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Langmuir 2007, 23, 8460-8473
Relationship between Swelling and the Electrohydrodynamic Properties of Functionalized Carboxymethyldextran Macromolecules Elise Rotureau, Fabien Thomas, and Je´roˆme F. L. Duval* Laboratory EnVironment and Mineral Processing, Nancy-UniVersity, CNRS, BP 40-F-54501 VandoeuVre-le` s-Nancy Cedex, France ReceiVed February 13, 2007. In Final Form: April 28, 2007 The electrostatic, hydrodynamic, and swelling properties of a well-defined, functionalized carboxymethyldextran (CMD) polysaccharide are investigated in aqueous NaNO3 solution over a broad ionic strength range. The impact of the polycarboxylate charge and molar mass of the CMD macromolecules on their electrohydrodynamic features is thoroughly examined by combined protolytic titration, dynamic light scattering, and electrokinetic analyses. Electrophoretic mobility data obtained for sufficiently high electrolyte concentrations reveal a typical soft particle behavior. Upon decrease of the ionic strength, mobilities strongly increase in magnitude while significant electrostatic swelling takes place, as reflected in a decrease in the diffusion coefficients. CMD entities undergo conformational transitions from compact random coil at large ionic strengths to swollen coil and possibly a wormlike structure at lower NaNO3 concentrations. The magnitude of the variations in size and mobility with electrolyte concentration strongly depends on the overall charge of the CMD entity as well as on its molar mass. These factors control the stiffness of the constituent polymer chains and thus the degree of macromolecular permeability (“softness”). Using the soft-diffuse interface formalism previously developed for the electrohydrodynamics of charged permeable macromolecules, a quantitative analysis of the electrophoretic mobility data is presented. The measured values of the diffusion coefficient and space charge density Γ°, as evaluated independently from the modeling of potentiometric titration curves, are taken into account in a self-consistent manner. It is found that large CMD entities of low charge densities are the most permeable to flow penetration with a limited heterogeneous electrostatic stiffening of the chains, whereas small CMD entities of larger Γ° significantly expand upon lowering the ionic strength, giving rise to a strong anisotropy for the spatial distribution of polymer chain density.
1. Introduction Biofilms, for example, those formed on plant roots, soil, or river sediments, may be viewed as complex, heterogeneous matrices that are composed largely of extracellular polymeric substances (EPS). The principal components of EPS are proteins and charged polysaccharides, and they play a key role in determining the physicochemical properties of biofilms,1,2 including mechanical stability, sorption capacity, and nutrient transfer. It is now well-established that EPS, and especially charged polysaccharides, possess a high ability to complex and immobilize metals such as copper3-5 from the surrounding aqueous phase. This property is largely due to the finite electrostatic charge of the biopolymers over a large pH domain and to their accessible and highly reactive binding sites. Numerous studies have been performed to measure the propensity of various metallic ions to adsorb onto/complex with natural polysaccharides such as sugar beet pulp,6 hyaluronate,7 or alginate.8 The mechanisms underlying the interactions between the metals and polysaccharides are strongly dependent on (i) the physicochemical conditions (pH, ionic strength) of the aqueous environment, (ii) (1) Branda, S. S.; Vik, A.; Friedman, L.; Kolter, R. Trends Microbiol. 2005, 13 (1), 20-26. (2) Flemming, H.-C.; Wingender, J.; Moritz, R.; Borchard, W.; Mayer, C. Spec. Publ.sR. Soc. Chem. 1999, 242, 1-12. (3) Deiana, S.; Gessa, C.; Palma, A.; Premoli, A.; Senette, C. Org. Geochem. 2003, 34 (5), 651-660. (4) Yun, Y.-S.; Volesky, B. EnViron. Sci. Technol. 2003, 37 (16), 3601-3608. (5) Le Cloirec, P.; Andres, Y.; Faur-Brasquet, C.; Gerente, C. ReV. EnViron. Sci. Bio/Technol. 2004, 2 (2-4), 177-192. (6) Reddad, Z.; Gerente, C.; Andres, Y.; Le Cloirec, P. EnViron. Sci. Technol. 2002, 36 (10), 2242-2248. (7) Magnani, A.; Silvestri, V.; Barbucci, R. Macromol. Chem. Phys. 1999, 200 (9), 2003-2014. (8) De Stefano, C.; Gianguzza, A.; Piazzese, D.; Sammartano, S. Anal. Bioanal. Chem. 2005, 383, 587-596.
the protolytic properties (nature and quantity of charges, ionexchange equilibria) of the biomacromolecule, (iii) the chemical affinity of the binding site, and (iv) the spatial structure of the “carrier” macromolecule, namely, its size, conformation, and flexibility. Milas and Rinaudo9 pointed out the large diversity of ordered conformations for ionic polysaccharides especially upon variation of the nature of the counterions, the ionic strength, or the temperature. The nature of the glycosidic linkage within the polysaccharide chain governs its solubility, the dynamics of interaction with adjacent water molecules,10 its conformation, and thus its rigidity properties.11 In view of these features, a mandatory prerequisite for understanding the interactions between polysaccharides and metallic ions is a rigorous analysis of the degree of polysaccharide permeability and therewith the coupled stiffness of the chains. For the sake of illustration, alginic acid has a high affinity for divalent ions because of its so-called “zigzag” structure. As a result, metal loading leads to a change of the macromolecular conformation following the formation of metal bridges and ultimately results in aggregation according to the “egg-box model”.12 As mentioned, structural aspects of the macromolecule are important but not sufficient to account for the metal binding properties. Indeed, the nature of the amphoteric charges of the macromolecule, their spatial distribution, and the resulting electrostatic potential distribution (that generally extends beyond the gyration radius of the macromolecule) impact in a dramatic manner on the adsorption of metallic ions. Experimental evidence that illustrates this may be found, for example, in refs 13-15. A common way to quantify the contribution of the (9) Milas, M.; Rinaudo, M. Curr. Trends Polym. Sci. 1997, 2, 47-67. (10) Zamparo, O.; Comper, W. D. Carbohydr. Res. 1991, 212, 193-200. (11) Sutherland, I. W. Microbiology 2001, 147 (1), 3-9. (12) Lamelas, C.; Avaltroni, F.; Benedetti, M.; Wilkinson, K. J.; Slaveykova, V. I. Biomacromolecules 2005, 6 (5), 2756-2764.
10.1021/la700427p CCC: $37.00 © 2007 American Chemical Society Published on Web 07/10/2007
Properties of Functionalized CMD Macromolecules Scheme 1. Chemical Structure of a Synthesized Carboxymethyldextran Macromoleculea
a For the sake of clarity, we have not represented all the hydroxyl groups that may undergo substitution by carboxylic groups according to the reaction indicated in eq 1.
electrostatics is to include in the appropriate adsorption isotherm expressions a Donnan potential term.13-15 However, as discussed in a previous analysis,16 the very notion of Donnan potential becomes necessarily inaccurate when operating at low ionic strengths and/or for small macromolecules, although such situations are commonly encountered in practice. Last but not least, the dynamics of interaction between metallic ions and polysaccharides is very much dependent on the hydrodynamic permeability of the biomacromolecule (mostly submitted to convective and electrical forces in nature), which cannot be viewed as a hard colloidal object.17 To our knowledge, only a few analyses consider both the hydrodynamic and electrostatic properties of organic colloidal compounds such as polysaccharides in the presence or absence of metallic ions. Electrokinetics, in particular, electrophoresis, has shown to be a very valuable tool in this respect.16-18 Recent electrokinetic theories for soft systems further allow advanced quantitative interpretation of raw electrokinetic data in terms of charge density, hydrodynamic permeability, and structural and chemical heterogeneities.19 In the current paper, we chose carboxymethyldextran (CMD) as a model colloidal polysaccharide ligand for metallic ions. The CMD biopolymer can be easily obtained by fixing carboxylated groups in a controlled manner onto chains of dextran (Scheme 1). The latter, produced by bacteria of the species Leuconostoc mesenteroides, are originally neutral, hydrophilic, and flexible and mainly contain R-1,6-glucoside linkages.20,21 We report here (13) Koopal, L. K.; Saito, T.; Pinheiro, J. P.; van Riemsdijk, W. H. Colloids Surf., A 2005, 265 (1-3), 40-54. (14) Pinheiro, J. P.; Mota, A. M.; Benedetti, M. F. EnViron. Sci. Technol. 2000, 34 (24), 5137-5143. (15) Pinheiro, J. P.; Mota, A. M.; Benedetti, M. F. EnViron. Sci. Technol. 1999, 33 (19), 3398-3404. (16) Duval, J. F. L.; Slaveykova, V. I.; Hosse, M.; Buffle, J.; Wilkinson, K. J. Biomacromolecules 2006, 7 (10), 2818-2826. (17) Duval, J. F. L. Electrophoresis of soft colloids: basic principles and applications. In EnVironmental Colloids and Particles: BehaVior, Separation and Characterization; Wilkinson, K. J., Lead, J., Eds.; Wiley: Chichester, 2007; Chapter 7. (18) Duval, J. F. L.; Wilkinson, K. J.; van Leeuwen, H. P.; Buffle, J. EnViron. Sci. Technol. 2005, 39 (17), 6435-6445. (19) Duval, J. F. L.; Ohshima, H. Langmuir 2006, 22 (8), 3533-3546. (20) Neelov, I. M.; Adolf, D. B.; McLeish, T. C. B.; Paci, E. Biophys. J. 2006, 91, 3579-3588. (21) Rief, M.; Oesterhelt, F.; Heymann, B.; Gaub, H. E. Science 1997, 275, 1295-1297.
Langmuir, Vol. 23, No. 16, 2007 8461 Table 1. Number-Average and Weight-Average Molar Masses (Denoted as Mn and Mw, Respectively) and Polydispersity Index (Ip) of the Unmodified, Uncharged Dextran Precursors as Determined by SEC-MALLS polymer
Mn (g/mol)
Mw (g/mol)
Ip
T40 T500
29 000 370 000
43 000 520 000
1.5 1.4
an experimental and theoretical electrokinetic analysis of CMD macromolecules as a function of their degree of substitution of carboxylated groups and molar mass. The effect of ionic strength on the electrohydrodynamic features of the macromolecule is also tackled in detail. The electrostatic and hydrodynamic information is retrieved from the experimental data by means of the model recently developed by Duval et al.19 based on a diffuse representation of the soft (permeable) interfacial region between the macromolecule and the solution side. The analysis is further supported by independent potentiometric titration and light scattering measurements. The strategy followed in this study has already proven successful in accounting for the electrostatic and hydrodynamic features of humic acids18 and succinoglycan polymers.16 The experimental data coupled with modeling have enlightened the complex behavior of this semirigid polysaccharide and pointed out the occurrence of aggregation upon lowering pH or increasing ionic strength. 2. Experimental Section 2.1. Materials. NaNO3, HNO3, and NaOH aqueous solutions were prepared in ultrapure Milli-Q water from analytical grades purchased from Aldrich (St. Quentin Fallavier, France) and were used without further purification. Carboxymethyldextran (CMD) was prepared from dextran produced by bacteria of the species Leuconostoc mesenteroides and obtained by Amersham Biosciences (Uppsala, Sweden). CMD macromolecules were synthesized after chemical modification of two precursors commercially available: a low mass (unmodified) dextran denoted as T40 and a larger called T500. The respective molecular weights and polydispersity indexes for T40 and T500 were determined by size exclusion chromatography coupled to multiangle laser light chromatography (SEC-MALLS). The corresponding results are reported in Table 1. CMD was obtained and then purified following a synthesis route published in refs 22 and 23 that consists of a step-by-step carboxymethylation of dextran (Dex(OH)3) according to the reaction scheme (1) NaOH, H2O, 0 °C
[Dex(OH)3]x+y 98 (2) ClCH2COOH, H2O, 60 °C
[Dex(OH)3]x[Dex(OH)2 (OCH2COONa)]y (1) where the symbols x and y are those indicated in Scheme 1. The overall yield, defined as the ratio of the recovered weight at the end of the procedure over that theoretically expected, was estimated at ∼80%. The degree of substitution, denoted as DS, indicates the number of added carboxylic functions for 100 glucosidic units. The magnitude of DS is controlled by the number of successive carboxymethylation reactions performed. The hydroxyl groups located at the C2 positions within the glucopyranose units along the dextran chain are predominantly substituted, but polysubstitution within a given monomer unit remains possible.24 DS values (or, equivalently, the carboxylic charges) were determined after careful analysis of potentiometric titration data collected on the synthesized CMD samples. The underlying theoretical procedure is detailed in section 3.1. For comparison purposes, three CMD macromolecules of various DS values and molar masses were prepared and will be (22) Chaubet, F.; Champion, J.; Maı¨ga, O.; Mauray, S.; Jozefonvicz, J. Carbohydr. Polym. 1995, 28, 145-152. (23) Mauzac, M.; Josefonvicz, J. Biomaterials 1984, 5, 301-304. (24) Krentsel, L. B.; Ermakov, I. V.; Yashin, V. V.; Rebrov, A. I.; Litmanovich, A. D.; Plate, N. A.; Chaubet, F.; Champion, J.; Jozefonvicz, J. Vysokomol. Soedin., Ser. A Ser. B 1997, 39 (1), 83-89.
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Table 2. Characteristic Parameters of the Different Synthesized Carboxymethyldextran Macromolecules: Precursor Polymer Used for Synthesis, Number of Carboxymethylation Reactions, Number-Average Molecular Weight (Mn), Maximum Space Charge Density (Γ°max), and Degree of Substitution (DS)
polymer
precursor polymer
number of carboxymethylation reactions
Mn (g/mol)a
Γ°max (mM)b
DS (%)b
CMD26% T40 CMD15% T500 CMD37% T500
T40 T500 T500
1 1 2
33 400 410 000 460 000
-88.5 -11.7 -16.2
26 15 37
Obtained by SEC-MALLS. b Obtained by potentiometric titration at 100 mM NaNO3 ionic strength and pH > 7.5 (complete dissociation of the carboxylic sites).
curves according to multiexponential fitting algorithms, as implemented in the CONTIN or Multiple Narrow Mode options of the Malvern software package. Concerning the delicate cases of multimodal and broad distributions, as encountered at very low ionic strengths, great care was taken to ensure repeatability of the measurements before attempting any interpretation. Electrophoretic mobilities were measured by laser Doppler electrophoresis, also known as phase analysis light scattering (PALS). The rate of change of the phase shift between the scattered light and a reference beam is correlated to the particle velocity and thus allows for evaluating the particular electrophoretic mobility.
3. Theory
a
15% 37% denoted in the following as CMD26% T40 ,CMDT500, and CMDT500. The DS subscript and superscript in the notation CMDX refer to the molecular precursor used as a reactant in eq 1 and the substitution degree of the synthesized CMD, respectively. The molar masses of the obtained CMD were determined by SEC-MALLS and are detailed in Table 2, where numbers of carboxylic charges are also collected. Polysaccharide solutions were prepared 24 h prior to experiments in filtered (0.22 µm) Milli-Q water and subsequently stored at 4 °C. For measurement of the diffusion coefficient and the electrophoretic mobility of the CMD macromolecules, aqueous dispersions 15% were prepared with concentrations of 1 g L-1 for CMDT500 and 37% 26% CMDT500 and 4 g L-1 for CMDT40 . This increase in concentration is necessary to obtain good quality data because the size of CMD26% T40 is close to that fixed by the detection limit of the apparatus used for the size and electrokinetic measurements (see section 2.3). For all electrokinetic and size experiments, the pH of the CMD suspensions was adjusted to 6 by addition of the appropriate aliquots of 0.1 M HNO3 and 0.1 M NaOH. 2.2. Potentiometric Titration. Potentiometric titrations were performed at 25 ( 0.1 °C using a computer-monitored titrator (736 GP Titrino, Metrohm) equipped with an automatic buret. Titrations of 50 mL of polymer solution were conducted under magnetic stirring and a nitrogen stream to avoid carbon dioxide dissolution in the solution. Potentiometric curves were measured at two NaNO3 electrolyte concentrations (0.01 and 0.1 mol L-1) maintained constant all along the experiments. The pH was measured with a pH electrode (Radiometer-GK2401C) calibrated with four commercial pH buffers (Titrinom, Merckeurolab). Titrant solutions (0.1 mol L-1 HNO3 and 0.1 mol L-1 NaOH) were gradually added by variable increments either after a fixed time delay of 10 min in standard experiments or once the pH drift was less than 0.2 mV min-1 in specified cases. To visualize the possible occurrence of hysteresis, titration curves were recorded in the following order: from natural pH (∼6-7) to basic pH and from basic pH to acidic pH. The consumption of protons by CMD was calculated at every pH after subtracting the corresponding contribution from the background electrolyte, as theoretically evaluated. For the quantitative analysis of the data, the ionic strength was calculated taking into account both background electrolyte and free [H+] and [OH-]. 2.3. Light Scattering and Electrokinetic Measurements. Diffusion coefficients and electrophoretic mobilities were measured at 25 ( 0.1 °C using a Zetasizer Nano ZS instrument (He-Ne red laser (633 nm), Malvern Instruments). The apparatus is equipped with an automatic laser attenuator and an avalanche photodiode detector. The position of the latter is located 173° relative to the laser source, so that backscattering detection is ensured. Experiments were driven by the Dispersion Technology software provided by Malvern Instruments. The solutions were pre-equilibrated for a period of 10 min before the measurement. For monomodal particle distribution, the diffusion coefficient, also called “cumulants mean”, is of particular interest. It is obtained from the second-order term in the Taylor expansion of the logarithm of the correlation function. Concerning the narrow polydisperse samples, the distribution of the diffusion coefficient was evaluated by fitting the measured correlation
3.1. Protolytic Titration. The theoretical formalism used in this study for the quantitative interpretation of the potentiometric titration data of CMD at various pH values and NaNO3 electrolyte concentrations is that extensively discussed in ref 18. We recall here the basic ideas as well as the governing, coupled equations of which the consistent numerical evaluation in spherical geometry allows interpretating experimental data in terms of protonation/ deprotonation constants, maximum space charge densities (i.e., when all active sites are deprotonated), and chemical heterogeneities. These characteristic parameters are then used for the electrohydrodynamic analysis of the measured electrophoretic mobilities of CMD (section 3.2). The charge as obtained by potentiometric titration, denoted as Qo, is defined by the spatial integration of all local charges over the spherical macromolecular volume, that is
Qo ) 4π
∫0δFfix(r,pH)r2 dr
(2)
where r is the radial position and δ is the radius of the ionpermeable macromolecule (Figure 1). The amphoteric nature of the ionogenic sites distributed within the macromolecule leads to a pH-dependence for the local space charge density that is denoted as Ffix in eq 2. The radial dependence of Ffix is determined from the local distribution of the protons within the macromolecule, which is in turn mediated by that of the electrostatic potential, denoted as ψ(r). The latter is computed on the basis of the nonlinear Poisson-Boltzmann equation. After straightforward rearrangement, it is written for a symmetrical z:z electrolyte of bulk concentration c∞
d2y(r˜) dr˜
2
Ffix(r˜,pH) 2 dy(r˜) - sinh[y(r˜)] ) r˜ dr˜ 2Fc∞z
+
(3)
where r˜ is the dimensionless position r˜ ) κr with κ being the reciprocal screening Debye length defined by κ ) (2F2c∞z2/ RTor)1/2 with F being the Faraday number, T being the absolute temperature, and or being the dielectric permittivity of the medium. The term y(r˜) represents the dimensionless electrostatic potential, y(r˜) ) zFψ(r˜)/RT, where R is the gas constant. The boundary conditions associated with eq 3 are
y(r˜ f ∞) ) 0
(4)
which defines the choice for the reference potential of a sufficiently dilute system (as is the case here), and another condition specifies the value of the electric field at the center of the macromolecule (r ) 0), that is
dy(r˜) dr˜
|
r˜)0
) 0 (symmetry condition)
(5)
To go further in the analysis, an expression for the local proton adsorption isotherms Ffix(r,pH), corrected by the appropriate
Properties of Functionalized CMD Macromolecules
Langmuir, Vol. 23, No. 16, 2007 8463
Figure 1. (A) Schematic representation of a soft diffuse particle and (B and C) illustration of the physical meaning of the key parameters introduced in the modeling of the macromolecular electrohydrodynamic properties.
electrostatic contribution, is required. For macromolecules for which the charge is determined by protonation/deprotonation of a single type of amphoretic group, we may write the LangmuirFreundlich isotherm in the form
1 Ffix(r,pH) ) Γ°max 1 + 10m(pK-pH) exp[-my(r˜)]
(6)
o stands for the maximum space charge where the quantity Γmax density of the sites upon complete dissociation (i.e., for pH . pK) and m is the Freundlich (heterogeneity) parameter of the sites. The latter determines the width of the distribution function for the corresponding affinity constant. In eq 6, the K constant accounts for the chemical (specific) contribution to the protoncarboxylic site interaction whereas the electrostatic (nonspecific) component is subsumed in the exponential Boltzmann term exp[my(r˜)] which involves the local potential within the macromolecule. Since the electrokinetic behavior of the soft macromolecules depicted in Figure 1 depends on space charge densities and not on the total amount of charges carried by the particle, we choose to reason in terms of the overall charge density Γ° (or, equivalently, volumic concentrations of charges, Γ°/F) defined by the ratio Qo over the volume 4πδ3/3. The quantitative interpretation of the titration measurements requires the evaluation of Γ° as a function of pH and ionic strength via the nonlinear and coupled eqs 2-6. This is done according to a numerical iterative scheme detailed in ref 18 which makes use of (i) the analysis of the data collected at electrolyte concentrations c∞ f ∞ for which y(r˜) f 0 (in such cases, one may neglect the electrostatic term in eq 6 and retain the only chemical contribution which can be further straightforwardly integrated according to eq 2) and (ii) the derivative of the protolytic data with respect to pH, that is, the affinity constant spectrum dΓ°(pH)/dpH. By merging experimental and theoretical data over the whole range of pH and ionic strengths according to the procedure discussed in ref 18, one may determine o the constants K, Γmax , and m using a least-square methodology. For that purpose, the necessary evaluation of the potential distribution at every pH and ionic strength investigated (ranges 4 < pH < 9 and c∞ ) 10 and 100 mM NaNO3 are examined here) is obtained by solving the set of coupled nonlinear finite differences equations associated to eqs 3-6 according to a globally convergent Newton-Raphson algorithm.25 For the sake of simplicity, we have made in the preceding development the tacit assumption that the polymer segment chains are homogeneously distributed in space. This means that the polymer segment density distribution function, denoted as f(r), corresponds to a step function with bulk values for 0 < r < δ and 0 for r > 0 (step function representation of the interface). For situations where this approximation is unrealistic, the impact of concentration gradients at the interface (heterogeneous soft layer) on the titration data can be taken into account by an appropriate choice of the function f(r) that is then necessarily involved in eq 6 (see section 3.2) as detailed in ref 19. This is to say that the dependence of the fixed charge density on the position is due to electrostatics as mentioned before but also to structural chain arrangements. It is stressed that, for a heterogeneous soft layer with a gradual distribution of the polymer segments density from the center of the macromolecule to its periphery, the radius of the macromolecule denoted above as δ corresponds to the outmost radius where one still finds a finite polymer concentration.19 In eq 2, the upper integration limit may then be replaced by ∞ essentially because Ffix(r,pH), linearly related to f(r), satisfies the asymptotic condition Ffix(r,pH) f 0 (see section 3.2).19 This soft diffuse representation of the interfacial region macromolecule/electrolyte constitutes the basis for the quantitative analysis of the electrokinetic data collected over a broad range of ionic strengths (c∞ ) 0.1-100 mM NaNO3) at pH ) 6. The theoretical formalism used for that purpose has been largely discussed in previous
(25) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical recipes in Fortran: The art of scientific computing, 2nd ed.; Cambridge University Press: New York, 1986.
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communications,19 and we recall its fundamental basis in the next section. 3.2. Electrokinetics. The electrokinetic properties of bioparticles (e.g., bacteria and yeasts) or environmental polyelectrolytes (e.g., humic acids and succinoglycan) are governed by a complex interplay between electrostatic and hydrodynamic processes.19 As demonstrated in numerous experimental and theoretical studies,17 the analysis of the electrohydrodynamics of those systems cannot be performed according to theories that are strictly valid for hard colloidal objects and the concept of zeta potential for such systems loses its conventional meaning. Instead, one must necessarily resort to more advanced models26-29 where flow penetration within the particle is explicitly taken into account. Recently, Duval et al. derived the governing set of electrohydrodynamic equations that account for the migration of a spherical19 or rodlike16 permeable, heterogeneous particle under the action of an externally applied electric field. The evaluation of the electrophoretic mobility of a diffuse soft particle relies on the complex numerical solving of the electrostatic and transport equations for the particle and ions that are distributed around and within it. The formalism extends the approximate analytical model originally developed by Ohshima which is applicable for homogeneous particles (step-function representation of the interface) when electric double layer polarization/relaxation effects are negligible.28,29 In addition, it includes the protolytic features of the particular amphoretic groups as determined from independent potentiometric titration data (see previous section). The electrokinetic modeling of a diffuse soft particle may be done for any radial polymer segment distribution f(r) obtained from either empirical, analytical, or computational considerations. In the following, in the absence, a priori, of concrete information on the local spatial distribution of the charged CMD chains, we take the following expression for f(r) as adopted in ref 19
f(r) )
{
ω r-δ 1 - tanh 2 R
(
)}
(7)
The parameter R has the dimension of a length and determines the degree of inhomogeneity of the distribution f for the polymer segment density (Figure 1). The dimensionless parameter ω is determined in such a way that the total amount of polymer segments remains constant upon variation of R and/or δ. It is therefore given by the expression
2 ω ) δ3/ 3
∫0∞{1 - tanh[r -R δ]}r2 dr
(8)
Assuming homogeneous distribution of the amphoteric groups along the CMD chain, the local space charge density is simply given by
f(r) Ffix(r,pH) ) Γ°max m(pK-pH) 1 + 10 exp[-my(r)]
(9)
and eq 6 is recovered from eq 9 by setting R/δ f 0. The friction forces exerted by the polymer segments on the hydrodynamic flow within the macromolecule are tackled along the lines of the Debye-Bueche theory30 that assimilates the polymer segments to so-called resistance centers. For relatively high water content within the macromolecule, the spatial distribution of the friction (26) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 258 (1), 56-74. (27) Ohshima, H. AdV. Colloid Interface Sci. 1995, 62 (2/3), 189-235. (28) Ohshima, H. J. Colloid Interface Sci. 1994, 163 (2), 474-483. (29) Ohshima, H. J. Colloid Interface Sci. 2000, 228 (1), 190-193. (30) Debye, P.; Bueche, A. M. J. Chem. Phys. 1948, 16, 573-579.
coefficient, denoted as k(r), is then formulated by k(r) ) ηλ02f(r), where η is the dynamic viscosity of water and λ0 is the nominal softness parameter which is approached as R/δ f 0. The quantity 1/λ0 has the dimension of a length, and it characterizes the typical penetration degree of the flow within the macromolecule. For the diffuse distribution of polymer chains (i.e., R/δ * 0), the hydrodynamic permeability or, equivalently, the softness of the macromolecule is a function of the radial position.19 The quantities λ0 and R are the key parameters that may be retrieved from consistent analysis of the electrokinetic data, the size δ, and the space charge densities Ffix(r,pH) stemming from the quantitative interpretations of the light scattering and potentiometric data, respectively. They provide valuable information on the swelling/shrinking behavior of the particle upon variation of the physicochemical characteristics of the medium and also may yield useful information on the occurrence or not of aggregation processes.16,18
4. Results and Discussion 4.1. Effect of Ionic Strength on the Size of the CMD Macromolecules. The changes of the diffusion coefficient upon variation of the electrolyte composition (ionic strength and pH) can be used to estimate the size and also to draw conclusions about the possible occurrence of aggregation, conformational changes, gelation,31 or swelling/shrinking processes32,33 of the macromolecules. The dependence of the diffusion coefficient D on solution ionic strength, denoted as I, at pH ) 6 is shown in Figure 2 for the various CMD macromolecules synthesized. As might be expected for a microgel-type particle, a large decrease of D is observed upon decreasing I. This decrease in D indicates a significant electrostatic stiffening of the polymer chains that constitute the microgel-assimilated CMD particle, which in turn leads to an increase in the hydrodynamic size with decreasing salt concentration or, equivalently, with increasing intramolecular repulsion. We recall that the equilibrium extent of swelling of such a particle is determined by the condition of zero overall osmotic pressure, denoted as Π, and defined by34
Π ) Πelec + Πmixing + Πelast
(10)
where Πelec, Πmixing, and Πelast are the electrostatic, polymersolvent mixing (chain entropy), and chain elastic components to the overall osmotic pressure. A closer inspection of the dependence of the diffusion coefficient on I and of the related dispersity spectra leads to discriminating between three distinct ionic strength regimes that we now describe. In regime 1, for ionic strengths larger than 10 mM, D reaches a maximum value that is nearly independent of I. The associated scattered intensity distribution is monomodal, and the corresponding particle population is denoted as φ1. Within this range of ionic strengths, the macromolecular charge is completely screened, the electrostatic potentials within and around the particle approach zero, as confirmed by electrokinetics (see section 4.3), and the intramolecular repulsive interactions are insignificant. As a result, the conformational structure of the microgel particle is determined by a balance between Πmixing and Πelast only, which excludes any significant impact of the electrolyte concentration. As for unmodified (and uncharged) dextran macromolecules, 15% 37% the entities CMD26% T40 , CMDT500, and CMDT500 may then be assimilated to spherical, random coil particles of hydrodynamic (31) Rinaudo, M. Macromol. Biosci. 2006, 6, 590-610. (32) Schiessel, H. Macromolecules 1999, 32 (17), 5673-5680. (33) Schiessel, H.; Pincus, P. Macromolecules 1998, 31 (22), 7953-7959. (34) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953; p 672.
Properties of Functionalized CMD Macromolecules
Langmuir, Vol. 23, No. 16, 2007 8465 Table 3. Characteristic Electrohydrodynamic Parameters as Obtained from the Titration, Dynamic Light Scattering, and Electrokinetic Analyses of the Different Synthesized Carboxymethyldextran Macromolecules polymer
pKa
ma
δexp (nm)b
δtheo (nm)a
1/λ0 (nm)c
CMD26% T40 CMD15% T500 CMD37% T500
4.15 4.20 4.18
0.9 1 1
6.3 ( 0.5 26.0 ( 0.5 27.8 ( 0.5
6.05 23 28
0.8 3.7 3.4
a Evaluated by quantitative analysis of the protolytic data (see sections 3.1 and 4.2 in the text). b Mean hydrodynamic radius as evaluated from the diffusion coefficient data at I > 10 mM NaNO3 (regime 1 in Figure 2) and the Stokes-Einstein equation. c Inferred from the modeling of the electrophoretic mobility data (see sections 3.2 and 4.3 in the text).
Figure 2. Diffusion coefficient versus NaNO3 ionic strength as 37% , measured by dynamic light scattering at pH ) 6 for (A) CMDT500 15% 26% (B) CMDT500, and (C) CMDT40 . The filled points pertain to the values of the cumulants mean (fast diffusion mode), and the empty points pertain to the values of the diffusion coefficient associated to the slower diffusion mode. In the inset of Figure 2A are represented the typical diffusion coefficient distributions that define regimes 1, 37% ). 2, and 3 (an example is given for CMDT500
radii δ that may be estimated using the Stokes-Einstein equation. Results are reported in Table 3. The δ values obtained for the modified macromolecules are larger than those measured for the uncharged/unmodified precursors (dextran T500: δ ≈ 17.5 nm; dextran T40: δ ≈ 5 nm). This is in line with the functionalization of the precursor dextran macromolecules (eq 1), that is, the
introduction of additional chemical groups, which leads to decreasing the chain entropy with, as a result, an accompanying swelling and size increase. This swelling process is of a different nature than that which originates from electrostatic stiffening of the chains. The obtained relationship D ≈ M -0.5 with M being the molecular weight (see Table 2) confirms the flexible nature of the CMD chains.34-36 On decreasing the ionic strength (regime 2), the diffusion coefficient associated with the particle population φ1 strongly decreases, which is in line with the occurrence of significant swelling of the macromolecules. This stiffening of the polymer chains, which were originally in a collapsed state (regime 1), is attributed to the strong intramolecular repulsions that find their origin in the electrostatic interactions between neighboring charged ionogenic groups distributed throughout the macromolecule. The corresponding increase in the electrostatic component of the osmotic pressure drop, Πelec, that is closely related to that for the local electrostatic potentials results in an entropy loss for the counterions confined within the macromolecule. For these counterions, it would be thermodynamically more advantageous to abandon the polyelectrolyte volume, because in this way they would gain significant entropy of translational motion. However, this is impossible due to the concomitant charge separation and potential generation. The counterions are therefore forced to remain confined in the polyelectrolyte. As a result of the increased osmotic pressure, chains expand in space to counterbalance that entropy loss until the equilibrium situation as determined by the condition Π ) 0 is reached. In regime 2, the diffusion coefficient follows a bimodal distribution as illustrated in the inset of Figure 2A. The associated double-exponential correlation function is then interpreted in terms of two modes: a fast diffusion mode (population φ1) and a slow one. In the following, the population of larger particles, that is, of lower diffusion coefficients, is denoted as φ2. The corresponding scattered intensity represents up to 50% of the total measured intensity, which basically means that population φ2 remains in the numerical minority as compared to population φ1. This is the direct consequence of Rayleigh’s equation37 that predicts a six-power dependence of the scattered intensity with respect to the macromolecular size. In a third regime that corresponds to ionic strength levels below 0.1 mM, the measured diffusion coefficients are ∼1 order of magnitude lower than those determined at larger ionic strengths (regime 1), even though satisfactory reproducibility is more difficult to obtain, as reflected in the larger experimental error (35) Callaghan, P. T.; Pinder, D. N. Macromolecules 1981, 14 (5), 13341340. (36) Von Meerwall, E.; Tomich, D. H.; Grigsby, J.; Pennisi, R. W.; Fetters, L. J.; Hadjichristidis, N. Macromolecules 1983, 16 (11), 1715-1722. (37) Kerker, M. The scattering of light and other electromagnetic radiation; Academic Press Inc.: New York, 1969.
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bars. Note also that size measurements for CMD26% T40 below 0.1 mM are not reported because the scattered intensity becomes two low to achieve experimental data of good quality. In most dynamic light scattering studies on semidilute polyelectrolyte solutions with low amounts of added salt (regimes 2 and 3), the slow mode of diffusion is observed, but controversy still exists on its physical interpretation.38-44 Some authors attribute this mode of diffusion to that of large multichain aggregates or domains. It is, however, surprising here that CMD polyelectrolyte entities undergo attractive forces and thus form aggregates because of (i) the low amplitude collected for the scattered intensity, (ii) the low polymer concentration used for our experiments (dilute regime), and (iii) the important intermolecular repulsive interactions that are expectedly resulting in a large energy barrier for the macromolecules to overcome before any effective encounter is possible. This has led some authors45 to evoke the inappropriateness of the Derjaguin-Landau-Verwey-Overbeek (DLVO) interaction potential for polyelectrolytic systems that systematically predicts at low electrolyte concentrations longrange repulsive interactions (homo-interactions). Other explanations were advanced, for example, by Schmitz et al.,46 who mentioned the possible formation of “temporal aggregates” as a result of fluctuating dipolar interactions due to asymmetric distributions of counterions around polyelectrolyte chains. It is not the purpose of the current study to discriminate between the possible origins of the slow mode of diffusion as observed for semidilute polyelectrolyte suspensions. We find it, however, valuable to underline the presence of such a diffusion mode for the dilute regime as examined in our study, and where chains overlap is expected to play a marginal role. In the following (sections 4.2 and 4.3), the analysis will focus on the particular population φ1 which undergoes electrostatic swelling upon lowering the ionic strength level. In region 3, the population φ2 with the slower mode of diffusion becomes predominant because only one peak remains in the D distribution spectrum. Under these conditions, putting aside the above elements on the nature of the slow mode of diffusion, the CMD polysaccharides possibly undergo stretching due to intrachain electrostatic repulsions. The resulting significant swelling process of the macromolecule is then possibly accompanied by conformational changes with a gradual transition from a random, expanded coil structure (population φ1, regimes 1 and 2) toward a wormlike shape (regime 3). It is noted that, unlike the diffusion coefficient associated with population φ1, the diffusion coefficient related to φ2 barely depends on electrolyte concentration. This suggests that particles φ2 present a stronger anisotropy than particles φ1 (strongly stretched or wormlike polyelectrolyte chains) and they are thus likely to undergo lesser significant size variations upon lowering the ionic strength. We precise here that our assumption of a dilute regime for the CMD suspension as analyzed in this study (1-4 g L-1, see section 2.1) is motivated by the recent rheological measurements performed by Rotureau et al.47 on unmodified/ neutral dextran macromolecules. These authors demonstrated the dilute nature of suspensions of concentrations below 30 and 60 g L-1 for T500 and T40, respectively. We shall in a forthcoming analysis examine more in detail the impact of polymer volume (38) Koene, R. S.; Mandel, M. Macromolecules 1983, 16, 973. (39) Koene, R. S.; Mandel, M. Macromolecules 1983, 16, 220. (40) Drifford, M.; Dalbiez, J.-P. Biopolymers 1985, 24, 1501. (41) Forster, S.; Schmidt, M.; Antonietti, M. Polymer 1990, 31, 781. (42) Sedlak, M. J. Chem. Phys. 1994, 11, 1. (43) Sedlak, M. J. Chem. Phys. 1996, 105, 10123. (44) Tanahatoe, J. J.; Kuil, M. E. J. Phys. Chem. B 1997, 101, 10839. (45) Ise, N. Angew. Chem., Int. Ed. Engl. 1986, 25, 323. (46) Schmitz, K. S.; Lu, M.; Gauntt, J. J. Chem. Phys. 1983, 78, 5059. (47) Rotureau, E.; Dellacherie, E.; Durand, A. Eur. Polym. J. 2006, 42, 10861092.
Rotureau et al.
Figure 3. Plot of the diameter 2δ as a function of ionic strength 37% 15% , (curve b) CMDT500 , and (swelling regime) for (curve a) CMDT500 26% (curve c) CMDT40 . The points correspond to the size as evaluated from diffusion coefficient data (Figure 2) and the Stokes-Einstein relationship. The dashed lines are power-law regressions of the data (log-log representation).
fraction on the electrohydrodynamic features of functionalized CMD polyelectrolyte particles that carry larger charge densities than those reported here. The preliminary results indicate that intermolecular electrostatic interactions are negligible or have a marginal impact for 1-2 g L-1 polymer concentrations and 10-100 mM ionic strengths. The decay of the diffusion coefficient from regime 1 to regime 3 is ∼1 order of magnitude for the three CMD entities. The electrolyte concentrations c∞1T2 and c∞2T3 that mark the transitions between regimes 1, 2, and 3 are nearly identical for high molar mass CMD (c∞1T2 ≈ 5 mM and c∞2T3 ≈ 0.1 mM for CMD15% T500 and ). However, the transition from regime 1 to regime 2 CMD37% T500 occurs at larger for the low molar mass coil polymer CMD26% T40 ionic strength (c∞1T2 ≈ 10 mM). This feature is consistent with o the range of space charge densities Γmax reported in Table 2 and with the (electrostatic) origin of the swelling process of the polyelectrolyte coil evoked before to account for the variation o , the larger the of the diffusion coefficient. The larger Γmax electrostatic component of the overall osmotic pressure and the larger the ionic strength at which coil expansion becomes significant upon lowering I. In Figure 3, we report explicitly the dependence of the size of the particles φ1 in regime 2, as estimated from the Stokes-Einstein relationship, as a function of the solution ionic strength. Within experimental error, a power law of the type δ ∼ (c∞)-1/β is obtained for the three macromolecules examined, with β ≈ -0.33, -0.39, and -0.19 for CMD26% T40 , 37% CMD15% T500, and CMDT500, respectively. For polyelectrolytic particles, a simple analysis of the electrostatic and elastic contributions to the overall interfacial pressure drop leads to δ ∼ (c∞)-1/3,48 which is in reasonable agreement with the experimental data. The electrostatic component stems from the osmotic pressure drop that arises from the difference in ionic concentrations outside and inside the particle, whereas the elastic contribution originates from the entropy loss in the expanding coil process. (48) Lietor-Santos, J. J.; Fernandez-Nieves, A.; Marquez, M. Phys. ReV. E 2005, 71, 042401.
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We now comment on the quantitative determination of the protolytic properties of the various CMD macromolecules studied. 4.2. Protolytic Characteristics of the Synthesized Carboxymethyldextran Macromolecules. Potentiometric titration data obtained for 100 and 10 mM NaNO3 electrolyte concentrations are reported in Figure 4. The data were quantitatively interpreted following the theoretical scheme described in section 3.1. In this range of ionic strength, the CMD macromolecules are assimilated to homogeneous, spherical particles, as indicated by the diffusion coefficient data that point out the absence of significant electrostatic stiffening of the macromolecular chains (collapsed state, regime 1). The conversion of the measured data (expressed in charge per unit mass) into a volumic charge density Γ° was carried out taking into account the molecular weights and hydrodynamic radii reported in Tables 2 and 3, respectively. Generally, even small variations in the radius δ considered for the quantitative interpretation of the curves (Γ°-pH) impact on the theoretical predictions, since they lead to cubic variations of o and thus to possible significant changes in Ffix(r,pH) and Γmax y(r) (eqs 2-6). Despite this, the values determined from theoretical analysis are in good agreement with those obtained from the dynamic light scattering data in regime 1. Good agreement between theoretical predictions and experimental data could be further met for CMD37% T500 over the whole range of pH values examined. Discrepancies appear, however, for pH < 4.5 when dealing with the protolytic features of CMD15% T500, and they are even more marked for CMD26% . In this pH region, forward and T40 reverse titrations did not reveal any significant hysteresis, thus ruling out the possibility of hydrolysis. Possibly, slight shrinking of the CMD macromolecules could take place as a result of a decrease in intramolecular repulsions following a decrease of the charge. For a given pH, this would result in larger space charge densities and thus in an increasing difference between titration data at 100 and 10 mM, as observed in Figure 4. This size decrease should, however, remain rather small, since diffusion coefficient measurements upon lowering the pH did not vary much between pH 9 and 4. Nevertheless, systematic convergence between the modeled and measured protolytic data was reached for pH values larger than 4.5, that is, in the pH range relevant for the electrokinetic measurements (performed at pH ) 6). In passing, we note that the DS values mentioned so far for the CMD macromolecules have been evaluated from the maximal charge density found at 100 mM ionic strength and high pH values. In the modeling, we considered one type of site, as derived from the observation of one maximum in the spectrum of proton affinity constants. This is consistent with fixation of only the carboxylic groups during the functionalization step of uncharged dextran precursors. The values obtained for the constants pK, m, o are reported in Table 3. The median intrinsic stability and Γmax constant pK ) 4.20 is in good agreement with the pK value for carboxylic groups (ranging between 4 and 5 as determined for polyacrylates in NaNO3 aqueous solution49). From the obtained parameter m ≈ 0.9-1, it is believed that only one type of carboxylic site is responsible for the macromolecular charge or, said differently, that the protonated functional carboxylic groups have a chemical environment that barely varies with their position along the macromolecular chain. This is in agreement with the fact that, predominantly, hydroxyl groups in C2 positions are substituted during the functionalization reaction scheme in eq 1.24 As expected from the classical double layer theory, the amount of titrable charges increases when increasing the ionic strength
Figure 4. Volumic charge concentrations Γ°/F evaluated for (A) 37% 15% , (B) CMDT500 , and (C) CMD26% CMDT500 T40 as a function of pH for two ionic strengths of NaNO3 electrolyte (10 and 100 mM). The points correspond to the experimental data with accompanied error bars, and the dotted lines correspond to the theoretical predictions obtained on the basis of the formalism detailed in section 3.1 with the model parameters given in Tables 2 and 3. The master curves 37% 15% (I f ∞) for CMDT500 and CMDT500 correspond with those computed for I ) 100 mM.
(49) De Stefano, C.; Gianguzza, A.; Piazzese, D.; Sammartano, S. React. Funct. Polym. 2003, 55 (1), 9-20.
for a given pH value. This is due to the screening of the fixed charges located in the macromolecule by the neighboring free
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ions. The difference between the titration data measured for various ionic strengths is directly correlated to the magnitude of the local electrostatic potential distribution, as explained in the theoretical section. The long-dashed curves in Figure 4 refer to the purely chemical component of the isotherm, also called the 37% master curve. For CMD15% T500 and CMDT500, that both correspond o , the master curve to a rather low space charge density Γmax basically matches the computed titration data at 100 mM, meaning that, at this ionic strength level, the macromolecular charge is completely screened, as further confirmed by the examination of the local potential distribution. On the other hand, it is observed that charge screening is incomplete for CMD26% T40 at 100 mM and 4.5 < pH < 7.5 where significant discrepancies are observed between the master curve and the isotherm computed at 100 mM. This is in agreement with the larger space charge density as derived for CMD26% T40 and with the still significant electrostatic potentials (even at such ionic strength levels) reached within and around the particle. For pH values that largely exceed the pK values, all carboxylic sites are deprotonated and the impact of the ionic strength on the titration curves is either absent (in the 37% cases of CMD15% T500 and CMDT500) or considerably reduced (as for 26% CMDT40 ). 4.3. Electrohydrodynamic Properties of Functionalized CMD Macromolecules. The measured electrophoretic mobilities are reported in Figure 5 as a function of ionic strength at pH ) 6. Mobilities are given in the dimensionless form µ j defined by
µ j)µ
eη orkBT
(11)
with kB being the Boltzmann constant and e being the elementary charge. For practical reasons, measurements were mainly performed in the ionic strength range 0.1 mM e I e 100 mM, which encompasses regimes 1 and 2 discussed in section 4.1 in relation with the dependence of the diffusion coefficient on electrolyte concentration. Modeling of the electrokinetic data was performed on the basis of the theoretical formalism outlined in section 3.2. From the conclusions laid out in section 4.1, the CMD macromolecules were assimilated to spherical, random coil polyelectrolytes over the whole range of ionic strengths investigated in the electrokinetic analysis. Furthermore, we disregard the impact of particle population φ2 (in minority as compared to population φ1) on the measured electrophoretic mobility. The heterogeneity in the polymer segment density distribution, as governed by the swelling degree of the particle as a whole, may be taken into account in the electrokinetic model, if required by the experience. It is subsumed in the ratio R/δ, where R is the typical decay length for the polymer segment density distribution f(r) from the center of the particle to its outer periphery. As a general comment, the electrophoretic mobility is negative at pH ) 6 for every I, which is in agreement with the negative charge that stems from the carboxylate groups distributed along the polymer chains (see Figure 4). The mobility further decreases in magnitude with increasing ionic strength as a result of the reduction of the magnitude of the local electrostatic potentials (increase of the macromolecular charge screening). For sufficiently large ionic strengths (above 10 mM), µ asymptotically reaches a finite nonzero value (that can be observed with precision with a linear plot of µ versus ionic strength), which is the typical signature of soft particle behavior with a given degree of flow penetration.27 That limiting value is solely determined by the overall fixed charge density Γ° at the pH of interest, by the characteristic penetration length of the flow within the macro-
molecule (1/λ0), and by the size of the latter, according to the relationship19,28
µ j |If∞ )
eΓ° (1 - sech(λ0δ)) orkBTλ02
(12)
At the pH of the experiments (pH ) 6), Γ° practically equals Γ°max. Analysis of the electrohydrodynamic data is done according to two procedures that we successively detail below. (i) In a first set of calculations, the numerical evaluations of the electrophoretic mobilities are carried out taking into account the volumic charge density Γ°max and the size δ of the particles (see Tables 2 and 3, respectively) as estimated from independent analysis of the protolytic titration (section 4.2) and diffusion coefficient data (section 4.1) at large ionic strengths where particle swelling is absent (regime 1 in Figure 2). For that purpose, the unique unknown parameter 1/λ0 is adjusted by least-square methodology to retrieve the experimental data at large ionic strengths (eq 12). The collapsed state of the polyelectrolyte chains in this range of electrolyte concentrations further justifies the use of a smeared-out polymer segment density profile that corresponds to a step-function representation of the interface. This is done by setting R/δ ) 0, that is, considering an homogeneous distribution for the density of polymer segments within the particle (Figure 1). This choice is, however, not a matter of discussion, since it was previously demonstrated19 that electrophoretic mobility plateau values obtained at large ionic strengths (eq 12) remain independent of the spatial distribution of the polymer segment density within the particle if the latter contains a sufficiently large amount of water, as is generally the case for polyelectrolytes in good solvent. Any discrepancies between the experimental and theoretical data (as possibly appearing at lower ionic strength levels as a result of particle swelling and accompanying interfacial gradients for polymer segment density) are accounted for by the appropriate adjustment of the heterogeneity parameter R/δ. Theoretical results and experimental data are shown in Figure 5. The key parameters 1/λ0 and R obtained from the analysis are reported in Table 3 and Figure 5, respectively. For the sake of comparison, the predictions based on the approximate analytical expression derived by Hermans and Fujita50 (H-F) are given for CMD15% T500 19 those and CMD37% T500. As extensively discussed elsewhere, predictions are clearly inadequate for fitting the experimental data obtained for sufficiently low ionic strengths. This is so because the underlying theory does not take into account the polarization and relaxation effects of the double layer by the applied field. These effects become increasingly significant upon an increase of the local electrostatic potential, and the H-F equation is further strictly applicable for homogeneous soft particles. In the case of CMD37% T500, agreement is achieved between the experimental data and theoretical predictions over the whole range of ionic strengths by using 1/λ0 ) 3.4 nm and R/δ ) 0. Upon closer inspection, the mobility values seem to present a slight discontinuity at 0.1 mM, which corresponds exactly to the critical concentration that marks the transition from regime 2 to regime 3 (Figure 2A). The experimental points at 0.1 mM were recovered by setting R/δ ) 0.4 in the computation. An increasing diffuseness in the spatial repartition of the polymer segment density (increasing values of R/δ) results in a decrease of the mobility (in magnitude), as explained in ref 19. This feature is essentially of hydrodynamic origin: the more extended the chains, the larger the friction exerted by the polymer chains on the flow (50) Hermans, J. J.; Fujita, H. K. Ned. Akad. Wet., Proc. 1955, 58B, 182-187.
Properties of Functionalized CMD Macromolecules
Figure 5. Dependence of the (dimensionless) electrophoretic 37% mobility on NaNO3 ionic strength at pH ) 6 for (A) CMDT500 , (B) 15% 26% CMDT500, and (C) CMDT40 macromolecules viewed as spherical polyelectrolytic entities. The points correspond to the experimental data, and the dotted lines correspond to the numerical predictions obtained on the basis of the formalism detailed in section 3.2 with the hypothesis of heterogeneous polymer segment density distributions. The dimensionless parameter R/δ that reflects the extent of diffuseness of the soft interface is systematically indicated, and the degrees of hydrodynamic flow penetration 1/λ0 are collected in Table 3. The plain line corresponds to the predictions obtained with the approximate analytical expression by Hermans and Fujita,50 where the space charge density has been replaced by the quantity Γ°max evaluated for I f ∞.
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within the outer part of the particle, and thus the lower the magnitude of the mobility. For CMD15% T500, a similar situation is observed: the data are consistent, within experimental error, with theoretical predictions at R/δ ) 0 and 1/λ0 ) 3.7 nm for ionic strengths down to 0.1 mM. At and below this ionic strength, ∞ which again corresponds to the critical concentration c2T3 for 15% CMDT500 (Figure 2B), an important discontinuity is found and corresponding mobilities are fitted with an increase in the (heterogeneity) dimensionless parameter R/δ up to 0.8. The extent of macromolecular swelling while decreasing the electrolyte concentration from infinity to c∞2T3 is, to a certain degree of approximation, indicated by the dimensionless parameter (Dc∞.1 37% ∞ )/D ∞ - Dc∞)c2T3 c .1 that takes the values of 0.59 for CMDT500 and 15% 37% 0.79 for CMDT500. Despite the larger Γ°max for CMDT500 (and therefore the therein larger intramolecular repulsions taking place from c∞ . 1 to c∞ ) c∞2T3) as compared to that for CMD15% T500, the latter swells more significantly. This is due to the stronger stiffness (or decreasing elasticity, see eq 10) of the chains in CMD37% T500, which is reflected in a lower overall hydrodynamic permeability. This point is further discussed below. The res37% pective swelling extents of CMD15% T500 and CMDT500 are qualitatively in line with the size increase of the particle, which is proportional to R/δ (Figure 1). For the particular profile chosen for the analysis (eq 7), it is basically given by 1 + 2.3R/δ, with the position r ) δ + 2.3R corresponding to that where the volume fraction of polymer segment is ∼1% of that in the bulk (see Figure 1C). It is recalled here that any realistic radial profiles (of which eq 7 is one) which satisfy the conditions defined in ref 19 must predict the increase in size with swelling. The polysaccharide with the higher volumic charge density Γ°max, that is, CMD26% T40 , presents lower electrophoretic mobility 37% values as compared to CMD15% T500 and CMDT500. This a priori counterintuitive feature is explained by the more important curvature effects on the setting of the equilibrium potential (lower values of κδ)19 and more importantly by a lower hydrodynamic penetration length 1/λ0 ) 0.8 nm. Interestingly, when decreasing I starting from I ≈ 100 mM, the corresponding increase (in magnitude) of µ j is more pronounced than that obtained for the 37% high molar mass polyelectrolytes CMD15% T500 and CMDT500. This is in line with the increased electrostatic driving force for the migration of the CMD26% T40 particle, which is governed by Γ° max. In other words, screening of the ionic charges at a given salt level 37% 26% is more efficient for CMD15% T500 and CMDT500 than for CMDT40 , which is consistent with the conclusions derived from the analysis of the protolytic data. The electrophoretic mobilities are in line with predictions based on a strictly homogeneous spherical polyelectrolyte for I g 10 mM, which corresponds exactly to the range of electrolyte concentrations where the diffusion coefficient remains constant (regime 1, Figure 2C). For ionic strengths below c∞1T2 (0.1 e I e 10 mM), the modeling shows that CMD26% T40 exhibits a significant heterogeneity for the radial polymer chain distribution as reflected in the increasing values of R/δ (0 < R/δ < 0.4). This is consistent with the electrostatic swelling as observed in regime 2. The theoretical predictions reveal that, under the condition of homogeneous chain distribution R/δ ) 0, a mobility minimum is observed in the range 0.1 e I e 10 mM. The presence of that minimum is the direct consequence of electric double layer polarization/relaxation by the applied electric field, as typically encountered for sufficiently low ionic strengths and large space charge densities.19 According to expectation, that minimum is displaced toward lower ionic strengths upon an increase of R/δ. This arises because increasing R/δ leads to a stronger hydrodynamic braking force and thus less
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pronounced asymmetry in the ionic cloud around the particle with respect to the polar plane (r,θ ) π/2), with θ being the angle as depicted in Figure 1. The fact that mobility minima are not experimentally observed along the theoretical predictions is a good indicator of the occurrence of significant changes in the polymer chain distribution when varying the ionic strength. Those changes are accounted for by the electrokinetic modeling according to a soft diffuse representation of the interface. As for 37% CMD15% T500 and CMDT500, a significant decrease (in magnitude) of the mobility is obtained at 0.1 mM. Comparison with the results for the high molar mass polyelectrolytes allows us to infer that the c∞2T3 value for CMD26% T40 is approximately 0.1 mM. The corresponding mobility values are recovered from the model by setting R/δ values (0.4 < R/δ < 0.8) that are indicative of very loose polymer chains (Figure 1C). (ii) In the case of homogeneous swelling, which corresponds to R/δ ) 0 over the whole range of ionic strengths, the volumic charge density Γ° may change significantly because of the corresponding volume variations of the particle. Such dependence of Γ° on c∞ was disregarded in the preceding computations, since normalization of Γ° by the appropriate particular volume is not physically relevant when considering a priori a diffuse (heterogeneous) interfacial system for which conservation of the overall number of polymer segments is the key condition to respect (see eq 8). A careful look at the results reported in Figure 5A and B pertaining to the high molar mass polysaccharides reveals that a quantitative interpretation of the mobility data in regime 2 is possible if we set R/δ ) 0 (we exclude from the discussion here the discontinuities at c∞2T3), which is obviously inconsistent with the large variations in the diffusion coefficient and size as measured in regime 2. Therefore, we report in Figure 6 for the three macromolecules analyzed the computed electrophoretic mobilities taking into account the appropriate normalization of the volumic charge density according to
Γ°max|c∞ ) Γ°max|c∞f∞(δ|c∞f∞/δ|c∞)3
(13)
where Γ°max|c∞f∞ and δ|c∞f∞ are reported in Tables 2 and 3, respectively, and the dependence of δ|c∞ on c∞ is expressed by the power laws determined in Figure 3. As explained above, eq 13 tacitly assumes homogeneous swelling, that is, numerical simulations are now performed by setting R/δ ) 0 with the value of 1/λ0 as obtained from the high ionic strength regime. Unsurprisingly, the theoretical results show a decrease in the magnitude of the mobility as a result of decreasing Γ° or equivalently increasing δ|c∞ upon lowering the ionic strength. Since hydrodynamic permeability is known to be dependent on the swelling ratio, all experimental mobility data (regime 2) were recovered by adjustment (increase) of the flow penetration length 1/λ0. The results are displayed in Figure 7. It is stressed here that in the hypothesis of heterogeneous swelling as adopted in the calculations reported in Figure 5 (point (i) above), the relationship between swelling and the permeability properties is subsumed in the polymer segment density profile (eq 7) that governs not only the local distribution of fixed (immobile) charges within the particle but also that of the local friction coefficient (or equivalently local permeability). From Figure 7, it can be readily seen that the penetration length 1/λ0 is a linear function of size 37% for the CMD15% T500 and CMDT500 particles (panel A), which is line with empirical relationships that are valid for homogeneously swollen porous systems,51-53 that is, 1/λ0 ∼ δ. The proportionality (51) Carman, P. C. Trans. Inst. Chem. Eng. 1937, 15, 150-166. (52) Cohen-Stuart, M. A.; Waajen, F. H. W. H.; Cosgrove, T.; Vincent, B.; Crowley, T. L. Macromolecules 1984, 17, 1825-1830.
coefficient is basically a measure of the “tortuosity” or “compactness” of the particle as a whole (analogous to the Kozeny parameter for porous media52). On the contrary, the quantity 1/λ0 pertaining to the CMD26% T40 particles increases exponentially with δ|c∞. This indicates a complex swelling process where the uniform distribution of the polymer segment density as met at large ionic strengths is gradually disturbed upon decreasing c∞ (or increasing δ), which results in heterogeneous spatial repartition for the assimilated resistance centers (see the calculations discussed in (i) above and Figure 5). On the basis of Figures 5-7, the high molar mass polyelec37% trolyte particles, CMD15% T500 and CMDT500, likely undergo uniform (isotropic) swelling behavior when decreasing the ionic strength from c∞ ≈ c∞1T2 ) 5 mM down to c∞ ) c∞2T3 ≈ 0.1 mM. At the characteristic concentration c∞2T3 that marks the transition between diffusion regimes 2 and 3, heterogeneities in the spatial repartition of the polymer segment density are possible, as judged by the abrupt increase in the interfacial diffuseness parameter R/δ. For the smaller particles CMD26% T40 , the analysis of the electrohydrodynamic data reveals a continuous increase of R/δ ∞ upon lowering the electrolyte concentration from c∞ ≈ c1T2 ) ∞ 10 mM to c∞ ) c2T3 ≈ 0.1 mM. Interpretation of the data according to the assumption of homogeneous swelling leads to violating the thereby expected linear relationship between 1/λ0 and δ. For these reasons, it is believed that CMD26% T40 exhibits heterogeneous swelling, meaning that water uptake upon size increase is nonuniform throughout the particle. It is possible to derive an effective hydrodynamic permeability, defined as the square of the ratio between the typical hydrodynamic penetration length over the radius of the macromolecule (λ0-1/δ)2. This dimensionless quantity, called the Darcy number and denoted as Da, represents the typical cross-sectional area available for flow normalized by the overall cross section of the 15% particle. For CMD37% T500 and CMDT500, Da may easily be estimated from the swelling data (regime 2) of Figure 7 and we obtain Da ) 0.11 and 0.40, respectively. These values deserve some comments in relation with the use of the Stokes-Einstein (SE) relationship for converting the measured diffusion coefficient into hydrodynamic radii. In ref 54, Routh et al. correctly underlined the inappropriateness of the Stokes drag expression (fB ) -6πηδU B , with U B being the particle velocity) for very permeable particles. Indeed, for deriving the SE relationship, the Navier-Stokes equation is solved subject to a no-slip boundary condition at the particle surface, which is a condition that may be severe for soft particles. Consequently, Routh et al. computed numerically a correction factor, δactual/δSE, as a function of the Darcy number, which allows a correct estimation of the size of the particles as a function of their hydrodynamic permeability. They showed that δactual/δSE f 1 for Da , 1, whereas deviations are observed for Da . 1 (δactual/δSE > 1), which conforms to intuition. As pointed out by Routh et al., the evaluation of δactual for permeable particles may be done providing that δSE and the Darcy number Da are a priori known. Within the framework of our analysis, Da is precisely the parameter to be determined from adequate interpretation of the mobility data: increasing the size of the particle from δSE to δactual (because of the inadequacy of the SE equation54) leads to decreasing the volumic charge density and decreasing the theoretical mobility. Recovering the experimental data asks for increasing the penetration length, which in itself may impact on the Da. The newly computed Da (53) Scheidegger, A. The Physics of Flow Through Porous Media; University of Toronto Press: Toronto, 1960. (54) Routh, A. F.; Zimmerman, W. B. J. Colloid Interface Sci. 2003, 261, 547-551.
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Figure 7. Plot of the hydrodynamic penetration length versus particle size in the electrolyte concentration regime where swelling takes place (regime 2). The values 1/λ0 are determined from the fit of the experimental mobility values with a proper account of the scaling of the volume charge density with respect to the increased particular 15% 37% volume. (A) open circles, CMDT500 ; filled circles, CMDT500 . (B) 26% CMDT40 .
Figure 6. Dependence of the (dimensionless) electrophoretic 37% mobility on NaNO3 ionic strength at pH ) 6 for (A) CMDT500 , (B) 15% , and (C) CMD26% macromolecules viewed as spherical CMDT500 T40 polyelectrolytic entities. The points correspond to the experimental data. The dotted lines (a) represent the numerical predictions obtained on the basis of the formalism detailed in section 3.2 with R/δ ) 0, and Γ°max and 1/λ0 as indicated in Tables 2 and 3, respectively. The dotted lines (b) represent the numerical predictions obtained on the basis of the formalism detailed in section 3.2 with R/δ ) 0, variable Γ°max (scaled with the particular volume, eq 13), and 1/λ0 as indicated in Table 3.
must then be critically analyzed in relation to the abscissa of Figures 2 and 3 in ref 54 to check if it is in agreement with the ratio δactual/δSE. The above arguments highlight the necessity to iteratively and consistently determine δactual and Da from mobility and diffusion coefficient data collected for soft particles with a priori unknown permeability.18 We have followed this tedious method for a number of experimental points (µ,δSE) in regime 2 and noticed that the so-determined Da did not change much from the value obtained from the data of Figure 7. This feature may be qualitatively explained as follows. The mobility may be written in the general form
µ j)
eΓ° + g(κδ,y(r)) orkBTλ02
(14)
where the first term corresponds to the high ionic strength limit (eq 12) for λ0δ . 1 (which is verified for the particles investigated here) and g is a complex function that basically accounts for the
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impact of electrostatics on the electrophoretic mobility (see the analytical expressions by Ohshima in ref 27). When lowering the ionic strength (i.e., in the presence of swelling), the concomitant increase in size, the decrease in Γ°, and the increase of 1/λ0 favor the first term as compared to the second, which decreases as a result of the decrease in the local potentials y(r). The mobility then basically verifies
µ j∼
1 λo2δ3
(15)
The variation dµ j /dδ is then given by
dµ j /dδ ∼ -
3 ≡ -3Da (λoδ)2
(16)
Equation 16 expresses that any deviation of the theoretical mobility as a result of increasing δ from δSE (to which corresponds a certain fitting parameter 1/λ0 called 1/λ0,SE) to δactual may be accounted for by increasing 1/λ0 from 1/λ0,SE to 1/λ0,actual ) 1/λ0,SE(δactual/δSE) with a constant Da as a result (Da ) (λoδ)-2 SE ) (λoδ)-2 ). actual Coming back to the estimation of Da, we have (in regime 2, swollen particles) Da ) 0.11 and 0.40 for CMD37% T500 and CMD15% , respectively. In the range of electrolyte concentraT500 tions where swelling is absent (regime 1), the Darcy numbers are, as expected, significantly lower, namely Da ≈ 0.015 and 15% 0.020-0.026 for CMD37% T500 and CMDT500, respectively. For 26% CMDT40 , evaluation of the Da along the lines described above is relevant for the high concentration regime where data are consistent with an homogeneous distribution of the polymer segments density within the particle: we obtain Da ≈ 0.0160.017. In view of these results, the sequence of hydrodynamic permeability at large ionic strengths is given by CMD26% T40 ≈ 15% < CMD . This nominal permeability sequence is CMD37% T500 T500 determinant in governing the extent of swelling at lower ionic strengths, since it indicates the extent of hydrodynamic compactness of the macromolecule and therewith the coupled stiffness (elasticity) of the chains. Let us first compare the CMDs of quasi-identical molar mass but significantly different substitution degrees. We notice then that the macromolecule CMD15% T500 with the lower number of moieties introduced during synthesis is more permeable to hydrodynamic flow. This indicates the interplay between sterical hindrance and hydrodynamic paths for the flow within the macromolecules: the larger the number of these moieties for a given macromolecular size and molar mass, the lower the spatial entropy of the chains and the larger the resistance to flow penetration. On the contrary, the effective permeability of CMD26% T40 is of the same order of magnitude as despite its lower number of functionalized that for CMD37% T500 groups along the chains. On the basis of the argumentation laid out before, one would expect a lower permeability for CMD37% T500 than that for CMD26% T40 . This may be explained by the significant impact of the excluded volume interactions (repulsive intramolecular forces) on the hydrodynamic permeability. Indeed, larger molar masses (as for CMD37% T500) will lead to stronger excluded volume interactions and thus larger and/or more numerous flow penetration pores/paths that would (if solely considered) result in increased effective permeabilities. In terms of the DebyeBueche theory,30 the electrohydrodynamic model19 applied here is based on increased excluded volume interactions leading to larger sizes of the polymer resistance centers and thus to an increase of the flow penetration length 1/λ0, as predicted by the
Figure 8. Schematic representation of the swelling behavior of (A) 37% 15% /CMDT500 and (B) CMD26% CMDT500 T40 , as inferred from the analyses of the electrohydrodynamic and diffusion coefficient data.
Brinkman equation.55 In the case of CMD26% T40 , that swellingdriven process is less important in magnitude than that for CMD37% T500. Hence, the lower number of moieties and lower excluded volume interactions for CMD26% T40 would lead to a similar effective hydrodynamic permeability as that for CMD37% T500, where sterical hindrance connected to the number of moieties is larger (unfavorable for flow penetration) but excluded volume interactions are larger (favorable for flow penetration). This probably results in chains that are strongly entangled for 37% CMD26% T40 , whereas those for CMDT500 remain in a looser configuration. If we now compare the Da for CMD26% T40 and CMD15% T500, the higher effective permeability of the latter is in all likelihood connected not only to the lower DS but also to the therein larger excluded volume interactions (i.e., larger molar mass). The above considerations on the respective hydrodynamic permeabilities for the three macromolecules analyzed and on the related conformational states of their constituting chains show that the electrostatic swelling processes (upon decrease of the 37% 15% ionic strength) for CMD26% T40 and for CMDT500/CMDT500 will affect in different ways the polymer segment density distribution. Indeed, as inferred from electrokinetic modeling, the 15% swelling of CMD37% T500/CMDT500 is homogeneous, whereas that of 26% CMDT40 is nonuniform. This feature suggests that CMD26% T40 resembles a core-shell particle with a relatively hard core (in 15% the hydrodynamic sense of the term), while CMD37% T500/CMDT500 present chains that are in a loose state and this chain status does not change much from the center to the periphery of the particle except perhaps at c∞2T3. As a result, electrostatic swelling takes place in a uniform way throughout the particle. Those arguments, sketched in Figure 8, are further qualitatively in line with the effective hydrodynamic permeability sequence discussed above. As a final remark, we note that the homogeneous swelling for (55) Brinkman, H. C. Research 1949, 2, 190.
Properties of Functionalized CMD Macromolecules 15% the large macromolecules (κδ . 1, as for CMD37% T500/CMDT500) coincides with the fact that the potential distribution therein may legitimately be considered as homogeneous (y(0 < r < δ) ) yD, with yD being the dimensionless Donnan potential), whereas heterogeneous swelling occurs in the situations where strong potential gradients exist within the particle (κδ ∼ 1, as for CMD26% T40 ).
5. Conclusions The relationship between electrostatic/hydrodynamic properties of synthesized carboxymethyldextran (CMD) macromolecules, their propensity to swell upon decrease of the ionic strengths, and the resulting impact on the spatial distribution of the polymer chains are thoroughly examined by detailed analyses of protolytic, electrokinetic, and diffusion coefficient data. It is found that CMDs of higher molecular weights undergo electrostatic swelling which does not lead to a significant anisotropy in the spatial distribution of the chains. This is due to a loose state of those chains and an accompanying large overall hydrodynamic permeability. On the other hand, smaller and more dense CMD entities present a lower hydrodynamic permeability and thus a stronger heterogeneous chain distribution, as typically obtained for core-shell systems. The results detailed in this work for CMD macromolecules reveal a larger flexibility of the chains as compared to that for succinoglycan16 and a larger overall
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hydrodynamic permeability, thus resulting in considerable conformational changes (chain stretching and electrostatic swelling) upon decrease of the ionic strength. Besides, unlike the case for succinoglycan, no aggregation processes come into play upon reducing intermolecular repulsion, that is, decreasing pH or increasing ionic strength. The study underlines the complex interplay between the determined electrostatic and hydrodynamic parameters, the structural variation of the macromolecules when varying ionic strength, and the intrinsic chemical properties of the macromolecule (degree of substitution, molar mass). In view of the preponderant presence of carboxylate groups in natural organic matter, the functionalized polysaccharides investigated here are, because of the diversity of their space charge densities and molecular weights, appropriate model systems for examining the interactions between metallic ions and soft permeable particles. This will be the purpose of a forthcoming analysis where special attention will be paid to the relation between particular conformational state/electrohydrodynamic features and underlying metal-complexation properties. Acknowledgment. The authors wish to thank the national program ToxNuc-E for financial support in the form of a postdoctoral grant attributed to Dr. Elise Rotureau. LA700427P
