Humic Substances Are Soft and Permeable: Evidence from Their

Their motion in an electric field is evaluated by a rigorous numerical evaluation of the governing electrokinetic equations for soft particles. The hy...
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Environ. Sci. Technol. 2005, 39, 6435-6445

Humic Substances Are Soft and Permeable: Evidence from Their Electrophoretic Mobilities† J EÄ R O ˆ M E F . L . D U V A L , ‡,§ K E V I N J . W I L K I N S O N , * ,‡ HERMAN P. VAN LEEUWEN,§ AND JACQUES BUFFLE‡ CABE (Analytical and Biophysical Environmental Chemistry), University of Geneva, Sciences II, 30 Quai E. Ansermet, Geneva 4, CH-1211 Switzerland, and Department of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands

Due to the complexity of the humic substances (HS), mathematical models have often been employed to understand their roles in the environment. Since no consensus exists with respect to the structure and conformation of the HS, models have alternatively given them properties corresponding to impermeable hard spheres or fully permeable polyelectrolytes. In this study, the hydrodynamic permeability of standard HS (Suwannee River fulvic, humic, and peat humic acids) are evaluated as a function of pH and ionic strength. A detailed theoretical model is used to determine the softness parameter (λo), which characterizes the degree of flow penetration into the HS on the basis of measured values of electrophoretic mobilities, diffusion coefficients, and electric charge densities. Their motion in an electric field is evaluated by a rigorous numerical evaluation of the governing electrokinetic equations for soft particles. The hydrodynamic impact of the polyelectrolyte chains is accounted for by a distribution of Stokes resistance centers and partial dissociation of the hydrodynamically immobile ionogenic groups distributed throughout the polyelectrolyte. The results demonstrate that the studied HS are small (radius ca. 1 nm), highly charged (500-650 C g-1 when all sites are dissociated), and very permeable (typical flow penetration length of 25-50% of the radius, depending on pH). The HS also coagulate slightly when lowering the pH of the solution. Modeling of the HS as hard spheres with a charge and slip plane located at the surface is thus physically inappropriate, as are a number of analytical theories for soft particles that hold for low to moderate electrostatic potentials and large colloids. The shortcomings of these simpler approaches, when interpreting the electrophoretic mobilities of HS, are highlighted by comparison with rigorous theoretical predictions.

Introduction It has long been recognized that humic substances (HS) influence the transport and binding of trace elements in †

This paper is part of the Charles O’Melia tribute issue. * Corresponding author phone: +41 22 379 6051; fax: +41 22 379 6069; e-mail: [email protected]. ‡ University of Geneva. § Wageningen University. 10.1021/es050082x CCC: $30.25 Published on Web 05/07/2005

 2005 American Chemical Society

aquatic environments (1). To gain a better understanding of the complexation properties of HS, a great deal of attention has been devoted to determining their physicochemical properties, including size, charge, and structure especially under conditions that are relevant for environmental systems (1-7). Nonetheless, no consensus has been reached in the literature with respect to the structure and conformation of HS (8, 9). For complexation studies, HS have most often been treated as a collection of homologous, small ligands with a few complexing sites per molecule (1, 10). In reality, HS are composed of a large number of similar but not identical molecules (typical example in Figure 1A). Their unique properties are due in large part to the fact that the HS are polyfunctional (11) and polyelectrolytic (12, 13) and able to form molecular aggregates (14). Furthermore, the size of individual HS places them at the limit of the domain of (macro)molecules and colloids. For these reasons, they have been modeled as hard spheres or cylinders (3, 15-17), permeable Donnan gel phases (13, 18), and branched (19) or linear (20) polyelectrolytes. More recently, due to novel applications of nuclear magnetic resonance spectroscopy (NMR), fluorescence correlation spectroscopy (FCS), atomic force microscopy (AFM), pyrolysis mass spectrometry, and numerical modeling, the representation of a HS as predominantly consisting of small and relatively simple near spherical molecules (1-2 nm) able to form reversible aggregates has been more or less accepted (5, 6, 9, 21, 22). Nonetheless, two radically different means of modeling the molecules still persist: one in which HS are viewed as hydrodynamically impermeable hard spheres (referred to as (i) in this paper), and another where they are considered as spherical fully permeable polyelectrolytes (ii). Recent studies by AFM (21) appear to support the correctness of a third intermediate approach in which HS are considered as semipermeable spherical colloids (iii) (18, 23). To our knowledge, no experimental studies supported by solid quantitative interpretation are available to unambiguously validate one of the above molecular representations (i), (ii), or (iii) for the HS. For example, some theories for ion binding to HS rely on an arbitrary choice between a permeable and a hard sphere (24). In some cases, the relevance of one model or another has been justified on the basis of experimental curve-fitting exercises (e.g., titration curves) carried out after the adjustment of numerous variables, some of which have no clear physicochemical basis, in particular when the physical and chemical heterogeneity of the HS is considered. A serious limitation of our understanding of the HS is that no detailed analysis of experimental data has been performed with techniques that are suitable for probing directly and accurately the permeability of the HS. Very recently, electrophoretic mobilities and diffusion coefficients of HS were systematically determined over a wide range of pH and ionic strengths using capillary electrophoresis (CE) and FCS, respectively (25). Generally speaking, the electrophoretic mobility (µ) of a colloidal particle moving in a liquid under the action of an applied dc electric field is a key quantity that provides information on the electrical and hydrodynamic properties of the particle. Depending on the size, charge, and permeability of the particle, the mathematical treatment of the fundamental electrokinetic equations is more or less sophisticated. For the particular case of small, highly charged and likely permeable HS, the use of simple analytical expressions that have been derived previously (hard (26-30) or soft (31) spheres) is inappropriate, and a numerical treatment of the fundamental equations is required. VOL. 39, NO. 17, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 1. (A) Statistical molecular representation of a Suwannee River fulvic acid (2). (B) Schematic representation of a soft particle, composed of a hard-core and a permeable charged polyelectrolyte layer, moving with a velocity, B U, in an electrolyte subject to a dc electric field, B E. The spherical coordinate system and corresponding unit vectors are also given. For humic substances, a ) 0, and the model corresponds to that of a spherical polyelectrolyte. Here, rigorous analysis of the electrophoretic mobilities reported in ref 25 is carried out on the basis of an electrokinetic model (32), which expands the formalism originally developed by Ohshima (31). The novelty of the approach is that it unites the theory for electrophoresis of hard spheres (33) with that for spherical polyelectrolytes (34-37) by taking into account the possibility of an inhomogeneous distribution of the hydrodynamically fixed and partially dissociated ionogenic groups within the polyelectrolyte layer. A key parameter in the model is the softness parameter (λo), which characterizes the degree of flow penetration (i.e., permeability to liquid flow) of the particle. In the limiting case where λo f ∞, the particle is hard whereas for λo f 0 the particle is fully permeable. Intermediate values of λo indicate partially permeable particles with 1/λo corresponding to the characteristic hydrodynamic penetration length. Using diffusion coefficients and titration data determined for HS at various pH and ionic strengths (5, 6, 25), a quantitative interpretation of the electrophoretic mobilities of HS can be successfully performed by numerical evaluation of the relevant electrostatic and transport equations with λo as the only unknown adjustable variable. The results clearly show that HS are fairly permeable colloids and that the molecular representation (iii) is undoubtedly the closest to reality. The respective dependencies of λo on pH and ionic strength were analyzed for three model humic substances in connection with their typical structural features. 6436

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Theory In this section, we briefly present the basic physical ideas and key equations underlying the electrokinetic modeling that is used to quantitatively interpret the electrophoretic response of the HS as a function of the solution composition. It is beyond the scope of the current paper to present the full details of the theory, the mathematical derivations, and the numerical method that were employed to solve the fundamental transport and electrostatic equations. Instead, this is the goal of a parallel paper (32). In the following, the general case of a diffuse soft particle composed of a hard (impenetrable) core and a permeable diffuse polyelectrolyte layer is first considered. In interfacial modeling, the term “diffuse” indicates that the possibility of inhomogeneous distribution of charged sites within the polyelectrolyte layer is explicitly taken into account (38). The model is then adapted to the specific case of the HS based upon key experimental data on their mobilities, diffusion coefficients, and protolytic properties over a wide range of pH and ionic strengths. Electrokinetic Model for a Diffuse Soft Particle. A spherical, core/polymeric shell particle that moves with a velocity U B in an unbounded electrolyte, in which a uniform dc electric field B E is applied, is first considered (Figure 1B). The origin of the spherical coordinate system (r, θ, φ) is placed at the center of the particle, and the polar axis (θ ) 0) is set parallel to B E. The radius of the particle hard core is denoted

by a, and δ represents the thickness of the polymeric shell that is, to some extent, permeable to ions and solvent molecules. In the limit a f 0, the particle core vanishes and the model reduces to that of a spherical polyelectrolyte, while for δ f 0, the other limiting case of a hard sphere is attained. The electrolyte is composed of N types of ionic mobile species with valency zi, a bulk concentration n∞i , and limiting ionic mobility λoi (i ) 1, ..., N). Monovalent fixed charges are distributed throughout the shell with a volume density that is denoted by Ffix(r), which is assumed to be dependent only on r (spherical symmetrical function). We adopt the model of Debye and Bueche (39), which considers the charged polymer segments as resistance centers that exert frictional forces on the liquid flowing through the gel-like layer. For a polyelectrolyte layer with a sufficiently high water content, the Brinkman equation (40) states that the friction coefficient (k) (acting against hydrodynamic flow) is a function of r (38):

k(r)/ko ) Ffix(r)/Fo ) f(r)

(1)

where f represents the radial distribution of the fixed charges within the polyelectrolyte layer, ko is the friction coefficient, and Fo is the volume charge density of that layer for a homogeneous distribution of sites (f(r) ) 1). Due to the spherical symmetry of the problem, the function f must satisfy the condition:

|

df(r) dr

r)a

)0

(2)

The softness parameter λo is defined by (31):

λo ) (ko/η)1/2

(3)

where η is the dynamic viscosity of water. λ-1 o has a length dimension and essentially characterizes the degree of flow penetration into the soft layer. On the basis of symmetry considerations (41), the liquid velocity b u(r) at the position r relative to the particle (u b(r) f -U B as r f ∞) may be written in the vectorial form:

(

)

1 d(rh(r)) 2 b u(r) ) - h(r)E cos θ, E sin θ, 0 r r dr

(4)

where h is a function of r satisfying the boundary condition (31):

µ h(r f ∞) f r 2

(5)

and µ is the electrophoretic mobility defined by µ ) U/E. The theoretical calculation of µ requires consistent numerical evaluation of (a) the velocity profile b u(r) or equivalently h(r), as defined by eq 4 (hydrodynamics); (b) the distribution of the local dimensionless equilibrium potential, denoted by y(r) (electrostatics); and (c) the radial function φi)1,...,N(r), which represents the local variation of the electrochemical potential of the ion i due to polarization of the double layer by the externally applied field B E. Following the strategy adopted by Ohshima (31), it is possible to derive the function h(r) from the Navier-Stokes equation written in the form:

Lr{Lrh(r)} - λ2of(r)Lrh(r) - λ2o -

e 1 dy(r) η r dr

[

df(r) h(r) dr

N

∑n z

∞ 2 i i

r

+

]

dh(r) dr

)

exp{-ziy(r)}φi(r) (6)

i)1

where e is the elementary charge and Lr is the differential operator Lr ≡ (d2/dr2) + (2/r)(d/dr) - (2/r2). The dimension-

less potential y(r) is defined as eψ(r)/kBT, where kB is the Boltzmann constant, T is the absolute temperature, and ψ(r) is the local electrostatic potential. The latter is given by the Poisson-Boltzmann equation:

d2y(r) dr2

+

2 dy(r)

+

r dr κ2 N

∑

zi2n∞i

{∑ N

zin∞i exp{-ziy(r)} +

i)1

}

Fo f(r) ) 0 (7) F

i)1

where κ is the reciprocal Debye length defined by κ ) N {∑i)1 F2zi2n∞i /(RT)}1/2 and () or) (i.e., the dielectric permittivity of water). The functions φi)1,...,N(r) are defined by the differential equations (31):

i ) 1, ..., N: Lrφi(r) )

{

}

dy(r) dφi(r) 2fih(r) zi dr dr e r

(8)

with fi being the drag coefficient of ion i given by fi ) zi2eF/ λoi (F is the Faraday constant). The boundary conditions associated with eqs 6-8 are given in the Supporting Information where the numerical method employed to solve the nonlinear system of coupled equations is briefly outlined. The corresponding Fortran program is available on request. The main assumptions of the above formalism are as follows: (a) the Reynolds numbers of the liquid flow inside and outside the polyelectrolyte layer are small so that the liquid may be regarded as incompressible; (b) the electrophoretic velocity, U B , is proportional to the applied field, B E, which is correct for the field E used here (in electrophoresis of the first kind, as considered here, terms in E of order higher than 1 may be neglected); and (c) the relative permittivities, r, inside and outside the polyelectrolyte layer are the same, which is reasonable for polymeric shells with sufficiently high water content. These assumptions are in line with the conditions under which the electrophoretic experiments were carried out (25) and the nature of the HS that were investigated. On the basis of the preceding theory, the hydrodynamic permeabilities can be evaluated from the experimentally accessible electrophoretic mobilities if the parameters relative to the particle size (a, δ), the charge distribution (Fo and f(r)), and the electrolyte (zi, n∞i , λoi ) are known. Application to the Electrophoresis of HS. In this section, we derive and justify the choice of the size and charge parameters for three HS (International Humic Substances Society (IHSS), standard Suwannee River fulvic (FA), humic (HA) and peat humic (PHA) acids) studied by CE and FCS (25) over a wide range of pH and ionic strengths. It must be stressed that all HS are both polydisperse and chemically heterogeneous (i.e., HS solutions are always mixtures of a large number of homologous molecules with similar but not identical properties). Typical Sizes of HS (a and δ). HS are assumed to be roughly spherical, polyelectrolytic entities (5-7) (Figure 1A), without a hard core (a ) 0) that would be stricto sensu inaccessible for any solvent molecules or ions. Such a choice is supported by the small size of the HS (1-2 nm). HS are then viewed as spherical polyelectrolytes with spatially distributed sites that may dissociate, depending on the solution composition. The effective hydrodynamic radius (δH) of a sphero-colloid can be estimated from its diffusion coefficient (D) (5, 6, 25) using the Stokes-Einstein equation:

δH ) kBT/6πηD

(9)

This relation will be used below, but it should be emphasized VOL. 39, NO. 17, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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TABLE 1. Parameters Used for Analysis of the Titration Data (42) on the Basis of the Henderson-Hasselbalch Equationa

p Ki

ci (in mequiv/g of C) Foi/F (in mol m-3)b

ni

a

ci

2

[A-] ∝ ∑

i)1

FA HA PHA FA HA PHA FA HA PHA FA HA PHA

1 + 10

pKi - pH

carboxylic functional groups (i ) 1)

phenolic functional groups (i ) 2)

3.79 4.42 4.21 10.91 9.36 8.47 -3380 -2946 -2902 3.22 3.83 3.20

9.51 9.66 9.86 1.32 1.92 0.83 -409 -596 -284 1 1.10 1

, with [A-] as the concentration of or-

n ganic charge. b Values given at pH 10.7 (δH ) 0.88, 0.91, and 0.94 nm for the FA, HA, and PHA, respectively, see Figure 2).

that it has a number of limitations. Since eq 9 only strictly holds for hard spheres, any δH computed for soft particles is nothing more than the radius of an equivalent hard sphere with the diffusion coefficient, D. Proper determination of δH for permeable particles (like HS) would require more involved models where the relationship between δH and the liquid velocity profile within the particle is clearly established. In addition, due to the inherent polydispersity of HS and the fact that neither monomeric humics nor their aggregates are strictly spherical, any δH value obtained from eq 9 should be considered as an average value only. Since the reliability due to the above factors is similar to the experimental error on the measurement of D, eq 9 will be employed semiquantitatively on an orientative level with the above restrictions in mind. Space Charge Density (Fo) and Spatial Distribution ( f (r)) of the Sites. Titration data for the FA, HA, and PHA could be accurately fitted using a Henderson-Hasselbalch model based on two binding sites (carboxylic and phenolic) of given concentrations (ci) and average proton binding constants (pK h i) (Table 1) (42). A statistical parameter, ni (g1), was included to account for distributions around the average pKi values (Table 1). From the average radii (δH), the average space charge densities (per unit HS) for each type of site (Foi) were estimated using molar masses of 991, 1136, and 1264 g mol-1 for the FA, HA, and PHA, respectively (25). The relatively high values of Foi (Table 1) are in good agreement with values reported previously (3, 4, 23) and justify the use of the Poisson-Boltzmann equation (eq 7) that is based on the assumption of a smeared-out electrostatic potential (i.e., individual double-layer effects are insignificant). By estimating that an average humic molecule carries about 5-10 elementary charges (depending on the pH and on the nature of HS investigated), it is possible to assume that for the electrolyte concentrations considered here (0.5-150 mM), κl , 1, since the average separation distance between two neighboring sites, l, is necessarily smaller than the hydrodynamic diameters of the molecules (ca. 1.6-2.0 nm). At the higher electrolyte concentrations examined here, κl ≈ 0.2 and the continuum (Poisson-Boltzmann) model is likely less appropriate. Fortunately, at these ionic strengths, the charges are well screened so that electrostatic effects play a much less important role in the mobility determinations (i.e., according to Ohshima’s theory, mobilities tend toward a constant value for a flat geometry; 31). Due to partial dissociation of the proton binding sites, the charge depends 6438

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on pH such that, following the Henderson-Hasselbalch equation (see Table 1), the total space charge density (Fo) may be written:

Fo(r, pH) )

F o1 1 + 10(pK1-pH)/n1 exp{-y(r)/n1} Fo2

+

1 + 10(pK2-pH)/n2 exp{-y(r)/n2}

(10)

where the Boltzmann factor, exp{-y(r)}, accounts for the local concentration of protons within the polyelectrolytic entity and the indices 1 and 2 refer to the carboxylic and phenolic sites, respectively. The charge Fo(r) and potential y(r) distributions are computed using the Poisson-Boltzmann equation (eq 7) assuming the site distribution (f) inside the polyelectrolyte is radial. As shown in Figure 1A, this latter assumption is clearly not valid for any individual HS molecule. However, it is likely that the HS mixture behaves as an “average” polyelectrolyte with an internal homogeneous site distribution (see next section). Aggregation and Swelling. In most theories on the electrokinetics of soft materials (see ref 31 and references therein), with the exception of the recent work by Hill et al. (43), polymer segments and their charged sites are considered as uniformly distributed within the polyelectrolyte layer. In that case, f(r) is a step-function:

f(r) ) R

(11)

where R ) 1 for a e r e δ and R ) 0 for r > δ. Recently, Duval and co-workers (38, 44) demonstrated a substantial influence of the spatial distribution of ionogenic sites on the electrokinetics of a gel-like polyelectrolyte layer/water interface. Based on a comparison between theory and experimental data, it was established (45) that the swelling/shrinking of a polyacrylamide gel was strongly coupled to the spatial gradient of site concentrations at the interface. For a polyelectrolyte, electrostatic arguments would suggest that the size should decrease with decreasing pH due to a reduction of intramolecular repulsion. For the HS, an increased size is observed, especially below the carboxylic pKa. These results strongly suggest that aggregation is occurring, perhaps simultaneously, due to a decreased intermolecular repulsion (5, 6, 25). Furthermore, the size of the HS only weakly depends on the electrolyte concentration (46). These observations, in addition to the known branched structure of the HS, are consistent with the interpretation that aggregation predominates over electrostatic swelling for the HS. Considering the high charge densities of the HS, the step-function (smeared-out site distribution) mentioned above (eq 11) is used for the calculations. The results support this choice, a posteriori. The electrophoretic mobility of the HS corresponds to a distribution of the electrophoretic mobilities of the individual HS molecules and thus takes polydispersity into account. The average value can be considered as the equivalent of the mobility of a statistically homogeneous mixture of HS molecules, with a smeared-out ionogenic site distribution, as expressed in eq 11. This matter does deserve further attention, particularly with respect to the way the mobility distribution is related to the molecular structures of the individual HS; however, such a detailed analysis is beyond the scope of the current paper. Based upon the above considerations, the only unknown parameter required for a complete characterization of the HS is the (hydrodynamic) permeability (λo). In this paper, this parameter was determined by adjusting the reported electrophoretic mobilities as a function of pH and ionic strength (25).

Quantitative Analysis of the Electrophoretic Mobilities of HS Electrophoretic Mobilities as a Function of the Ionic Strength. Permeabilities. The hydrodynamic radii of FA, HA and PHA, calculated from FCS data (25) (eq 9), remained constant within the ionic strength range (10-150 mM) where µ was measured. Values for µ were obtained at pH 10.7 for different ionic strengths (I) in Na2CO3. Bulk concentrations (n∞i ) of the pertinent ions Na + (i ) 1), HCO3- (i ) 2), and CO32- (i ) 3) are given by

n∞i )

χiI 3 + 10pKa2-pH

,

(12)

with χ1 ) 2 + 10pKa2-pH, χ2 ) 10pKa2-pH, χ3 ) 1, and pKa2 ) 10.3. Limiting ionic mobilities (λoi ) were obtained from the literature. The experimental data together with the theoretical predictions are given in Figure 2 for the FA (panel A), HA (panel B), and PHA (panel C). Mobilities are given in the dimensionless form:

µ j)µ

eη kBT

(13)

The experimentally determined values of δH and Fo1 + Fo2 that were used in the calculations are also mentioned in the captions. Mobility values are in excellent agreement with theory for a unique value of λo over the entire range of I, consistent with the observation that the structure of the HS remains nearly constant for ionic strengths between 5 and 150 mM. Furthermore, λo is found to be comparable for all three examined HS. Based on the diffusion coefficient data obtained at variable ionic strengths and pH 10.7 (25), a value -1 of λ-1 o ≈ 0.32 nm was found, similar to the value λo ≈ 0.27 nm found from the diffusion coefficient data obtained when pH was varied (I ) 5 mM). In light of the subsequent analysis of µ as a function of pH (see corresponding section below), the value of λ-1 o ≈ 0.27 nm is certainly the most reliable. As compared to the values of hydrodynamic radii (δH), the value obtained for λ-1 o suggests a fairly high degree of permeability for each of the HS investigated (λ-1 o /δH ≈ 0.31 for the FA and HA, and λ-1 o /δH ≈ 0.29 for the PHA). The slight difference between the FA/HA and the PHA might be related to the more ramified and more hydrophobic structure of the PHA as compared to the other HS, such that the underlying resistance to flow penetration is larger. Quantitative examination of the mobility data unambiguously confirms that the electrokinetic response of the HS with varying I was solely determined by a charge screening, excluding any structural changes. The characteristic feature of a soft particle is that µ asymptotically reaches a nonzero constant value at high ionic strength. This constant depends on the (total) charge density due to immobile groups (noted as Ftotal o ) and the permeability (λ-1 o ). It is simply given by (31)

µf

Ftotal o ηλ2o

as I f ∞

(14)

Substituting Ftotal ) Fo1 + Fo2, the asymptotic values of µ j are o -1.50, -1.46, and -1.26 for the FA, HA, and PHA, respectively. Within the range of ionic strengths that were considered, these constant values were not reached. The theory previously outlined predicts that the asymptotic mobilities would be reached for ionic strengths of several moles per liter, concentrations for which it is practically difficult to obtain accurate mobility values. At constant ionic strength, the mobilities for the different HS of interest show a negative

FIGURE 2. Reduced electrophoretic mobilities (eq 13) as a function of total Na2CO3 ionic strength at pH 10.7 for the FA (panel A), HA (panel B), and PHA (panel C). Closed circles are the experimentally measured mobilities while the open circles/squares are the theoretical predictions (with the plain/dashed curves as guidelines for the eye). Two sets of model parameters, obtained using D (I, pH 10.7) (open circles) and D (pH, I ) 5 mM) (open squares) data, were used for calculations. In panel A, curve a: δH ) 0.88 nm, Ftotal ) o -623.4 C g-1, λ-1 j f -1.50 for I f ∞); curve b: δH o ) 0.27 nm (µ ) 0.95 nm, Ftotal ) -625.3 C g-1, λ-1 j f -1.65 for I o o ) 0.32 nm (µ f ∞). The curves c and d in panel A correspond to mobilities computed with λ-1 o ) 0.27 nm (10%. In panel B, curve (a): δH ) 0.91 nm, Ftotal ) -568.1 C g-1, λ-1 j f -1.46 for I f o o ) 0.28 nm (µ ∞); curve b: δH ) 0.97 nm, Ftotal ) -572.3 C g-1, λ-1 j o o ) 0.32 nm (µ f -1.62 for I f ∞). In panel C, curve a: δH ) 0.94 nm, Ftotal ) o -504.7 C g-1, λ-1 j f -1.26 for I f ∞); curve b: δH o ) 0.27 nm (µ total -1 ) 1.01 nm, Fo ) -503.9 C g-1, λo ) 0.32 nm (µ j f -1.42 for I f ∞). shift from PHA to HA to FA. This shift is consistent with their as measured by titration. respective Ftotal o VOL. 39, NO. 17, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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{∑ N

i)1

FIGURE 3. (A) Dimensionless equilibrium potential distributions for the FA at various electrolyte concentrations (indicated) and pH 10.7 (δH ) 0.88 nm, Ftotal ) -623.4 C g-1). (B) Dimensionless Donnan o potential (yD), potential at the center of the HS (yc), and potential at the outer edge of the HS (yo) as a function of ionic strength (in Na2CO3) for the FA (open circles), HA (plain circles), and PHA (open squares) at pH 10.7. To illustrate the sensitivity of the numerical outcome with respect to the value λ-1 ≈ 0.27 nm determined above, o theoretical curves were calculated for λo values corresponding to 0.27 nm ( 10%, while maintaining constant the values of all other parameters (curves a and b, Figure 2A). Similar results were obtained for the HA and PHA (not shown). Relatively small differences in λo resulted in changes in µ that were significant as compared to the typical experimental error. In other words, reconstruction of the µ-I curves was strongly dependent on λo, suggesting that the values obtained for λo are likely accurate to a precision of about 0.01 nm. The typical hydrodynamic penetration length found for the humic particles is on the order of one-third of their hydrodynamic radius. This rather large permeability suggests that a large proportion of water molecules inside the humic particles have bulk properties and that the use of bulk permittivities in the modeling aspects of this paper is likely justified. Spatial Distribution of the Electrostatic Potential. Figure 3A illustrates the typical equilibrium potential distributions calculated for the FA as a function of ionic strength at pH 10.7 (similar results are obtained for the HA and PHA, not shown). For the sake of illustration, the corresponding Donnan potentials (yD), the potentials at the outer edge of the HS (noted yo ≡ y(r ) δ)), and the potentials at the center of the HS (noted yc ≡ y(r ) 0)) are reported in Figure 3B. The Donnan potential corresponds to the potential difference between the medium phase and the bulk phase HS. It is determined by the charge balance between mobile ions and fixed charges and is given by the transcendental equation: 6440

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zin∞i exp{-ziyD} +

1

F o1

+ F 1 + 10(pK1-pH)/n1 exp(- yD/n ) 1 Fo2 ) 0 (15) 1 + 10(pK2-pH)/n2 exp(- yD/n2)

}

For small soft colloids such as the HS, yD has no real physical meaning since it is effectively only reached for polyelectrolyte particles for which the size largely exceeds κ-1, which is not the case for the HS studied here (κδH ≈ 0.05-1.2 for I ) 0.5-150 mM). As expected, the screening of charges with increasing I resulted in a reduction of the local potentials, y(r). Even at relatively large I (≈ 0.1 M), the potentials were still significant (yc ≈ - 3.6) and thus incompatible with the use of any analytical formula expressing mobilities based on the DebyeHu ¨ ckel approximation within the formalism (i), (ii), or (iii), as mentioned in the Introduction. From Figure 3, it is also clear that the electric fields are the largest inside the polyion. On the basis of electrostatic considerations only, the picture of a hard sphere with all the charges located at its “surface” (dashed line in Figure 3A) provides an extremely poor description of the real potential distribution and the ensuing electrokinetic behavior of the HS. The potentials yD, yo, and yc are largest for the FA, corresponding to the most charged HS investigated. Comparison between the Electrophoretic Mobilities Associated with Models (i-iii). The mobility data of the PHA modeled as a semipermeable particle (iii) were compared with those derived from the model of (i) a hard-sphere with a slip plane and charge located at r ) δ and (ii) a fully permeable polyelectrolyte (Figure 4A). Within the framework of (i), the space charge density Ftotal would be converted into o a surface charge density (σtotal o ) and the electrophoretic mobility of the hard sphere would then be given by the simple analytical expression (30):

2 µ j ) ζf(κδ) 3

(16)

where f(κδ) varies from 1 to 1.5 as κδ changes from 0 to ∞. f(κδ) was estimated from an approximate expression given by Ohshima in ref 47. ζ represents the (dimensionless) electrokinetic potential and is given by numerical evaluation of the following transcendental equation valid for a symmetrical electrolyte (of valency z) (48, 49):

κδ 4 2 sinh (zζ/2) + tanh (zζ/4) - σtotal o /(orkBT/eδ) ) 0 z z (17) Values for ζ and µ j were computed for z ) 1. This choice is motivated by the highly negative potentials (Figure 3), which limit the impact of the divalent co-ion, CO32- , on the double layer composition. This choice is also supported by the numerical results for yD (Figure 3B), which can be fairly well approximated by eq 15 using z3 ) 0. It should be added that eq 16 does not reflect any polarization of the double layer by the electric field. Therefore, the corresponding mobilities necessarily overestimate the numerically computed values obtained from the model of O’Brien and White (50). Comparable results were obtained for the FA and HA. Within the framework of (ii) (fully permeable polyelectrolyte), numerical computations based on exact numerical -1 theory were performed for λ-1 o . δH, with a value of λo set at 100 nm. This value corresponds to a plateau in the µ versus λ-1 o plot that indicates a freely drained spherical polyelectrolyte with no hydrodynamic interactions between resistance centers (32).

effect taken into account), are also presented (Figure 4A, curve c). This expression, valid for low potentials (yc e 1 or equivalently ψc e 25 mV) is given by:

Ftotal o

µ j)

(κ-1λo)2F

∑

zi2n∞i

g(κ, λo, δ)

(18)

i

where the function g is described by

( )( ()

g(κ, λo, δ) ) 1 +

{( )(

1 λo 3 κ

2

1 + e-2κδ -

-1 2 1 λo 1 + (κδ) × 2 3 κ (λ /κ) - 1 o

)

1 - e-2κδ + κδ

)

}

λo 1 + e-2κδ - (1 - e-2κδ)/κδ - (1 - e-2κδ) κ (1 + e-2λoδ)/(1 - e-2λoδ) - 1/λ δ o (19)

FIGURE 4. (A) Comparison of the theoretical predictions for a spherical semipermeable polyelectrolyte (curve b, exact numerical theory, λ-1 o ) 0.27 nm), for a hard sphere (curve a, eqs 16-17) and a fully permeable colloid (curve d, exact numerical theory with λ-1 o ) 100 nm). Curve c is obtained from the approximate eqs 18 and 19. The open circles are the experimental mobilities of Figure 2C. Model parameters: δH ) 0.94 nm, pH 10.7, Ftotal ) -504.7 C g-1 o (PHA). (B-D) Flow streamlines corresponding to the hydrodynamic flow within and/or around a PHA viewed as a semipermeable (panel B), hard (panel C), and fully permeable (panel D) (I ) 10 mM, δH ) 0.94 nm, pH 10.7). Panels B, C, and D refer to curves b, a, and d in panel A, respectively. The circles represent the HS of normalized radius KδH. The flow streamlines are obtained from the numerical solution (r, θ) of the equation Ψ(r, θ) ) constant with Ψ the stream function given by Ψ(r, θ) ) -h(r)E{sin θ}2 and h is the radial function defined by eq 6. λ-1 o

λ-1 o

Results obtained for (i) ) 0 (curve a) and (ii) f∞ (curve d) did not quantitatively account for the experimental data (Figure 4A) but, as intuitively expected, constituted the lower and upper limits of the rigorous model (iii) for a semipermeable spherical polyelectrolyte. Figure 4A (curve b) also revealed a maximum (in magnitude) for the mobility, as computed using exact numerical theory, for ionic strengths around 2 mM. This maximum is characteristic of a double layer polarization by the applied field (43). It results from an asymmetry of the ionic atmosphere of countercharges around/in the polyelectrolyte and effectively acts as a retarding force. The larger the charge carried by the polyelectrolyte, the larger is the value of yc and the more limiting is the polarization for the electrokinetic mobility. Considering the values of yc for the HS of interest (Figure 3B), it may seem surprising that the maximum does not occur for ionic strengths higher than 100 mM. The main reason for this observation is that the double layer polarization is strongly counteracted by the significant penetration flow inside the HS polyion, as illustrated by the flow streamline plot (Figure 4B). Flow streamlines obtained for (i) a hard sphere (Figure 4C) and (ii) a fully permeably polyelectrolyte (Figure 4D) are also represented in order to clearly demonstrate the fundamental differences among model representations (i), (ii), and (iii). Results obtained using the analytical expression derived by Ohshima (31) for a spherical polyelectrolyte, albeit within the restricted Debye-Huckel approximation (no polarization

Not surprisingly, the normalized absolute mobilities calculated on the basis of eqs 18 and 19 are larger than those obtained from numerical evaluation of the electrokinetic equations over the whole range of I, although they merge together at high ionic strengths where the Debye-Hu ¨ ckel approximation becomes reasonable. The theoretical predictions based on eqs 18 and 19 fail to match the exact numerical results because the electrostatic potentials (yc) for the investigated humic substances are quite large (> ca. 25 mV; see Figure 3B) (i.e., in a range where double layer polarization may play a significant role). The above results clearly show that the electrophoretic behavior of HS cannot be quantitatively interpreted using the simple analytical theories and formalisms of (i) or (ii). The ionic strength dependency of the experimentally measured mobilities can be described by the following phenomenological relationship:

µ j ) yc(I)F(κδ, I, λo, Ftotal o , ...)

(20)

where F is a complicated function that depends on all of the electrostatic and hydrodynamic parameters introduced previously. Within the range from 10 to 120 mM, where flow penetration strongly counterbalances polarization effects, F remains approximately constant (Figure 5). Since the potentials yD and yc exhibit similar (logarithmic) dependencies on I (see Figure 3B), one could replace the physically relevant potential yc by yD (which has no physical reality for HS) in eq 20. The use of yD here is motivated by its simple evaluation from eq 15 whereas the determination of yc requires the numerical solution of the Poisson-Boltzmann equation. The Donnan potential yD depends on the ionic strength (eq 15) and under the experimental conditions of ref 25 is fairly well approximated by

yD ≈ ln

(

{ ( )} )

Ftotal Ftotal o o + 1+ 2en∞ 2en∞

2 1/2

(21)

corresponding to the Donnan potential for a soft layer with charge density (Ftotal o ) drained by a 1:1 electrolyte (z3 ) 0) at bulk concentration, n∞ ) I. In the range 10-120 mM, 2 (Ftotal o /2en∞) . 1 so that combining eq 20 and eq 21 yields,

µ j ∝ ln(I)

(22)

which is in good agreement with the experimental data (inset Figure 5). It should be stressed that eq 22 is given a posteriori VOL. 39, NO. 17, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 5. Reduced mobility as a function of the Donnan potential (yD) for the PHA (δH ) 0.94 nm, pH 10.7). The dashed curve is a guide for the eye, and the plain curve a linear interpolation of the points corresponding to -6 < yD < -3.5. Domain 1: the flow penetration is predominant in determining µ (µ f -1.26 when I f ∞). Domain 2: polarization of the double layer is strongly counterbalanced by flow penetration. Domain 3: polarization becomes increasingly important as yD decreases (see text for more details). In the inset, an illustration of the logarithmic dependence of µ with respect to the ionic strength is given for the three HS: FA, HA, and PHA (eqs 20-22 in the text). from the rigorous numerical results. It has merit only in accounting for the qualitative logarithmic dependence of µ with respect to I and takes a deceptively simple form. To summarize the preceding discussion, (a) HS do not behave as hard spheres or fully permeable particles but rather as semipermeable polyelectrolytes; (b) the dependence of the electrophoretic mobility on the ionic strength is primarily determined by a screening of the charges and not by structural changes; and (c) numerical evaluation of the electrokinetic equations is required for a quantitative interpretation of the HS mobility data. As a last remark in this section, it should be stressed that the notion of a zeta-potential for a HS is physically irrelevant. Electrophoretic Mobilities as a Function of pH. As mentioned before, small but significant decreases in the diffusion coefficients of the HS were observed when lowering the pH of the solution from 10 to 4 (5, 6, 25). The corresponding increase in δH suggested the formation of small aggregates due to a reduction in the intermolecular electrostatic repulsion (5). This view is supported by recent surface tension measurements (51), which show that the aggregation of HS is facilitated at low pH. Because the structure of the HS strongly depends on pH, the softness parameter may also be expected to do so. Mobility data of the FA, HA, and PHA for 5 mM ionic strength and various pH values (25) are accurately reproduced using the exact numerical theory and a systematic adjustment of λ-1 o (Figure 6). The space charge densities Fo1,2 required for the calculations are those determined from the titration data. Numerical simulations were performed by taking into account the ionic charges, bulk concentrations (which depend on the pH), and the limiting ionic mobilities, λoi (i ) 1, ..., N), of the 6442

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FIGURE 6. Permeability λ-1 o and hydrodynamic radius δH for the FA (panel A), HA (panel B), and PHA (panel C) as a function of pH at I ) 5 mM. The δH values (open circles) are calculated from the FCS measurements reported in ref 25. The dashed lines are based on linear and fourth-order polynomial interpolations of the radii (δH) for the PHA and FA/HA, respectively (see text). The values for λ-1 o (precision ( 0.01 nm, see text) were obtained after numerical analysis (exact numerical theory) of the electrophoretic mobilities reported in ref 25. electrolytic buffers used in the experiments (25). When δH were not available at precise pH values, estimates were obtained by linear or polynomial interpolation of the raw FCS data (Figure 6). For the case of the PHA (Figure 6C), the hydrodynamic radius showed a 10% increase as the pH was lowered from 11 to 4. The increased radius, at pH values corresponding to a decreased intra- and intermolecular repulsion, corresponds to an aggregate of at least 2-3 PHA units (25, 46). An approximate 50% increase in the degree of permeability λ-1 o was also observed over this pH decrease. Qualitatively, the increased permeability can be understood by referring to the notion of hydrodynamic path. For a given aggregate composed of several, probably entangled PHA units, the fluid can flow not only through the permeable units but also among them via (hydrodynamic) paths. As the number of units

increases during aggregation, the number of hydrodynamic paths increases, as does the overall permeability. Although the mobility analysis was performed by assuming that the aggregates maintained a spherical symmetry (unlikely), λ-1 o can be considered to be an “effective” hydrodynamic penetration length. The value of λ-1 o determined for PHA at high pH was consistent with that derived from the earlier analysis as a function of electrolyte concentration (pH 10.7). For the FA and HA (Figure 6, panels A and B, respectively), the situation was slightly different. In the pH range of 5-9, similar aggregation/permeability features were observed as for the PHA. On the other hand, from pH 9 to pH 11, δH increased significantly for the FA and slightly for the HA. The increase in size may be explained by an increase in the intramolecular repulsion, resulting in a swelling of the polyelectrolyte, in line with the respective magnitudes of the total charges carried by the FA and HA. At low pH (4-5), it is rather difficult to distinguish between aggregation and electrostatic processes. Within experimental error, the hydrodynamic radii for the HA and FA flatten and remain practically constant. The accompanying strong decrease in permeability (determined by λ-1 o ), as observed for the FA, suggested that the corresponding aggregates became more compact. This observation could be due to the formation of H-bridges, as suggested from the location of the maximum in λ-1 o around the pK value for the carboxylic sites (Table 1). This explanation is also supported by experimental data on HS disaggregation kinetics (52). In the limit of a low volume fraction of soft material (40), the Brinkman equation predicts that the penetration length λ-1 o is proportional to the radius of the resistance centers (polymer segments) that constitute the polyelectrolyte. Similar functionalities follow from empirical expressions (53, 54) including

λ-1 o )

[1 Cφ- φ]

1/2

R

FIGURE 7. Plots of λ-1 o vs the physical radius δa (eq 24 in the text) for the FA, HA, and PHA (indicated) within the pH region corresponding to the occurrence of aggregation (I ) 5 mM). The physical radius of the HS (δa) was estimated by subtracting 0.12 nm from δH. This value corresponds to the difference in radii determined for a FA in water and DMSO (25).

(23)

where φ is the mean volume fraction of the resistance centers in a spherical polyelectrolyte-type microgel of radius R, and C is a constant that relates the effective radius of those centers to R. Assimilating the primary units of the (supposedly spherical) humic aggregates to hydrodynamic resistance centers, we may use the phenomenological relationship (23) in the form:

λ-1 o ) βδa

(24)

where δa is the physical radius of the aggregate and β is a constant that decreases with the “tortuosity” and “compactness” of the structure as a whole (Kozeny type parameter; 55). The physical radii were estimated as δa ) δH - 0.12 nm, corresponding to the difference between the radius of the FA determined in water and DMSO using fluorescence polarization experiments (25). Satisfactory agreement with eq 24 was obtained when λ-1 o was plotted against δa in the pH region where aggregation occurred (Figure 7). Analysis of λ-1 o at constant δa yielded the sequence of decreasing permeability FA > HA . PHA, consistent with the known ramified structure and high hydrophobicity of the PHA, as compared to the FA and HA. The slope β “quantifies” the change in permeability for a given increment in δa (i.e., for a given aggregate growth) and reveals the same sequence of decreasing compactness FA > HA . PHA. Given the stronger electrostatic repulsion within a FA aggregate, one would expect it to be “loose” as compared to the HA and PHA, strongly suggesting that a molecular interpretation for the compactness sequence cannot be solely based upon electrostatic considerations. On the other hand, the sequence was consistent with the possibility of H-bridging between

FIGURE 8. Permeability λ-1 o for the FA as a function of ionic strength and pH. The required δH for the evaluation of the λ-1 o are those of Figure 6A. Dashed lines are guides for the eye. In the inset, corresponding plots of λ-1 o vs the physical radius δa (eq 24 in the text) were given in the pH region where aggregation occurs. monomers. In that case, the number of H-bridges should be higher for the FA because of its larger density of ionogenic sites. H-bonds would render the structure more compact and consequently limit the number of hydrodynamic paths. The interpretation is in line with the observed maximum in permeability around pH 5. In Figure 8, the results of the quantitative analysis of the mobility curves in terms of λ-1 o and δH as a function of pH at different ionic strengths (25) are collected for the FA. The data are consistent with Figure 6A in the sense that aggregation, rather than electrostatics, affects the structure of the FA for pH 5-9. The slight variations observed with ionic strength (inset Figure 8) may result from the dependence on I of the number of undissociated sites available for the VOL. 39, NO. 17, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 9. Dimensionless equilibrium potential distribution for the FA at pH 4 (δa ≈ 0.83 nm) as a function of ionic strength (indicated). In the inset, (reduced) potentials yc for the FA, HA, and PHA as a function of pH (I ) 5 mM). Dashed lines are guides for the eye. It is noted that the potential distribution is not only determined by the charge Ftotal but also by the size of the spherical colloid (see curves o for the PHA and HA and eq 7 in the text). anchoring of different units and the decrease in inter/ intramolecular repulsion, both giving rise to more compact structures (decrease of the slope β). Nevertheless, the reproducibility for the mobility data limits the possibility of making more firm conclusions. For pH values higher than 9, λ-1 o depended only weakly on the ionic strength and pH, a result that is consistent with the fact that the FA is mainly composed of single polyelectrolytic entities. On closer inspection, the small decrease of λ-1 o with increasing I may be attributed to a molecular shrinking resulting from reduced intramolecular repulsion (5, 6). In the region of low pH, the permeability of the FA entities also depended on the ionic strength. With increasing I, the maximum in the λ-1 o versus pH curves decreased and gradually disappeared, likely due to the screening of the interparticular repulsion (Figure 9), which resulted in more compact aggregates. For the sake of illustration, the dependence of the potentials (yc) is also given as a function of the pH for the three HS at I ) 5 mM (Figure 9).

Acknowledgments We thank Jason D. Ritchie and E. Michael Perdue of the Georgia Institute of Technology for graciously providing raw titration data and M. Hosse for the raw electrophoretic mobilities and diffusion coefficients for the standard humic substances. We appreciate the useful comments of four anonymous reviewers. This work was supported by the Swiss National Funds.

Supporting Information Available Details on the numerical method employed to compute the electrophoretic mobilities of a diffuse soft particle without any restrictions of size, charge, and double layer thickness. This material is available free of charge via the Internet at http://pubs.acs.org.

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Received for review January 14, 2005. Revised manuscript received March 22, 2005. Accepted March 24, 2005. ES050082X

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