Influence of kinetics on the determination of the surface reactivity... suspensions by acid-base titration

Influence of kinetics on the determination of the surface reactivity of oxide
suspensions by acid-base titration
M. Duca, F. Adekolab,c, G. Lefèvreb and M. Fédoroffb*
Present address: Laboratoire Central des Ponts et Chaussées, 58, Boulevard Lefèvre, 75015
Paris, France
Université Pierre et Marie Curie-Paris6, ENSCP, CNRS-UMR7575, Laboratoire
d'Electrochimie et de Chimie Analytique, 11, Rue Pierre et Marie Curie - 75231 Paris Cedex
05, France
On leave from Department of Chemistry, University of Ilorin, P.M.B. 1515, Ilorin, Nigeria
*corresponding author. E-mail address: [email protected], tel.: (33) (0)1 56 81 30 56,
fax: (33) (0)1 56 81 30 59
The effect of acid-base titration protocol and speed on pH measurement and surface
charge calculation was studied on suspensions of -alumina, hematite, goethite and silica,
whose size and porosity have been well characterized. The titration protocol has an important
effect on surface charge calculation as well as on acid-base constants obtained by fitting of the
titration curves. Variations of pH versus time after addition of acid or base to the suspension
were interpreted as diffusion processes. Resulting apparent diffusion coefficients depend on
the nature of the oxide and on its porosity.
Keywords: acid-base titration ; surface charge ; oxides ; titration kinetics ; surface acid-base
constants ; surface complexation
1. Introduction
The sorption properties of metal oxides and hydroxides have been the subject of many
investigations during recent years because they play an important role in the transport of toxic
and radioactive species in the environment, in catalytic processes, in corrosion inhibition...
The sorption properties of such compounds are generally described by the acid-base behavior
of superficial hydroxyl groups using one of the surface complexation models (1-pK [1] or 2pK [2] mono-site models, 1 pK multi-site model [3,4]). The acid-base properties are defined
by several parameters whose number depends on the model used: surface site density, acidity
constants, and electrostatic parameters. All these parameters are generally determined by
fitting the data acquired from titration experiments, in which a suspension of the solid is
titrated in order to determine the quantity of acid or base necessary to protonate or
deprotonate the surface hydroxyl groups. Hence, the accuracy of the determination of acidobasic parameters depends on the accuracy of the titration experiments. The quality of titration
data can be affected by several factors. A number of errors are associated with
instrumentation and measurements methods: poor calibration and drift of electrodes, junction
potentials, underestimated activity coefficients, effect of suspension on electrode response [5].
Other sources are intrinsic to the solid: solubility, presence of impurities, kinetics of the
processes at the interface, evolution of the solid. In previous investigations, we have
evidenced that transformation of -alumina into bayerite in aqueous solutions results in a
continuous modification of the acid-base parameters of the solid [6]. We also showed that the
solubility has a large effect on the calculated surface charge of alumina and a correction
method was developed [7]. All these factors may explain why a large scatter is observed for
acid-base parameters obtained by different authors on oxides with the same chemical
composition [8].
The present study is devoted to the kinetic aspect of titration. Some authors have already
drawn attention to this effect [9,10]. Sorption of protons during titration of oxides is often
described as a two-step process: a fast initial uptake followed by a slower process [11]. The
fast step is attributed to the reaction of protons with the superficial hydroxyl groups and the
time necessary to reach equilibrium is often considered to be less than a few minutes. The
slower process is poorly understood and numerous assumptions have been published, such as
a rearrangement of protons at the surface [11], their diffusion into micro-pores [10] or macropores [12], in the lattice of the oxide [13] or in the hydrated layer formed on the surface [10],
and also the diffusion of oxygen ions from the bulk of the solid to its surface [13].
In order to circumvent the slower step, "fast" titrations with a short delay between each
addition of acid or base, were proposed for non-porous solids (ZnO [14,15] or -Fe2O3 [10]
for example). Thus, the titration curve should only correspond to the reaction with surface
hydroxyl groups. However, the appropriate time necessary to separate the fast and the slow
steps cannot be easily determined.
The present study was initiated after several observations during titration experiments
performed with aluminum, silicon and iron oxides. Titration curves depend, often strongly, on
the titration speed: time interval between the addition of each aliquot of acid and base, and
volume of each aliquot. Most often a hysteresis appears when a reverse titration is performed.
As a consequence, titration speed has an influence on the calculated acid-base constants and
site density, denoting that equilibrium which is supposed to be met in surface complexation
models, is not achieved.
The aim of the present work was to study in detail the effect of kinetics on the validity of
the data obtained by titration, to determine the procedures leading to the minimum errors and
to understand, when possible, the causes of the kinetic effects observed. Several oxihydroxides, largely studied in the literature and with different morphologies were chosen: alumina, goethite, hematite and silica. For this purpose, acid-base titrations with various
speeds and procedures were performed. Influence of equilibration time on surface charge
calculation was studied. Modeling of charge vs. pH curves was undertaken so as to estimate
the influence of experimental procedures on calculated acidity constants. In order to associate
the results with the characteristics of the solids, we used chemical analysis, X-ray diffraction,
high-resolution scanning electron microscope, surface area and porosity measurements and Xray photoelectron spectroscopy.
2. Materials and Methods
Chemicals and solutions
The solids used in this study were -alumina (Merck 90), goethite (BASF), hematite (2
samples from Alfa of 99.8 and 99.945 % purity), and silica (Merck 100) (Table 1). A
hydration period of one day (ferric oxi-hydroxides and silica) or 15 days (-alumina) was
performed prior to titration. The hydration time for alumina was chosen in order to avoid an
evolution of surface reactivity during titration [6].
Other reagents were HNO3 and NaOH 0.1 M (Normadose © Prolabo), sodium nitrate p.a.
(Prolabo) and deionized water (conductivity > 16M).
Titration methods
“Continuous” titrations were performed using a Metrohm automatic system monitored by
a homemade software, with a combinated Ag/AgCl – glass electrode (Metrohm) calibrated
with disposable standard buffer solutions (Centipur © Merck). The test solution vessel (100mL PE bottle) with alumina (0.1 g), goethite (0.3 g), hematite (0.5 g) or silica (0.05 g)
equilibrated with 50 mL of 0.1 M NaNO3, with a continuous flow of argon to prevent CO2
uptake, was immersed in a water bath thermostated at 25.0°C  0.1°C. The solid/solution ratio
was chosen to obtain the same order of magnitude for surface area in all suspensions (ca. 5 to
15 m2). The added volumes of acid or base and equilibration time before the next addition
were varied according to the pH value since the stabilization of the measured potential was
faster for pH values far from the neutral range. Three protocols of such titration methods were
used (Table 2).
Continuous titrations with longer and constant intervals (1 hour) between each addition of
acid or base were also used.
“Batch” titrations were performed with -alumina by stirring 100 mg of powder in 50 mL
of a 0.1 M NaNO3 solution at 25°C  1°C during time intervals of 7, 18 and 48 h. The initial
pH was adjusted by addition of NaOH or HNO3. After the fixed time interval, the pH was
measured in a glove box with a flow of nitrogen.
Methods of Analysis
Surface areas and porosities were determined by nitrogen adsorption and desorption on a
Coulter SA 3100 instrument, after degasing 30 minutes at 120°C, using the BET model for
surface area, the t-plot for the estimation of micro-pore (diam. < 2 nm) volume and the BJH
model for the estimation of meso-pore (2< diam. < 40 nm) and macro pore volume and
distribution [16]. Scanning electron microscope observations were performed with a LEO
DSM 982 high resolution instrument, the powders being deposited on a carbon ribbon and
metallized with a Pt/Pd alloy. X-ray diffractograms (XRD) were recorded using the Co K
radiation. X-ray photoelectron spectroscopy (XPS) measurements were performed using a VG
CLAM2 analyzer with an unmonochromated aluminum X-ray source (Al K, 1486.6 eV).
Surface charge calculation and modeling
The surface charge of the solid was calculated from the quantity of added acid or base,
after subtracting a “blank”. This blank was determined by titration of a solution of the same
electrolyte without solid. Fitting of this blank curve allows to relate the measured pH value to
the concentration of protons in the whole pH range through apparent activity coefficients, and
without any need for junction potential correction nor theoretical activity coefficients.
The acidity constants and site density were determined by fitting the titration curves
according to a 2-pK model. Several models have been proposed to describe the distribution of
the electrostatic potential in the liquid layer adjacent to the solid surface [17]: the diffuse layer
model (DLM), the constant capacitance model (CCM) and the triple layer model (TLM). The
TLM was not used in this study, since it is based on seven adjustable parameters, whose
variations are difficult to associate with titration kinetics. Use of DLM model often leads to
non-convergence of the optimization procedure. Therefore, the CCM model was used to
compare the results of the different titration procedures. The computer code FITEQL 4.0 [18]
was used for that purpose, in comparison with a fitting by the least square method using a
home-made calculation code.
3. Results and discussion
Characterization of solids
The main morphological characteristics of the solids are summarized in Table 1. The
morphology of -alumina and its evolution with time of hydratation were studied in detail
elsewhere [6]. As observed by SEM, it is constituted of irregular approx. spherical particles
with an average diameter of 100 m. At high resolution, these particles show flat surfaces
with slits from 10 to 200 nm in width. X-ray diffraction patterns indicate that these particles
are constituted of crystals of approx. 3 nm large. The BET surface area of 152 m2.g-1 is much
larger than the surface of observed particles. T-plot calculation did not indicate any micropore volume (diam. < 2 nm), while BJH [16] analysis of the nitrogen sorption and desorption
isotherms led to a porous volume of 0.21 mL.g-1 (125 m2.g-1) in the meso-porous region (2-40
nm). Thus the major portion of surface area corresponds to meso-pores, which probably
constitute the interstices between crystallites forming the agglomerates observed by SEM.
After hydration during 15 days, X-ray diffraction and other methods detect the presence
of bayerite, but the SEM aspect and BET surface are slightly modified [6]. Platelets, probably
constituted by bayerite appear only after one month of hydration.
Goethite powder shows, at low resolution, agglomerates of needle-like particles. At high
resolution, these needles appear as elongated particles (length  600 nm, width  50 nm). A
detailed study [19] concluded that these particles are either single crystals or agglomerates of
a few single crystals. The measured surface area is close to the surface of particles and the
meso-pores correspond to the volume between particles.
Hematite is constituted of agglomerates of almost spheroidal particles, truncated by flat
surfaces, of approx. 50-100 nm diameter, for both purities. From X-ray diffraction, it appears
that these particles are constituted of single crystals. The observed porosity corresponds to the
volume between particles.
Silica is constituted of irregular particles of 50 m mean diameter. At high resolution, it
shows an irregular surface and holes which may be interpreted as ends of porosities, whose
diameter is estimated to be approx. 20 nm, which is close to the BJH determinations.
According to X-ray diffraction, this silica is amorphous.
Effect of titration speed
Examples of titration curves of alumina by an acid solution performed by continuous
titration at three different speeds and by batch titration with three different equilibration times
are presented in Fig. 1. Only the 5 to 8 pH range is represented in the figure for clarity.
Experimental conditions for continuous titrations are indicated in Table 2. The titration speed
varied in the order Titr1 > Titr2 > Titr3. The surface charge values are represented in atoms
per square nanometers of the solid surface and were corrected for the dissolution of alumina
When the titration speed is increased, the measured pH value for the same added volume
of acid increases. Fewer protons are consumed by the solid and the calculated surface charge
decreases, leading to the shift observed in Fig. 1. Batch experiments with 7 h and 18 h
intervals gave results varying in the same range as for continuous titration experiments. Only
the longest batch experiment (48 h) gave systematically higher charge values, indicating that
these values are closer to equilibrium, although the achievement of equilibrium is not sure,
even for these long times of contact. Furthermore, a deformation (“bump”) with speeddependent amplitude appeared within the 5.5-7.5 pH range, especially for the fastest
continuous titrations.
Analysis of pH evolution in titration experiments with long intervals between additions
In order to understand the origin of the slow surface acid-base kinetics, titrations were
performed with a long interval (1 h) between each addition of acid or base, with continuous
pH measurements during this time interval. Examples of results are shown in Fig. 2. The
positive time scale corresponds to acid additions, while the negative one corresponds to
additions of base. Curves show a fast variation of pH in the first 3-4 minutes, followed by a
slow variation until a constant value is reached. The first minutes correspond to the time
necessary to reach homogeneous concentration in solution and equilibrium at the electrodes,
as it can be observed for titrations without any solid. The slow variation of pH corresponds to
the release or fixation of protons from or onto the solid, until an equilibrium value is reached.
Nevertheless, a large proportion of protons have already reacted within the first minutes after
Differences in the shape of curves appear immediately. During the fast variation of pH,
we observe an “over-shooting” in the case of alumina and silica: after addition of acid, pH
rapidly decreases until a sharp minimum, it then slowly increases until a constant value. A
symmetrical variation is observed after addition of base. This phenomenon was not observed
or is much more limited for goethite and hematite. Differences also appear during the slow
step: the evolution of pH seems to be particularly slow in the case of hematite.
In order to have a more accurate knowledge of the kinetics of proton exchange with the
solid, we have calculated the variation of surface charge versus time. The surface charge Qt at
time t is calculated from electro-neutrality:
 
Q t  NO3  Na  
 
γ OH H 
 
where [NO3-] and [Na+] are the concentration of nitrate and sodium ions in solution
respectively, Kw is the ionic product of water, (H+) the activity of protons as measured by the
electrode, H and OH are the respective apparent activity coefficients of protons and
hydroxide ions, determined from "blank" titrations without solid.
The kinetics is represented by the variation of the achievement factor F versus time,
defined as:
Q t  Q0
Q  Q0
where Q0 and Q are the surface charge before addition of acid or base and at
equilibrium, respectively.
The kinetic curves could be best fitted according to a diffusion law in spherical particles
[20] after a fast step achieved within a few minutes after addition:
 6
F  1 - 2
 π
 Dπ 2 n 2 t 
 1  Ff   Ff
exp  
n 1 n
where t is time, r the radius of particles, D the diffusion coefficient, F the total
achievement factor, Ff the achievement factor of the fast step. Typical experimental results
and fitting curves are shown in Fig. 3. Apparent diffusion coefficients were calculated from
D/r2 ratios using mean particles radii taken from Table 1.
Results of mean apparent diffusion coefficients are presented in Table 3. Two groups of
solids appear from these results: alumina and silica, with apparent diffusion coefficients
ranging from 610-13 to 410-12 m2.s-1, goethite and hematite, with diffusion coefficients
ranging from 1.710-20 to 1.710-19 m2.s-1. The solids of the first group have large surface
areas (Table 1), much larger than the external area of their particles, and significant pore
volumes. It is likely that diffusion inside the pores of these materials is the controlling process
of the slow kinetic step. As a comparison, Kim et al. [21] have found an apparent diffusion
coefficient of  10-13 m2.s-1 for porous nickel hydroxide, while a simulation in a membrane
with pore diameters in the nm range gave 10-10 - 10-9 m2.s-1 values [22]. Presence of “overshooting” in the curves of Fig. 2 indicates that, when an aliquot of acid is added, the H+
concentration increases rapidly in the solution and then decreases slowly as H+ ions diffuse
into the porous solid.
For hematite and goethite, it is likely that a diffusion process inside a more compact
structure takes place. Yu et al. [23] determined a diffusion coefficient attributed to protons of
 710-22 m2.s-1 in a compact layer of CVD -alumina. In hematite, Berube et al. [9] reported
a diffusion coefficient of  310-25 m2.s-1. These authors assumed that diffusion of protons
occurred within a hydrated surface layer with an estimated thickness of not less than 2.6 nm,
but did not exclude other kinetic processes.
In order to explore the occurrence of an hydrated layer, we have performed X-ray photoelectron spectroscopy (XPS) for goethite and hematite. Decomposition of the O1s photoelectron band led to similar intensities for oxygen and hydroxide peaks in goethite (FeOOH),
which contained both of these species. In the case of hematite (Fe2O3), the O1s corresponds
almost entirely to oxygen, with a weak shoulder which may be attributed to traces of surface
hydroxyl groups. Thus, XPS results ruled out the possibility of a significant hydrated layer.
Although the kinetic process in hematite and goethite can be related to a solid state diffusion
and has a significant effect on titration experiments, knowledge of the real process needs
further investigations. Apparent diffusion coefficients are also related to the purity of the
oxide, as shown for two types of hematite (Table 3).
Another feature is the systematically higher diffusion coefficients found for pH>7
compared to those for pH<7. Since the diffusion coefficient of hydrogen ions is higher than
that of hydroxide ions (in water, 9.310-9 and 5.310-9 m2.s-1, at 25°C respectively [24]),
this effect cannot be assigned directly to the mobility of these ions, but to successive
interactions with species in solution, surface or solid. A higher diffusion coefficient in
alkaline pH than in acid pH was also observed in biological material (4.110-10 instead of
1.410-10 m2.s-1) and interpreted, in this case, as due to the buffering effect of amino acids
Absence or weakness of “over-shooting”, compared to alumina and silica, may give an
indication on the process. Complete absence of this effect is observed for hematite, when
aliquots of base are added. This means that when the base is added, OH- ions are immediately
uptaken by the solid. Further increase of pH indicates that OH- ions are then slowly released
back into the solution (or H+ ions slowly fixed by the solid), this release being controlled by a
diffusion process.
Effect of kinetics on surface charge and pH determination
Not waiting until stable pH values has an impact both on pH and on the surface charge,
which is calculated from the pH values. An example is given in the case of hematite (Fig. 4),
where pH varied by 0.4 units and surface charge by 0.1 at/nm2 in a time interval between 5
and 60 min. The influence on surface charge is considerable, if compared to the site density of
approx. 1.2 at/nm2.
As highlighted above, several authors [10, 14, 15] have attempted to use fast titration
procedures, with the aim of excluding the slow step attributed to diffusion inside the bulk of
the solid. Our results show that it is difficult to choose the appropriate time interval
demarcating the "fast" and the "slow" processes, since in the first minutes after addition the
processes at the electrodes are still not at equilibrium, while the slow step is already
proceeding. One way to select the fast process would be to fit the kinetic curves as proposed
in the present work in order to calculate the achievement factor of the fast step Ff. However, it
is necessary to follow the variation of pH versus time systematically after each addition of
acid or base.
A method often used in titrations is to wait until an accepted pH drift (often 0.01 min-1)
and retain the last pH measurement. This drift value was often used, whatever the titrated
solid, from low specific area oxides as ZnO (2-13 m2/g [14, 15]) or Fe2O3 (22 m2/g [10]) to
oxides whose specific area is greater than 100 m2/g as -alumina [26, 27, 28] or silica [29]. In
a few recent articles concerning -FeOOH (94 m2/g) [30] or -alumina (140 m2/g) [31],
authors have detailed the conditions of potential drift of the glass electrode (respectively < 0.2
mV/min and < 0.1 mV/h), but this is an exception.
To illustrate the influence of this method, we have represented in Fig. 5 an example of
the evolution of the calculated surface charge of hematite versus pH drift. The drift was
calculated from the derivative of pH versus time, after smoothing. In this case, measurement
of pH when the drift is 0.01 pH unit per minute would result in an error of 0.043 at/nm2 for
the surface charge.
Effect of the titration speed on modeling
A comparison of continuous titrations at 3 different speeds and 3 batch titrations at
different equilibration times was done for alumina. Since most of experimental points were
obtained in a pH range where the surface charge is positive, in a first series of calculations, we
have determined only the K+ acidity constant of the 2-pK model.
In a pH range where negative sites are negligible, the intrinsic acidity constant Ki+ can
be expressed as:
K i 
([MOH]tot -[MOH2 ])(H ) RT
[MOH2 ]
where [MOH]tot is the total density of surface sites, [MOH2+] the concentration of
protonated hydroxyl groups,  the electrostatic potential. [MOH2+] is equal to the surface
charge Q and [MOH]tot is equal to the maximum achievable charge Qmax.
In the constant capacitance model, the previous equation can be expressed as:
K i 
(Qmax -Q)(H ) RTC
where C is the capacitance of the double layer at the surface of the solid, in the constant
capacitance model.
This expression can also be written:
pH  pK i  log(
(Qmax -Q)(H )
where Q is expressed in C/m2. The parameters pKi+, Qmax and C were fitted by the least
squares method from the pH versus Q variation for the 6 titrations, for Q > 0.1 C/m2 (0.62
at/nm2 ). Since the surface sites are far from being saturated, there is a large uncertainty on
Qmax. In order to compare the acido-basic parameters pKi+ and C, we have fixed Qmax, whose
value must not depend on the titration speed. This value was chosen as 2.5 at/nm2 or 0.4 C/m2,
according to previous saturation experiments [7].
Another series of calculations was performed using FITEQL code [18], for the
determination of both pKi+ and pKi- in the constant capacitance model in the range 3.5 < pH <
10.5 and –0.2 < Q < 0.3 C/m2. Again, due to the large uncertainty on Qmax, this parameter was
fixed at 0.4 C/m2.
Both methods indicate that the calculated acid-base parameters depend on the titration
speed (Table 4). When pKi+ alone is calculated, its values increase when the speed of titration
is decreased or when the time of equilibration is increased. The capacitance value is lower in
batch titrations, but we need to recall that this parameter results from fitting and it does not
mean that the electrostatic conditions at the surface depend on the titration conditions.
pKi+ and pKi- also increase in the same manner, when calculated by FITEQL, in the case
of continuous titrations. For batch titrations, results show large variations without clear
relation to equilibration time. This is probably connected to the scatter in the experimental
batch results.
4. Conclusion
All experiments and calculations carried out in the present study showed that there is a
large influence of kinetics on the accuracy of surface charge and pH values obtained by acidbase titrations of suspensions. Moreover, these uncertainties in acid-base titrations end up
with the uncertainty in acid-base parameters. Kinetic effects largely depend on the nature of
oxide studied. Porosity is one of the parameters which slow down the equilibration of proton
exchange at the surface, but effects similar to diffusion in solid matter can also play a role.
Kinetic studies on the variation of pH versus time after addition of an acid or base to a
suspension should be performed each time a new solid is studied, in order to develop a
titration protocol capable of leading to the minimum uncertainties in pH and surface charge.
“Fast titration procedures”, proposed with the aim of limiting the measurement to the
“surface” reaction, seem not to be adequate. The utility of such procedure would necessitate
developing a method capable of differentiating between the "fast" step (attributed to the
surface) and the "slow" step (attributed to the bulk).
F. Adekola is grateful to the French government for the award of a fellowship (2005/2006).
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Table 1 Morphological characteristics of the solids used in the study
dimensions (nm)
BET surface
area (m2/g)
t-plot surface
area (m2/g)
t-plot micropore
volume (mL/g)
surface area
BJH pore
volume (mL/g)
Agglomerates 50-100
Agglomerates 100200 µm
<8 nm: 4%
20-80 nm:
<8 nm: 5%
20-80 nm:
<8 nm: 11%
10-20 nm:
<8 nm: 76% <8 nm: 11%
20-80 nm:
20-80 nm:
* unavailable data (amorphous solid)
BJH desorption
size distribution
Table 2 Experimental conditions used for the study of the influence of titration speed on the
variation of calculated surface charge vs pH for hydrated -alumina. In each pH range, the
volume of 0.1 M HNO3 additions and time interval between additions are indicated
pH 5.5-9
pH 4.6-5.5
pH 3.5-4.6
pH 2-3.5
0.01 mL
0.01 mL
0.02 mL
0.02 mL
5 min
3 min
3 min
2 min
8 min
5 min
4 min
2 min
15 min
8 min
5 min
2 min
Table 3 Apparent diffusion coefficients D (m2.s-1) of slow process calculated from titration
experiments with 1 hour interval between additions.
D pH<7
D pH>7
6.0 10-13
1.1 10-12
4.5 10-20
5.3 10-20
Hematite 99.8%
1.7 10-20
1.7 10-19
Hematite 99.945%
6.6 10-20
7.7 10-20
1.7 10-12
3.9 10-12
Table 4 Intrinsic acidity constants pKi+, pKi- and capacitance C of -alumina calculated from
titration curves with different speeds, and batches with different equilibration times.
Calculations were performed for pKi+ alone by least squares method (LS), for pKi+ and pKiby Fiteql code. The site density was fixed at 2.5 at/nm2.
C (F/m2)
C (F/m2)
Batch 7 h
Batch 18 h
Batch 48 h
Figure captions
Fig. 1 Comparison of continuous and batch titration curves of -alumina, represented as the
surface charge versus pH. Conditions of continuous titrations (Titr1, 2, 3) are indicated in
Table 2. Duration of batch titrations is indicated on the figure. Lines are guides to the eye and
do not result from fitting.
Fig. 2 Examples of pH variation versus time for continuous titration with 1 h interval between
additions. Positive time scale corresponds to additions of acid. Negative time scale
corresponds to additions of base.
Fig. 3 Examples of normalized surface charge (achievement factor F) variation versus time
after acid addition: experimental values and fitting by diffusion in spherical particles.
Fig. 4 Example of influence of time after addition of acid on pH and calculated surface charge
Q for hematite 99.8. pHinf and Qinf correspond to steady state of pH variation.
Fig. 5 Example of influence of pH drift on calculated surface charge Q for hematite 99.8. Qinf
corresponds to steady state of pH variation.