the reverse the most hydrophobic polymer (where y is 50% and x only 10%) shows an even smaller q^∗ value and a much stronger peak intensity I(q^∗). The same trend is observed with SANS (Fig. SI.1).

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Figure 5

Evolution of the scattering of tri-co-polyelectrolytes as a function of hydrophobicity (a), chemical charge fraction (b), polyelectrolyte concentration (c) and added salt (d).

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Figure 6

Log-log plot of q^* as a function of c_p, for a strongly hydrophobic and a quasi-hydrophilic polyelectrolyte.

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These results simply show that the solvent quality has a great influence on the solution properties. Depending on the proportion of the three monomers, the chain follows a continuous transition, from a partly pearl necklace collapsed conformation (Dobrynin and Rubinstein, 1999), to a stretched one described within the isotropic model through the scaling approach (de Gennes et al., 1976; Dobrynin et al., 1995). This is in agreement with the previous result (Essafi et al., 1995) found on three different di-copolymers, with the same f_chem = 55%: poly(ST-co-SNa), poly(ST-co-AMPS) and poly(AM-co-AMPS). But here with tri-copolymers, we observe a combined contribution to hydrophobicity of both ST (undubiously positive) and AM (null or negative, see in the Discussion a comparison between tri-copolymer and some di-copolymers).

3.2.2

3.2.2 For larger charge fractions f_chem

Note first that larger charge fraction means necessarily smaller styrene fraction y, since the sum of the neutral monomers decreases as f_chem increases: y = 1 − f_chem − x < 1 − f_chem.

For a constant charge fraction f_chem ≈ 50%, insert of (Fig. 5a) for SANS shows that the peak position does not vary strongly as the styrene content decreases from 23% to 04% (there is however unexplained decrease of 30% of the intensity).

For a constant charge fraction f_chem ≈ 70% (see S.I.2), the available styrene fractions are bound to be even lower: y is equal to 17 and 6. The effect is even less pronounced, as seen by SAXS on S_T(q)/c_p).

In summary of this sub-section, when the styrene fraction is below 20%, we reach a plateau regime where the chain behaves as hydrophilic. Note that the hydrophobicity can also be balanced either by the increase in charge fraction, or by the presence of non-negligible amount of AM.

In summary of Section 3.2, the effect of the hydrophobicity is moderate for styrene contents until 25%, while for styrene contents approaching 50% or larger, it is very significant. The maximum styrene content reached for a soluble copolymer was for AM20-ST55-AMPS25, for which q^∗ went as low as 0.055 Å⁻¹ (see Fig. S.I.3). For even larger content, e.g. for AM05-ST67-AMPS28, the copolymer was not soluble any more: the aqueous solution was demixed, as indicated by a strong upturn in the scattering, and the vanishing of the peak (see Fig. SI.3).

In the Discussion, we will focus on a more detailed comparison with other systems formerly reported in the literature (Fig. 7), using the q^∗ values, which sheds light on finer differences.

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Figure 7

Variation of q^* with styrene content y in different copolymers at c_p = 0.34 mol L⁻¹, compared with data PSSNa (Essafi et al., 1994), PSSNa (Essafi et al., 2009) and STAMPS (Essafi et al., 1995).

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3.3 Effect of the chemical charge fraction and acrylamide content at constant hydrophobic styrene content

Using the same set of data, we can also analyze the effect of the chemical charge fraction at constant hydrophobic content.

3.4 Effect of polyelectrolyte concentration

Another marker of hydrophobicity is the variation of q^∗ with polymer concentration c_p. It was studied by SANS and SAXS in a range between c_p = 0.04 and c_p = 0.68 mol L⁻¹. Fig. 5c shows the evolution of SAXS and SANS (Insert) profiles with polyelectrolyte concentration for the hydrophobic polyelectrolyte AM17-ST44-AMPS39. The same effect of c_p is observed as c_p increases: the peak position q^∗ shifts to higher q values and the normalized scattered intensity per monomer decreases. For the hydrophilic polyelectrolyte AM54-ST04-AMPS42, the evolution with c_p of SAXS profiles (Fig. SI.7) looks similar at first sight.

However, differences linked with hydrophobicity arise from a more quantitative analysis, namely the variation of q^∗ as a function of c_p: Fig. 6 shows that q^∗ scales as c_p^−α for both polyelectrolytes, but with distinct exponents α: 0.37 ± 0.02, for the hydrophobic polyelectrolyte, and 0.44 ± 0.03, for the hydrophilic one. This evolution indicates that the structures of both polyelectrolytes are different; the system is likely to behave as a solution of elongated chains in the case of hydrophilic polyelectrolyte (as in the string controlled limit), as well known, and as a solution of charged beads in the case of hydrophobic polyelectrolyte (collapsed spheres or bead controlled limit).

Our findings here are very similar to the ones for the model partially sulfonated PSS in water: the total structure function was measured by SAXS and SANS and it was found that q^∗ was scaling as c_p^–α, with α decreasing from 0.5 to less than 0.4 when reducing the chemical charge fraction of the chain f_chem (Essafi et al., 1994; Spiteri, 1997). Later, this was also observed by Baigl et al., through both SAXS and atomic force microscopy studies (Baigl et al., 2003a,b; Dan Qu et al., 2003).

3.5 Effect of ionic strength

Monovalent salt addition is expected to screen the repulsive Coulombic interactions, which makes the polyelectrolyte “peak” vanish. The scattered intensity at very low q (up to the position of peak) increases with salt concentration c_s, reflecting larger long-range concentration fluctuations and thus an increase in the compressibility of the system.

This is seen in Fig. 5d enabling to observe the effect of the hydrophobicity. For the more hydrophilic polyelectrolyte, with a smaller added salt concentration of 0.1 mol L⁻¹, the peak remains present at the foot of the low q upturn, but even for c_p = 0.34 mol L⁻¹ and c_s = 0.34 mol L⁻¹ NaCl (c_s/c_p = 1), the polyelectrolyte peak is slightly visible. For the hydrophobic case, Insert of Fig. 5d shows that a ghost of a peak is hardly visible; the sensitivity to salt is higher, inducing stronger concentration fluctuations (however there is no demixing). This reactivity is even more visible with yet no phase separation for the most hydrophobic polyelectrolyte, and 1 mol L⁻¹ NaCl (c_s/c_p ∼ 3), in Fig.SI.8.

Very similar observations have been made by Essafi et al. (1999) on AM-AMPS, and by Essafi et al. (1994) and Spiteri et al. (2007) on partially sulfonated PSSNa. Indeed the scattering behavior of tri-copolymers is akin to the one of partially sulfonated polystyrene for polyelectrolyte concentration variation as well as salt addition. Let us now make more precise comparisons.

4.1 Methodology: from SANS to SAXS

The scattering functions S_T(q) from SANS obtained after normalization to the contrast K_N² can be made superposed on those from SAXS after normalization to a contrast K_X². Calculating K_X² without accounting for the Na⁺ counterions, the result is satisfying. Including the counterions in the calculation does not change the ranking of the contrast values for the different copolymers neither makes strong changes of S_T(q). Hence, estimating condensation by comparisons between Neutron and X-rays measurements is delicate here and requires a larger and dedicated set of data to conclude. Anyhow, sodium counterions (Na⁺) are not the main origin of the SAXS contrast for extended or partly contracted linear chains.

4.2 Global comparison with former scattering from pearl-necklace-like systems

The literature mostly concerns partially sulfonated polystyrene (Essafi et al., 1999, 1994, 1995, 2009) which indeed exhibited such pearls (see Introduction). Except for the unique case of ZAC measurement of S₁(q), we can assume that most of the accessible information can be summarized by the values of q^∗.

The variation of q^∗ values with styrene content, is shown in Fig. 7 for both di-copolymers and tri-copolymers. Variation with other parameters (AM or AMPS content) more scattered, can be seen in Supporting Information (see Figs. SI.9 and SI.10). The q^∗ of tri-copolymers show, in gross approximation, a variation with styrene content rather close to the one for “bi-copolymers” case, namely poly(styrene-co-sodium styrene sulfonate), and for poly(styrene-co-sodium-2-acrylamido-2-methylpropane sulfonate) for one point. However:

• –

  a clear difference appears below 20% ST content, where q^∗ stays quasi-constant to the value at zero ST content. For these points, the AM content (see Table 3) is large, while the AMPS is always larger than the Manning threshold. This evidences an hydrophilic effect of AM, while AMPS content effect does not vary. Reciprocally, in the same interval we also notice a lower point in Fig. 7: it indeed corresponds to a much smaller AM content.

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  when ST approaches 40% ST content, for all samples where AMPS remains larger than 36%, AM is bound to be reduced: however its effect is still observed since q^∗ remains only slightly higher than for the di-copolymers, for two other pairs of points.

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  finally, above 55% ST content, AM content is even more decreased and there is no more hydrophilic effect: the q^∗ values overlap those of the di-copolymers. We note that for the largest styrene content, 67%, precipitation is observed.

A proposed evolution of the conformation of the tri-co-polyelectrolytes in aqueous solution as the hydrophobicity increases (% styrene increases) is given in Fig. 8 below.

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Figure 8

Proposed evolution of the tri-co-polyelectrolytes conformation in aqueous solution as the hydrophobicity increases (% styrene increase).

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Let us now attempt to more quantitative considerations using a theoretical model.

4.3 Detailed comparison with the Dobrynin - Rubinstein (DR) model

As briefly recalled in Introduction, the different models for strongly charged hydrophobic polyelectrolytes result in a different variation of q^∗ as a balance of a bad solvent effect and electrostatic repulsion. We will use specifically for our discussion the DR model, which we believe is the most relevant. The chain is a random walk of correlation blobs of size ξ, each of which is an extended configuration of electrostatic blobs of g_e monomers inside the blob diameter D. Therefore ξ is close to a geometrical distance between rod-like strands of chains and depends on their effective linear density of segments (g_e/D), which we will define as equal to B/a, defining B as in (Dobrynin et al., 1995) and a is the monomer size:

In the case of poor solvent, if we stay in the string-controlled regime, q^∗ will still be related to the inverse blob size, and one can introduce the corresponding expression of B which depends on the normalized distance τ $\left(\tau =\frac{\Theta -T}{\Theta }\right)$ to the theta temperature Θ (Dobrynin and Rubinstein, 1999):

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  1/A specific class of hydrophobic polyelectrolytes

The DR model describes a class of non-associative hydrophobic polyelectrolyte. The chains repel each other in solution. This is what observed in our study (for which hydrophobic content (styrene) is never larger than 50%): the solutions stayed clear and did not show any increase of viscosity. The scattering shows a clear repulsion, via the existence of a peak, which can be characterized via the variation of q^∗.

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  2/A specific behavior with polyelectrolyte and salt concentration

The salt effect on scattering, which is similar to the one for hydrophilic polyelectrolyte, is also in favor of independent chains with well-defined charge, as discussed just above.

The variation of q^∗ with c_p predicted by DR model is a power c_p^½ in Eq. (6). However, there can be crossover between the semidilute (c_p^1/2) and the dilute regime (c_p^1/3). Moreover, after the formation of pearls, a crossover is predicted between a string controlled and bead controlled regime. We observed a power law q^∗ ∼ c_p^{0.37±0.02 − 0.45±0.03}, close to what was also found for PS-co-SSNa solutions, which showed pearl necklace behavior conformation (SANS-ZAC measurements).

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  3/A balance between hydrophobicity and hydrophilicity

The expression of q^∗ (Eq. (7)) rules a balance between electrostatic energy (l_B/a). f_eff² and solvent quality $\tau =\frac{(\Theta -T)}{\Theta }$ . The first term is linked to the effective charge along the backbone. In the context of Manning-Oosawa theory, the effective charge is expressed as follows:

Let us evaluate f_eff in our system. This is equivalent to consider Manning condensation:

• –

  if we consider separately each of the AMPS sequences, with a ∼ 2.5 Å, Manning condensation will reduce each of their charge by a factor 2.5/7 ≈ 36%, so the global effective charge rate will be f_eff ≈ 0.36 z/(x + y + z).

• –

  if on the contrary we consider the whole sequence, since the segment length a is very close for the three polymers PAM, PAMPS and PS, the Manning threshold reduces the charge rate to 36% over the full copolymer length (as long as the AMPS monomers fraction is larger than 36%, which is the case here). The total effective charge is 36%, independent of the AMPS fraction z (z/(x + y + z) > 0.36).

Note that here we consider the charge fraction before the chain contraction (additional condensation may occur later due to the compacity of the chain, or around the pearls, and reduce the final value of f_eff, as observed).

Let us now evaluate the solvent quality τ: Since Styrene is strongly hydrophobic, the slight difference in hydrophilicity brought by AM and hydrophilicity brought by AMPS could be neglected in first approximation and the simplest is to assume that $\tau =\frac{(\Theta -T)}{\Theta }$ is driven by the Styrene content. Thus from Eqs. (7) and (8) we can check whether q^∗ is simply driven by a ratio R of Styrene content (τ^−1/4) over the AMPS content (electrostatics, f_eff^−1/4).

However, while Fig. 7 shows a strong effect of Styrene (ST) content as pointed above, it also displays a specificity of the tri-copolymers values of q^∗ with high AM content on (Fig. 7), as described above. The possible loss of hydrophilicity brought by the decrease in AMPS is masked by the increase in AM. The solubility is more improved by the mixing of PS units with hydrophilic AM units than with AMPS units: τ is driven by the Styrene content but also by the AM content. This agrees with the fact that polyacrylamide is quite hydrophilic (χ ∼ 0.5 (Li et al., 2012; Sen et al., 1999)). AMPS does not seem dominant on τ. Its role is mostly to bring electric charges but because of condensation, the charge rate brought remains 36% for all our samples.

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  4/A pearl necklace conformation

Such conformation is the main originality of DR model. Unfortunately, conformation can only be evidenced using Zero Average Contrast, which requires deuteriation of some of the tri-copolymers. However, we can propose, infering from the strong similarities with the PS-co-SSNa that such conformation is present for these tri-co-polyelectrolytes.