SUPPLEMENTARY INFORMATION For the paper

Metal flux and dynamic speciation at (bio)interfaces. Part I: Critical evaluation and compilation of physico-chemical parameters for complexes with simple

ligands and fulvic/humic substances.

Jacques Buffle*, Zeshi Zhang, Konstantin Startchev

Analytical and Biophysical Environmental Chemistry (CABE), University of Geneva,

Sciences II, 30 quai E. Ansermet, CH-1211 Geneva 4

* Corresponding author

1

A. DATA ON METAL IONS AND SIMPLE LIGANDS A1. Diffusion coefficients The values of these tables were obtained from the mentioned literature. Whenever possible, they were recomputed at infinite dilution by extrapolation to zero concentration. In a few cases, viscosity effects due to salt concentration were corrected for by using Stokes-Einstein’s law. Literature values were rejected when i) the method for data interpretation seemed to be non appropriate, ii) the reported value was rather different from other well-documented values for similar compounds, or iii) the given value did not fit a trend of values in a given series of homologous compounds. Table S1. Best values of diffusion coefficients of hydrated inorganic cations and anions in water, at T=298.15 °K and infinite dilution. Most values are taken from (1). Others refs are indicated as footnote

a: Inorganic cations b: Inorganic anions Hydrated metal

ion D (10-10 m2.s-1) Hydrated anion D (10-10 m2.s-1)

H3O+ 93.11 OH- 52.7 Ag+ 16.48 H2AsO4

- 9.05 Al3+ 5.41 Br- 20.8 Ba2+ 8.47 CO3

2- 9.2 Be2+ 5.99 HCO3

- 11.9 Ca2+ 7.92 Cl- 20.3 Cd2+ 7.19 ClO4

- 17.92 Co2+ 7.32 F- 14.8 Cr3+ 5.95 Fe(CN)6

3- 8.96 Cs3+ 20.56 Fe(CN)6

4- 7.35 Cu2+ 7.14 I- 20.45 Fe2+ 7.19 IO3

- 10.78 Fe3+ 6.04 MnO4

- 16.32 Hg2+ 8.47 NO3

- 19.02 Hg2

2+ 9.13 PO43- 6.1

K+ 19.57 HPO42- 7.59c

La+ 6.19 H2PO4- 9.59c

Li+ 10.29 HS- 17.3 Mg2+ 7.06 SCN- 17.58 Mn2+ 7.12 SO3

2- 9.59c

Na+ 13.34 HSO3- 13.31

NH4+ 19.57 SO4

2- 10.7 Ni2+ 7.05a HSO4

- 13.3 Pb2+ 9.45 S2O3

2- 11.32 Tl+ 19.89 Tl3+ 6.1b Zn2+ 7.03

a (2); b (3); c (4);

2

Table S2. Best values of diffusion coefficients (in 10-10 m2.s-1) of some organic ligands in water, at T=298.15K and infinite dilution (a,e) or in dilute aqueous solution. When not otherwise stated, values are taken from (10). Other refs. are given in the footnote. Units of 100.dlnD/dT are %/°K.

a: Carboxylic acids b: Amino acids + Urea c: Peptides

Name D Name D 100. lnT

d Dd

Name D

Acetate- 10.89a,b Glycine 10.6e 2.17f Diglycine 7.91 Citrate3- 6.23a,b α-alanine 9.15e 2.22f Triglycine 6.65 Malonate2- 8.45a,b β-alanine 9.29e 2.22f Alanylglycine 7.21 Oxalate2- 9.87a,b Sarcosine 9.67 Glycylleucine 6.23 HOxalate- 11.85c Threonine 8.16e Leucylglycylglycine 5.51 o-phthalate2- 6.96a,b Valine 7.50e 2.31f

m-phthalate2- 7.28a,b Norvaline 7.67 2.29f

Salicylate- 9.59a,b Leucine 7.24 2.30f Succinate2- 7.83a,b Asparagine 8.30 Tartarate2- 7.94a,b Norleucine 7.25 2.29f H2EDTA2- 5.83d Glutamine 7.62 Proline 8.79 Histidine 7.33 Phenylalanine 7.05 Tryptophane 6.59 Serine 9.28e Arginine 7.44e Isoleucine 7.75e 2.31f o-aminobenzoic

acid 8.40

Urea 13.78 a infinite dilution; b (1); c (6); d (11); e (12), values estimated by polynomial extrapolation of diffusion coefficient to zero concentration; f (13);

3

Table S3. Diffusion coefficients of metal complexes in water. Unless otherwise stated, T=298.15K and variable ionic strength. References corresponding to each value are given in footnotes. a value valid for both (+) and(±) compounds corrected for zero salt concentration. b T=293.15K

Metal ions Ligand ComplexesCd Co Cu Hg Ni Pb

Inorganic ligands ML3 6.7c,b OH-ML4 6.2d 5.7f,b 6.3k ML 11.1l ML2 12.3l ML3 10.4l Cl-

ML4 11.5l Organic ligands

Citrate ML 5.43g EDTA ML 5.4e 5.61h 5.7m 5.72n Leucine ML2 3.32o NTA ML 5.4f,b 7.3i

Tartratea ML2 4.14j,b c (14); d (15); e (16a, 16b); f (17); g (18); h (11,19a, 19b); i (20); j (21); k (22); l (23); m (24); n (11,24); o (26) :

4

A2. Stability constants of simple complexes with ionganic ligands Table S4. Logarithmic values of metal complexation equilibrium constants. For all ligands including OH- , they are expressed as M+nL⇆MLn, , i.e. βn=[MLn]/[M][L]n. For MHCO3, K = [MHCO3]/[M][HCO3]. Values collected for T=298.15°K in (27, 28, 29, 30) and fitted for ionic strength (I) dependance with the equation: logβI = logβI=0 + A√I + B.I + C.I3/2. (I in mol.dm-3)

ML ML2 logβ1(I=0) A B C logβ2(I=0) A B C

Other complexes (I=0)

Hydroxide Cd(II) 3.90 -0.949 - - Co(II) 4.30 - - - 8.80 - - - Co(OH)3

- logβ3=10.1 Cu(II) 6.40 -1.79 2.46 -0.761 13.36 -1.36 - - Ni(II) 4.10 - - - 8.90 - - - Ni(OH)3

- logβ3=11.5 Pb(II) 6.21 -0.610 0.351 0 11.28 -0.488 - - Zn(II) 4.68 - - - 11.10 - - - Zn(OH)3

- logβ3=13.92 Sulfate Cd(II) 2.43 -2.27 0.747 - 2.66 -1.03 - - Co(II) 2.52 - - - Cu(II) 2.27 -4.09 3.93 -1.73 Ni(II) 2.44 -3.36 1.31 -0.030 Pb(II) 2.74 -1.87 0.412 - 3.47 - - - Zn(II) 2.34 -3.53 2.64 -0.66 3.12 -2.30 0.62 -

Chloride

Cd(II) 1.94 -1.42 0.983 -0.163 2.48 -2.26 1.86 -0.370 CdCl3- logβ3=2.37

CdCl42- logβ4=1.7

Cu(II) 0.30 -0.845 0.354 - -0.26 - - -

Pb(II) 1.54 -1.74 1.42 -0.310 1.92 -1.72 1.20 -0.168 PbCl3- logβ3=1.80

PbCl42- logβ4=1.4

Zn(II) 0.47 -0.94 0.38 -

Carbonate Cd(II) 4.36 - - - 7.23 - - - CdHCO3

+ logK=10.69 Cu(II) 6.74 -2.82 1.84 - 10.10 -1.72 1.05 - CuHCO3

+ logK=12.13 Pb(II) 7.00 -3.34 1.66 - 9.00 -0.370 0.179 - Zn(II) 4.94 - - - 7.48 - - -

6

A3. Rate constants of metal ion dehydration in aqueous solution Table S5. Values of the rate constants for elimination of a water molecule from the inner water shell of metal cations. T = 298.15°K. Data taken from refs [a,b,c,d, see footnote]. Units: k-w=s-1; ∆H‡=kJ.mol-1; ∆S‡=J.K-1mol-1. Values in parentheses are either ranges of reported values or additional reported values. Monovalent cations Divalent cations Trivalent cations

Cation Log k-wa Cation Log k-w ∆H‡ ∆S‡ Cation Log k-w ∆H‡ ∆S‡

Cs 9.5 Ba 9.2a Al 0.1a(-0.8c) 112.9c 117.0c

K 9.2 Be 2.4a(4.5c) Bi >4c Li 8.5 Ca 8.3a Cr -5.6b(-6.3c) 109b 12b

Na 8.8 Cd 8.2c Fe 2.2b 64b 12b

Rb 9.3 Co 6.5a,b(6.1- 6.4) 33.4-49.7c Ga 2.6b(3.3c) 67b 30b

Cr 9.8c(8.5d) 12.5c -12.5c In ~4.3c(5.3) ~17c Cu 9.0a(9.0-9.9) 23.4c 25.1c La 7.3c(8.0d) Fe 6.6b(6.3-6.6) 41b 21b Sc 7.3d Hg 9.3b Ti 5.3b(4.8c) 43b 1b

Mg 5.7c 42.6c 8.4c Tl ≥9.5c Mn 7.3b 33b 6b V 2.7b 49b -28b

Ni 4.5b 57b 32b Pb 9.8d Sr 8.7a V 1.9b 62b -0.4b

Zn 7.5c(7.3-7.8) a (31); b (32); c (33); d (34);

7

A4. Rate constants for the formation of the complex ML, based on the reaction of M with L or its protonated forms Table S6a Reported values of the rate constants ka

L,exp and kaHL,exp (eq. 19) for the reactions:

M + L → ML, and M + HL → ML + H respectively. The n° of each complex refers to Fig 3a,c

Number Ligand Metal T (oC) I (M) logkaL,exp (M-1.s-1) logka

HL,exp (M-1.s-1) References

1 Acetate Ni 25 0 5.15 (5) 2 Acetate Pb 0.2 9.88 (6) 3 2-Aminomethylpyridine Ni 25 0.3 1.54 (7) 4 Ammonia Co 30 5.04 (8) 5 Butyrate Ni 25 0 5.10 (5) 6 CDTA Ni 25 0.3 5.28 (7) 7 β-chloropropionate Ni 25 0 5.13 (5) 8 L-2,3-Diaminopropionate Ni 25 0.3 3.28 (7) 9 L-2,4-Diaminopropionate Ni 25 0.3 3.45 (7) 10 HEDTAa Cd 25 0.5 9.57 (9) 11 EDTA Ni 25 0.3 5.28 (7) 12 Ethylenediamine Co 0.01 1.21 (25) 13 Ethylenediamine Ni 25 0.3 2.77 (7) 14 Floride Ni 25 0.5 3.90 (42) 15 Glutamate Zn 0.2 8.60 (43) 16 Glycinate Co 25 0.1 6.18 -0.60 (25,44) 16 Glycinate Co 25 0.2 6.48 (45) 17 Glycinate Cu 25 0.1 9.60 (45) 17 Glycinate Cu 25 0.2 9.53 (46) 18 Glycinate Ni 25 0.1 4.61 (44) 18 Glycinate Ni 25 0.2 4.32 (45) 19 Histidine Ni 23.7 0.1 3.34 (47) 20 IDA Ni 25 1.25 -1.11 (48)

8

21 Imidazole Ni 23.7 0.1 2.78 (47) 22 Lysine Ni 25 0.3 3.64 (7) 23 Malonate Ni 25 5.65 (49) 24 N,N,N-trimethylechylenediammonium Ni 25 0.3 2.70 (50) 25 NTA Cd 25 0.1 9.96 5.58 (51) 26 NTA Pb 0.2 7.77 (52) 27 Ornithine Ni 25 0.3 3.30 (7) 28 Oxalate Ni 25 0.1 4.88 3.70 (53) 29 Propionate Ni 25 0 5.10 (5) 30 Pyridine Ni 25 3.60 (54) 31 Pyridine-2-carboxylate Ni 25 0.3 1.48 (7) 32 Pyridine-2,6-dicarboxylate Ni 25 0.3 3.70 (7) 33 Pyrophosphate Ni 25 0.1 6.32 (55) 34 Thiocyanate Cd 25 3.0 8.70 (56) 35 Thiocyanate Zn 25 3.0 7.56 (57) 36 Triphosphate Ni 25 0.1 6.83 (58)

a reaction: HEDTA + Cd++ → HEDTACd

9

Computation of the formation rate constant of ML with protic ligands. An estimation of the formation rate constants, ka, for the association of M with a ligand which

may be either monoprotonated, HL, or non protonated, L, can be obtained by assuming that

the classical Eigen mechanism is operative (68). Extension to multiprotonated ligand and/or

to the generalized Eigen-Wilkins mechanism (eq. 10) is straightforward. The two paths for

ML formation with L and HL are

linked as follows:

Kos,ML k-wML

M + L ⇆ (M,L) → ML ⇅ ⇅

M + HL ⇆ (M,HL) → ML + H Kos,MHL k-w

MHL

The overall formation rate of ML, R, is thus:

R = k-wML[(M.L)] + k-w

MHL[(M,HL)] = ka [M][L] (S1)

where ka is the effective, overall rate constant for the formation of ML. By assuming that

equilibrium is always reached for the formation of both outer-sphere complexes (classical

Eigen mechanism), their concentrations can be computed from their equilibrium

complexation constants, Kos,ML and Kos,MHL. Then, eq. S1 can be expressed as:

R = [M](k-w

ML Kos,ML [L] + k-wMHL Kos,MHL[HL])

= [M][L](k-wML Kos,ML + k-w

MHL Kos,MHL β1H [H]) (S2)

where β1

H = [HL]/[L][H]. The effective, overall rate constant is thus:

ka = k-wML Kos,ML + k-w

MHL Kos,MHLβ1H [H] (S3)

Note that the dehydration rate constant of free M can usually be assumed to be independent of

the nature of the incoming ligand, and thus of its degree of protonation, i.e., k-wML = k-w

MHL =

k-w.

10

A5. Rate constants for the formation of successive metal complexes Table S6b Reported values of the rate constant ka,n

exp for the reaction: MLn-1 + L → MLn.

logkaexp (M-1.s-1) Number Ligand Metal T (oC) I (M) ML2 ML3 ML4 ML5

References

1 Acetate Hg 30 1.0 9.89 (58) 2 Ammonia Co 25 5.53 5.74 6.32 6.67 (8) 3 Choloride Hg 25 0 9.88 (59) 4 Cyanide Cd 0.1 10.53 (60) 5 Glycinate Co 25 0.1 6.30 5.90 (44) 6 Glycinate Cu 25 0.1 8.60 (45) 7 Glycinate Ni 25 0.1 4.75 4.63 (44) 8 Thiocyanate Zn 25 3.0 8.17 8.73 8.60 (57)

11

B. DATA ON FULVIC/HUMIC SUBSTANCES B1. Diffusion coefficients Figure S1: A. Diffusion coefficient (D) and hydrodynamic radius (rFS) of Suwanee river fulvic substances (SRFS) as function of pH, for 2 ionic strengths (5 mM: white circles; 50 mM: black squares) . D and rFS are related by the Stokes Einstein equation. {SRFS} = 10 mg/l. Modified from (35). B. Diffusion coefficient (D) and hydrodynamic radius (rHS) of Suwanee river humic substances (SRHS) as function of pH, for 2 ionic strength (5 mM: white circles; 50 mM: black squares) . {SRhS} = 10 mg/l. D and rHS are related by the Stokes Einstein equation. Modified from (35).

pH3 4 5 6 7 8 9 10

Diff

usio

n co

effic

ient

(10- 1

0 m

2 se

c-1 )

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0 0.82

0.94

1.11

1.29

r FS (n

m)

A

12

pH3 4 5 6 7 8 9 10

Diff

usio

n co

effic

ient

s (10

-10

m2

sec-

1 )

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

0.94

1.11

1.29

r HS (

nm)

B

Fig. S1B

13

Figure S2. Effect of ionic strength on the diffusion coefficient, D, and hydrodynamic radius, rFS, of Suwanee river fulvic substances at pH = 4.5 (black squares) and 6.8 (black circles). {SRFS} = 10 mg/l. rFS is computed from D by the Stokes Einstein equation. Modified from (35).

Ionic Strength (mM)0 20 40 60 80 100

Diff

usio

n co

effic

ient

(10-

10 m

2 s-1

)

2.0

2.2

2.4

2.6

2.8

3.0

r FS (n

m)

0.82

0.94

1.11

1.29

pH 4.5

pH 6.8

14

B2. Electrical potential profiles inside fulvics/humics and at their interface with solutions. The figures S3-1 and S3-2 provides the potential distance profiles between the centre of

fulvic molecules (supposed to be spherical) and the bulk solution, for two pH values and

various ionic strengths. The Figures S3-3 and S3-4 gives the change of the potential at the

center, ψc, and at the outer surface, ψo, of the fulvics, as function of ionic strength and pH.

Usually, the average potential, , for the whole fulvic molecule, is of interest. It is obtained

from the profiles of Figs S3-1 and S3-2, by integrating ψ on the sphere of radius r

__ψ

FS and

dividing by its volume, i.e:

∫=

==

FSrr

rFSdrrr

r 0

23

__)(3 ψψ S4

The following linear relationships (units in mV) are observed between and ψ__ψ c or and ψ

__ψ 0,

at variable ionicg strengths:

at pH 10.7: = 0.65 ψ__ψ c – 8.95 S5

= 1.05 ψ__ψ 0- 16.7 S6

at pH = 4: = 0.71 ψ__ψ c + 0.62 S7 __ψ = 1.15 ψ0 – 3.01 S8

Below the relationship = 0.68 ψ__ψ c + a will be used, where the coefficient 0.68 is an average of

those of eqs S5 and S7, and a is a parameter which depends on pH, but not on ionic strength. By

assuming that a is a linear function of pH in the pH range 4-10.7, one gets, for any pH and ionic

strength:

= 0.68 ψ__ψ c - 1.43 pH + 6.34 S9

This relationship is used below to get an empirical equation enabling to compute values of

at any pH and ionic strength.

__ψ

15

Figure S3-1 Equilibrium potential distribution inside and outside Suwanee river fulvic molecules, at pH = 10.7 and various ionic strengths indicated on the curves (in mM). rFS = 0.88 nm = radius of fulvic substances. Modified from (41).

-1 0 1 2 3 4 5 6-6

-5

-4

-3

-2

-1

0

(r-rFS)/rFS

135

85

45

25 5 mM

0-1

(r-rFS)/rFS

y =

Fψ/R

T

-150

-100

-50

0

ψ (m

V)

16

Figure S3-2 Equilibrium potential distribution inside and outside Suwanee river fulvic molecules, at pH = 4 and various ionic strengths, indicated on the curves (in mM). rFS= 0.83 nm = radius of fulvic substances. Modified from (41).

-1 0 1 2 3 4 5-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

4020

5 mM

(r-rFS)/rFS

y =

Fψ/R

T

-60

-40

-20

0

ψ (m

V)

17

Figure S3-3 Change with ionic strength (Na2CO3) of the potentials i) at the center of humic/fulvic substances, ψc, yc, and ii) at the outer edge of the humic/fulvic molecule, ψo, yo (y= Fψ/RT). pH = 10.7. PHA, HA and FA are a peat humic acid and Suwanee river humic acid and fulvic acid respectively. Modified from (41).

0 20 40 60 80 100 120 140-6

-5

-4

-3

-2

yc, ψc

yo, ψo

yoyc

r

PHA HA FA

Ionic strength (mM)

y c, yo

-120

-80

-40

ψc, ψ

o (mV

)

From the change of ψc with ionic strength, in figures S3-1, S3-2 and S3-3, the following

empirical relationship is obtained for fulvics:

ψc = b + c.logI S10

where I is ionic strength, and b and c are constant parameters which only vary with pH. The

parameter c varies little with pH and this variation is assumed to be linear in the range 4< pH

< 10.7 . With the data of Figs S3-1 to S3-3, we gets:

c = -5.37 + 4.64 pH S11

18

Figure S3-4 Change with pH of potential at the center of humic/fulvic substances, ψc, yc,. Ionic strength = 5 mM. PHA, HA and FA are a peat humic acid and Suwanee river humic acid and fulvic acid respectively. Modified from (41).

4 5 6 7 8 9 10 11-6

-5

-4

-3

-2

pH

y c = F

ψc/R

T

-140

-120

-100

-80

-60

FAPHA

HA

ψc (m

V)

The parameter b, in eq. S10 for fulvics is obtained by fitting the corresponding curve of Fig.

S3-4 with a polynomial function of pH:

b = - 328.9 + 190.2 pH - 41.60 pH2 + 3.670 pH3 - 0.1130 pH4 S12

By combining eqs S9-S12, one gets the general empirical expression:

= - 217.4 + 127.9 pH - 28.29 pH__ψ 2 + 2.496 pH3 - 0.07685 pH4

+ 3.159 pH.logI - 3.654 logI S13

Equation S13 enables to compute values at any pH or ionic strength in the domains

4 < pH < 11 and 0.005 M < I < 0.15 M. The difference with experimental data used to

find eq. S13 is less than 1 mV.

__ψ

19

B3. Site distribution of fulvic and humic substances related to the Differential Equilibrium

Function

The concept of Differential Equilibrium Function (DEF) is applicable to any metal

complexation in a mixture of a large number of different ligands, each site type, iS, forming a

1:1 complex, MiS, with M, with a stability constant iK. DEF does not make any assumption

on the driving force for the complexation reaction. Qualitatively, the titration curve of a

chemically heterogeneous complexant, like FS (Fig. S4A), can be thought of as resulting from

the successive saturation of a large number of different ligands, each one being present in

very small concentration. According to this picture, the curve of log ([MS]t/{FS}) = f(log [M]

should show a large number of jumps of very small amplitude (36). In practice, this is not so

because the iK values of the successive complexes are too close to each other for each ligand i

to be titrated separately. Consequently a continuous curve, devoid of jump is observed (Figs.

S4-A,C) and the effective equilibrium “constant”, K*, decreases along the titration curves

since weaker and weaker sites are titrated.

At a given point, j, of the titration of FS by M, where the free metal concentration is j[M], the strongest site types are saturated, the weakest site types are free, and intermediate

site types are partially occupied. A value of the effective equilibrium constant at point j, for

the reaction of M with the whole of the site types at this point, is given by (61):

t

jjt

jj

SdM

MSdK

][][

][* −= (S14)

where [MS]t = Σ [MiS], and [S]t = Σ[iS] are the total molar concentrations of the complexes

and of free sites respectively, at point j. jK* decreases when j[M] increases along the titration

curve, and the corresponding ensemble of points forms the so-called DEF, K*. Its interesting

features are the following: i) it is readily computed experimentally (38, 40, 62), ii) it is

simply related to the parameters of each complexing site type i (i.e., iK; iχ = mole fraction of

site type i; iθ = degree of site occupation by M; see Figure S4-C-F and related discussion),

and iii) it is simply related to [M] (36,37) by:

K* = (1-Γ)/Γ [M] (S15)

Combining eqs. 29 and S15 leads to:

** loglog}{][log KKFS

MSot Γ−Γ=⎥

⎦

⎤⎢⎣

⎡ (S16)

20

Figure S4 A,B Examples of experimental relationship between the values of K* (eq. S14-

S15) and the ratio of the metal bound to fulvic substances (expressed as [MS]t = molar

concentration of bound metal, over the fulvic concentration, expressed as dissolved organic

carbon concentration, DOC, in kg/dm3 of solution). Note that {FS} ~2 DOC. T = 25°C, I =

0.1M. A: Titration of a humic sample by Cu(II) (38). B: Results obtained for a large number

of different water fulvic samples. Each point corresponds to the K* of a different sample at

the corresponding value of [MS]t/DOC. Modified from (39) and (67).

-10 -8 -6 -4-2

-1

0

1-10 -8 -6 -4

0.5

A

log ([M])

log

([MS]

t/DO

C) (

mol

/kg

C)

log ( K* )

12 10 8 6 4

-2

-1

0

1B

0.5

Cu(II)pH 8-8.3

Pb(II)pH 4.5-5.0

log

([MS]

t/DO

C) (

mol

/kg

C)

log (K*)

21

where Ko* is a constant characteristic of M under the experimental conditions used (pH, T,

ionic strength). It has been found (Figure S4A) that this equation is valid for fulvic or humic

minor sites, over at least 3–5 orders of magnitudes of logK* depending on metal, and no limit

to its validity has yet been found in the low [MS]t/{FS} range (corresponding to large iK

values). Equation S16 is applicable both to laboratory titration of individual fulvic samples

(38, 63- 65; Fig. S4A) and for the comparison of the complexing properties of fulvic of

various origins (39, 65,66; Fig. S4B). Interestingly, the latter results show that both Γ and K*o

depend on the nature of the metal ion, but can be considered as rather independent from the

origin of the fulvics.

Note that Ko* is only a reference parameter equal to K* when the ordinate of Fig. S4

is nil. Its value (in dm3/mol) depends on the parameter and units used on the ordinate, e.g.

[M]t/{FS} (in mol/kg of fulvics) or [MS]t/DOC (in mol/kg of C) and should be selected

coherently with the latter.

It has been shown that K* is an average of all the stability constants iK, weighted by

the degree of occupation (iθ) and the mole fraction, Δiχ, of each site type, i (37,38,39,40).

∑

∑

=

=

Δ−

Δ−= n

i

iii

iiin

i

i KK

1

1*

)1(

)1(

χθθ

χθθ (S17)

where Δiχ = [iS]t/Σ[iS]t ([iS]t = total concentration of site type i), and iθ = [MiS]/[iS]t . K* is

related to [M] through eq. (S18) valid for all sites:

][1

][MK

MKi

ii

+=θ (S18)

As it is discussed below, for any [M] value (e.g. at any titration point of FS by M), the relative contribution of a site i to the K* value is maximum if iK = 1/[M] and about negligible (≤1% ) if either logiK ≥ - log[M] + 2 or logiK ≤ - log[M] - 2 . Figure S4-C gives a schematic representation of a typical titration curve for the minor sites of a chemically heterogeneous complexant. The horizontal axis coincides with the direction of the titration, from left to right. The increase in [MS]t or in [M] corresponds to the successive saturation of weaker and weaker sites and in turn to a decrease in the logiK of the sites effectively active at each titration point. Figures S4-DEF give a quantitative description of a complexant whose site distribution (Δiχ = f(logiK) ; fig. S4-D, curve 2 ) corresponds to

22

a linear DEF (eq. S16) with Γ = 0.5, i.e. a case often found in natural systems (Table 2 of main paper). Such a site distribution can be simulated by the Sips distribution function (37, 38). Figures S4-DEF show the contributions of each individual site to the various terms of eq. S17, at a given titration point corresponding to [M] = 10-10 M. The horizontal axis is drawn with decreasing values of iK, to allow comparison with fig. S4C. Figure S4-C: Definition of the strong (I), intermediate (II) and weak (III) site classes amongst the minor sites, at any particular titration point (corresponding to the free metal ion concentration [M]) of fulvic/humic complexants by M. Calculations are based on the assumption that the system follows eq. 29, with the same conditions as in figures S4-DEF. Tabulated numerical values are the percentages of the summation terms of eq. S17 corresponding to each site type class, with respect to that for all three classes. In the numerators, when no subscript is given under the symbol Σ, summation is done over site type class I, II or III, as indicated in the column of the table. Modified from (36).

log[MS]t

{FS}condition ofFig. S3-DEF

logM =-10| |

logK*=10logK*

log Ki Class Istrong sites

Class IIintermediate

Class IIIweak sites

iθ ~ 1(1 - ) ~ 0iθ

0.01< <0.99iθ

Σi i iθ θ Δ χ(1- ).

(%) 12 88 0

I+II+III

I+II+III

(%) 0 88 12

481216

iθ ~ 0(1 - ) ~ 1iθ

Σi i i iK. (1- ).θ θ Δ χ

Σi i i iK. (1- ).θ θ Δ χ

Σi i iθ θ Δ χ(1- ).

logM| |

23

At each titration point, three classes of site types may be discriminated on the bases on their value of iθ (Fig. S4-D): class I includes the sites for which values of iK are so large that they are fully saturated (i.e. iθ ~1 in this class); class II includes the sites whose iK values are not as high as those of class I, but high enough to form stable complexes; in this class, 0.01 < iθ < 0.99; class III includes the sites for which iK is so low that iθ ~0. The contribution of any site in controlling [M] (and K*) depends on a) iθ (most effective control being achieved when iθ = 0.5), and b) iK (eq. S2). iθ.(1- iθ) (Fig. S4-E) is one of the weighting factors of eq. S17. It is a measure of effect a) above: iθ.(1- iθ) is zero when the sites I are either saturated (iθ = 1) or totally free (iθ = 0) and it passes through a maximum at iθ = 0.5, for sites with iK close to iK= 1/[M] (= 1010 M-1 in fig. S4-CDEF). Clearly, with respect to this property, sites having - 2- log[M] < logiK < 2 - log[M] are the only important ones controlling [M] in the whole system (note that the above limits for iK depends somewhat on Γ). The term iθ.(1- iθ) however is not sufficient to compare the relative importance of the various sites to control [M]; the influence of their relative concentrations on the buffering action should also be considered. The latter increases with the mole fraction of occupied + unoccupied sites, Δiχ.The relationship between Δiχ and logiK is shown in fig. S4-D (curve 2). The actual contribution of each site to the buffering action is obtained by multiplying iθ.(1- iθ) by Δiχ. This yields curve 1 in fig. S4-F, which, when compared with fig. S4-E reflects the dissymmetry of the Δiχ distribution, giving more importance to the sites present in large concentrations. The role of logiK (effect b) above) is shown in fig. S4-F (curve 2) which reflects a bias in the contributions of the sites, favoring large iK values. Thus K* (eq. S17) is an average of the iK’s of all sites, weighted for their buffering action. Graphically it is represented by the ratio of the surface areas under curves 1 and 2 of fig. S4-F. K* = 1010 = 1/[M] is obtained from the figure. The important point is that, although dissymmetric site distribution (Δiχ= f(logiK)) has a “spreading” effect on the sites which play an important role on K* (compare figs S4-E and S4-F), this effect is not large, since site type of class II (1012 > logiK > 108) account for 88% of both the numerator and denominator of eq. S17 (fig. S4-C). It is also important to note that the width of class II sites may be affected by the value of Γ, but it does not depend on [M].

24

Figure S4-D,E,F: Changes in the various terms of eq. S17, corresponding to each site type i, as a function of their stability constant iK. (for explanations, see text). Curves calculated under the following conditions: [M] = constant = 10-10M; for each site, iK = iθ/[M](1-iθ) with iθ = [MiS]/[iS]t and Δiχ = [iS]t/Σ[iS]t. The overall complexation follows eq. S16, i.e. it obeys the Freundlich isotherm, and Δiχ values are given by a Sips distribution (eq. 30) with Γ = 0.5 and logKo

* = 8.5 and ΔlogK = 0.1. (Modified from 36; for calculation details, see 38).

14 12 10 8 60.0

0.1

0.2

0.3E

i θ.(1

−i θ)

14 12 10 8 60

1

2

3

4

5

F

log (iK /M-1)

104 .i θ(

1-i θ)

Δi χ

0

1

2

3

4

5

10-6.i K

i θ(1-

i θ)Δi χ

(1)

14 12 10 8 60.0

0.5

1.0

1.5

(2)(1)

D

i θ (1

) or

102 .Δ

i χ (2

) (1

) (2)

(2)

IIIIII

IIIIII

25

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