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Variety of Aggregation and Growth Processes of Lanthanum Fluoride as a Function of La/F Activity Ratio. 1. Potentiometric pF-Stat and DLS Study of LaF3 Nada Stubicˇar*

CRYSTAL GROWTH & DESIGN 2005 VOL. 5, NO. 1 113-122

Laboratory of Physical Chemistry, Chemistry Department, Faculty of Science, University of Zagreb, Zagreb, Croatia Received February 11, 2004;

Revised Manuscript Received September 7, 2004

ABSTRACT: The constant composition method, in particular the potentiometric pF-stat method, was proved to be applicable and convenient to study the kinetics of submicrometer size lanthanum fluoride particles besides that of macroscopic crystals. The state of ions in supersaturated solution (due to hydrolysis and other chemical reasons) (La-to-F ratio, the attained pH, ionic strength, temperature, the way and speed of mixing, etc.) was kept constant during the whole growth process. Steady-state kinetics was achieved after about 150% of growth as regards the initial mass of seeds. Because of very reproducible results, this method is prominent for the preparation of new materials with respect to morphology and microstructure, even in nanosize scale (parts 1 and 2), far from thermodynamic equilibrium and at nonstoichiometric condition. Depending on the activity ratio of La- to F-ions in supersaturated solution used for particle growth, i.e., whether lanthanum nitrate or potassium fluoride is in a great excess, the kinetics and the mechanism are different. For an excess of lanthanum nitrate over potassium fluoride and at the attained pH of about 4, the first-order rate of growth was determined and submicrometer scale particles were obtained and were characterized by the dynamic light scattering method. Fast relaxation decay times measured at scattering angles of around 90° and slow relaxation decay times measured at smaller scattering angles (maximum at around 35°) with the assumption of spherical particles gave an estimation of dH ∼ 8 nm and dH ∼ 24 nm for primary particles and larger vesicles, respectively. In this size region, the rod shape may be assumed also and we got for the biaxial ratio of an oblate ellipsoidal cross-section an aspect ratio of F ) b/a ) 3.0 ( 0.2. Approaching the stoichiometric condition in supersaturated solution and the isoelectric point at pH ) 5.6 ( 0.3, the power law kinetics prevailed and the order of growth was determined to be 4.0 ( 0.2. All the data from pH 4 to 6 can be well fitted to a linear fit (ln vg vs ln s, supersaturation) with the overall order p ) 3.3 ( 0.4 and a growth rate constant kp of about 7 × 10-12 m s-1. This rate order supposes the polynuclear growth mechanism. Layer-by layer (sheet-by-sheet) growth on the surface of anisotropic LaF3 crystals, several hundred microns in size, was observed on polarizing optical microscopic pictures. In these images, the systems prepared in the acidic pH region around 4 are shown as anisotropic dendrites (loose aggregates). Analysis of the static light scattering data proved that theoretically aggregates of sol particles are self-similar fractals with the universal fractal dimension determined to be Df ) 2.10((0.05) for the reaction-limited aggregation, as in a vast number of cases in the scientific literature. Introduction The results which will be reported in these two papers are the continuation of our work partially presented at the 12th and 14th ECIS Conferences,1,2 now with special emphasis on the preparation, i.e., kinetics and mechanism of growth and aggregation, and on the characterization of submicrometer and nanometer scale particles of lanthanum hydroxide and fluoride. Systems have been prepared from La(NO3)3 and KF solutions at 25 °C using the classical spontaneous precipitation, as well as by the steady-state pF-stat and pH-stat method. Particles being formed in the acidic solution of nonhydrolyzed or partly hydrolyzed La-ions at pH between 3.9 and 5.3 (part 1) and those in neutral or mildly alkaline solution of completely hydrolyzed La-ions at pH between 6 and 7.3 (part 2) (i.e., below and above the experimentally determined isoelectric point, IEP) are quite different. The pH of the supersaturated solution was attained depending on the concentration (or on the activity) ratio of La- to F-ions, i.e., whether La(NO3)3 * Address: Marulic´ev trg 19/II, P.O. Box 163, Hr-10001 Zagreb, Croatia. E-mail: [email protected] or [email protected]. Tel: ++3851- 48 95 529. Fax: ++385-1-48 95 510.

or KF was in a great excess, without acid/base and without neutral electrolyte addition. This means that systems were very far from the stoichiometric ratio 1/3. Roughly, it holds for the systems studied in part 1 that a(La)/a(F) . 1, and a(La)/a(F) , 1 for those presented in part 2. The pF-stat kinetics of crystal growth and of aggregation of the stoichiometric c(La3+)/c(F-) ) 1/3 systems is presented in part 1. Simultaneously with an increase in supersaturation, reasonably, the ionic strength was increased in the present study (without separate addition of the neutral KNO3 solution in each run). It was proved many times that growth of crystals of the sparingly soluble salts in mineral aquatic and in biological systems mostly occurs under nonequilibrium and nonstoichiometric conditions. At a given thermodynamic driving force for the crystallization (given supersaturation), the rate cannot be defined solely in terms of the activity product of lattice ions, but it depends on the ratio of lattice ion activities as well.3,4 The precipitation three-dimensional diagram (called Tezak’s precipitation body) for a pair of cation-anion, lattice ions [A] and [B], at a defined time, usually

10.1021/cg049938k CCC: $30.25 © 2005 American Chemical Society Published on Web 11/18/2004

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Crystal Growth & Design, Vol. 5, No. 1, 2005

approaching equilibrium state, was assigned by Nielsen as isograms for constant growth rate in a precipitation diagram. In “ideal” very dilute solutions, the lines of a constant ionic product, which divide the region with and without detected solid phases at a given time (that is never true thermodynamic equilibrium), are straight lines. Submicrometer particles in electrolyte solution (without special stabilizers like in this case) are hardly ever thermodynamically stable. The position of the line (coordinates) depends on the experimental technique of detection, except on the time of reaction. However, moving to higher concentrations the state of ions in solution may have been changed because of chemical reasons such as formation of complexes in the case of AgCl,3,4 ion-pairs in the case of CaCO3 and BaSO4,5,6 hydrolytic products in the case of Al(OH)3,7 and so forth. Along with the higher rate of growth, the lines became strongly bent curves. Moreover, exactly the same rate of growth may be achieved by different mechanisms of growth leading to different structural and morphological phase formation, as was presented by us for PbF2 sparingly soluble salt.8-10 The steady-state approach, known as the constant composition method11 (CCM), and the dual constant composition method,12 in particular the pF-stat method which we use, requires that the titrant addition rate, the temperature, and the composition of the working solution (that means the state of ions and the pH, which were checked several times during the run) remain constant. This means that the volume of the working solution has been changed only by a few percent (small volume increments were added) and the volumes added and withdrawn as samples at given times during the run were almost the same. Hence, this enables us to compare always the same interval of the percent of growth as regards the initial mass of seeds (always the same mass of seeds was added, in the limit of error) and to compare different rates and mechanisms of compositionally different systems (in different runs). In the case where hydrolysis plays a main role, the position of the solubility line at the particular pH gives us the first sight knowledge that a different mechanism of particle growth and aggregation is taking place at spontaneous precipitation, as we will present and see here in the precipitation diagram of LaF3. To elucidate the different mechanisms of growth, we used several experimental techniques for particle characterization of the samples during the run: polarizing optical microscopy (POM),13,14 static and dynamic light scattering (SLS and DLS),15-17 as well as small- and wide-angle X-ray scattering (SAXS18 and WAXS) and X-ray diffraction (XRD) in suspension (noninvasive techniques) and on dry powder. The latter two will be presented in part 2. While pF-stat crystal growth and POM give us information about kinetics of crystal growth, morphology, and anisotropy of the several hundred micron size LaF3 crystals, DLS and SLS studies are a rich source of information about the dynamics and structural properties of submicron-size LaF3 (confidently between 10 nm and 3 µm with the instrument used).17 Besides, it was proved by us2 that pF-stat kinetic study might be applied to this scale region as well. SAXS study (part 2) gives us information about nanosize LaF3 (1-10 nm) particles,18 which have

Stubicˇar Table 1. Characteristics of Lanthanum Fluoride Seeds DLS of the seed suspension: dH (wa ( sd) ) 360 ( 16 nm, polydispersity index dw/dn ) 1.01, pH ) 5.85 electrophoretic mobility µ ) 3.14 × 10-4 cm2/V s, zeta potential σ ) 38.95 mV SAA determination of the dry seeds (dried for 1 h at 100 °C) ) 56.13 m2/g (BET method using He/N2 ) 70:30) XRD analysis of the dry seeds: 6 most intensive peaks of the hexagonal LaF3 (JCPDF card no. 32-483)

been formed by the phase transformation from an amorphous gel. Experimental Section A. Materials and Methods. Chemicals KF and La(NO3)3 were of analytical grade, p.a., Merck, Darmstadt. The twicedistilled water, which was used throughout the experiments, was filtered through a membrane filter (0.1 µm Millipore, Bedford, MA), as were the prepared stock solutions employed. The pH value of the water used was 6.0 (without CO2). Both pF and pH potentiometric measurements at (298.2 ( 0.1) K were performed in the pF-stat kinetic experiments using a Fselective electrode (Metrohm, Switzerland) and a glass electrode combined with two saturated calomel electrodes as references. Namely, these were also essentially the pH-stat experiments; pH was measured during the run and at the end of the run. The addition rate of titrants (which were in the equivalent ratio 1:3) was proportional to the rate of growth. The increments of volume (mass) addition were governed by a decrease in the activity of fluoride ions, a(F-), in the working solution (ws). So the difference against the preset potential was 2 mV or maximum 3 mV (∼3 × 10-5 mol L-1 F-) in few cases. The average of the six ∆Vtit/∆t values (changes of the added volume of both components in time), which were recorded at the defined constant mass ratio ∆(m/m0) ) 2-7, i.e., 100-600% of growth, was taken in calculation of the rate. The error is about 2-3%. The initial mass of seeds added was in all cases m0 ≈ 12 mg (4 mg/mL). Seeds were prepared by spontaneous precipitation: dropwise alternating addition of titrants up to a given mass concentration. The experimental setup and the procedure have been described previously.2,8 That is, first the increments of KF reactant were added slowly to 200 mL of water, to check first the Nernstian response of the F-electrode, which is proved to be one of the most ideal electrodes. The incremental addition of La(NO3)3 up to the given supersaturation was followed, and then the seeds were added into the stable supersaturated solution and the rate measurements were started. Seeds were obtained by slow dropwise alternating addition of both components. Some of the characteristic properties of LaF3 seeds, already published,1 are given also in Table 1. The DLS measurements on the seeds and the analysis are shown in the results section of this work. The size and particle size distribution (PSD) of the submicrometer particles were determined by DLS (dH, hydrodynamic diameter determination) using the DLS-700S Otsuka (Japan) spectrophotometer with an Ar+ laser operating at λ ) 488 nm and at the optimum power of 15 mW,17 using the samples withdrawn at the definite times as was mentioned above. Aliquots of 10 mL without dilution were used for immediate DLS particle sizing in a φ ) 12 mm cylindrical cell placed in the immersion index matching liquid, di-n-butyl-phthalate, using a temperature control by the thermostat circulator (298.2 ( 0.1 K). DLS-700S operates by the homodyne method; i.e., interference of the Rayleigh scattered light by passing through two pinholes satisfies the coherent condition. The histogram method and Auto 1 type of particle size analysis, based on the non-negative least-squares procedure, were used for the calculation of particle size distribution, as was already reported.19

Effect of Activity Ratio on LaF3 Crystal Growth

Crystal Growth & Design, Vol. 5, No. 1, 2005 115

Table 2 complexes

log K (298.15 K, I f 0)

LaNO32+ OHLaOH2+ H2F2 (aq) HF2HF (aq)

1.5 -14.0 -9.0 6.77 3.75 3.17

complexes LaF2+ LaF+ dissolved solids: LaF3 (s) La(OH)3 (s)

log K (298.15 K, I f 0) 3.60 5.56 -17.95 -21.20

J ) [(∆Vtit/∆t)(ctit - cws)]/m0 SSA

Furthermore, particles in the samples were directly visualized in suspension and in the dry state (systems became concentrated after slow evaporation of water under the microscopic glass due to illumination with light) using the polarizing light microscope with a λ plate (see e.g. p 38 in ref 13), Leitz-Wetzlar (Germany), equipped with an automatic camera. B. Calculation of Supersaturation and pF-Stat Growth Rate. Kinetic crystal growth experiments were performed from the solutions of constant supersaturation. The activity of the ions, pH, ionic strength (I), and activity coefficient (y1,() were calculated using the computer program Mineql+ (ref 20). The constants listed in Table 2 (taken from refs 21-23) were taken into account. Taking these stability constants, the pH values calculated were higher either by 0.2 or maximum by 0.5 pH units than the pH values measured in acidic and in neutral or mildly alkaline solutions, respectively. This was considered as satisfactory. Although a diversity of values exists in the literature, the refinement of the constants is not necessarily needed for the pH region investigated in this paper (0.2 pH units are in the limit of the experimental error). However, the kinetic study of more alkaline solutions needs the introduction of the other hydrolytic stability constants (e.g., for polynuclear species). The driving force for crystallization is the difference in chemical potential for the given salt in supersaturated and in saturated solution:

∆µ ) νRT ln(

∏a

νi i

/KsoL)T,p1/ν

(1)

Here the supersaturation ratio is

∏a

s)(

νi i

/KsoL)1/ν

The pF-stat growth rate (J in mol s-1 m-2), which is proportional to the (dVtit/dt) times the effective concentration of the titrants (titrants were consumed to keep the supersatutation constant and for the growth of solid phase), was normalized to the mass of seeds and to the specific surface area of LaF3 seeds:

(2)

(3)

The SSA was determined to be 56.13 m2 g-1 (BrunauerEmmett-Teller (BET) method) (Table 1).1 The linear growth rate (vg/m s-1), that is, growth in one dimension perpendicular to the crystal surface,5 is

vg ) JVm

(3a)

Here Vm is the molar volume of the crystalline LaF3 (Vm ) 4.36 × 10-5 m3 mol-1). The rate vg is related to the overall order of growth process p by the empirical, power law equation, generally accepted and proved for a long time (based on the Burton-Cabrera-Frank theory for crystal growth, 1951)6,24-28 (and some references therein):

vg ) kpσp

(3b)

Here kp is the rate constant for polynuclear growth. Hence, in logarithmic form of eq 3b, ln vg vs ln σ or ln s, the slope is equal to the overall order of reaction p and the intercept to the ln kp. Equation 3b is based on the Christiansen and Nielsen model of the rate of nucleation-growth as a power law on supersaturation, where p is originally defined as the number of ions in the critical sized cluster (nucleus). The simplicity and consistency to analyze crystal growth data were demonstrated recently for many salts having relatively high solubility products.28 C. DLS Data Evaluation. The kinetics of growth of submicrometer size particles2 was double-checked also by the DLS measurements, using the change in median (50% frequency) mass (weight) average (wa) hydrodynamic diameter (dH) with time of the withdrawn samples, as was described above, expressed as ∆dH(median)i/∆dH(median)1. The measured autocorrelation function obtained by the homodyne method is given by

G2(τ) ) 1 + R|G1(τ)|2

(4)

And the relative supersaturation is

σ)s-1

(2a)

∏aiνi is the ionic activity product, νi is the stoichiometric coefficient of ion i in the formula of a sparingly soluble salt (ν ) Σνi), and KsoL is the standard thermodynamic solubility product. When s ) 1, σ ) 0, and ∆µ ) 0, the system is in equilibrium (the solution is only saturated with respect to the solid phase). Hence, the relative supersaturation σ for this particular (1:3) salt was calculated using the equation

σ ) [(c(La3+) c3(F-) y1,(12/KsoL](1/4) - 1

where τ is the correlation (decay) time and G1(τ) is the firstorder correlation function defined by

G1(τ) ) exp(-Γτ)

(5)

Γ ) q 2D t

(6)

The decay rate is

DLS experiments can be used for the study of relatively rigid rods, besides spheres.15,16 Generally the decay rate is then given by

(2b)

Here the average activity coefficient y1,( of the positive and negative univalent ions was calculated from the simple Debye-Hu¨ckel equation (without the Davies extension). The standard thermodynamic solubility product KsoL ) 1.12 × 10-18 was taken into account,21 although our value1 determined by the titration for the fresh precipitate was much higher, that is, KsoL ) (2.95 ( 0.02) × 10-17 at 298.15 K and I ) 0, calculated using the known Gran plot for the analysis of the titration curves, see e.g. Figure 1 in ref 1. Our value is in agreement with the value reported in the other primary publications.22,23 That greater value is reasonable because the greater the reactivity, the smaller the size and the more nonperfect the crystal particles, as is known.

Γ ) q2Dt + 6Dr

(7)

where Dt and Dr are translational and rotational diffusion coefficients of rods, respectively. And the scattering vector, q, is

q ) (4πn/λ) sin(θ/2)

(8)

It was not needed to perform the depolarized DLS (DDLS) experiments as long as qL < 4 (L is the length of rods). In this angular regime, only translational motion contributes significantly to the decay rate, Γ, and the contribution of rotation of the rods may be neglected.15,29 This holds up to L ≈ 400 nm using our instrumental setup. Hence, we calculated the

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Stubicˇar

Figure 1. Precipitation diagram of lanthanum fluoridehydroxide determined by the spontaneous precipitation by mixing KF-La(NO3)3 solutions, at 298.15 K and 60 min after mixing the components (time approaching the equilibrium state). Circles denote the lowest pair of concentrations with the solid phase detected by LS and the corresponding measured pH of these systems. Fitting (dashed) line I (of the points in the acidic pH region around 4) is parallel to the equivalence line (shifted toward smaller concentrations), and dashed fitting line II (of the points where pH is around 6) coincides with the theoretical solubility line for 3:1 salt. Stars denote some systems with the determined hydrodynamic diameters, dH, in nanometers. Squares denote concentrations of salts in the working supersaturated solutions (ws) used for the kinetic measurements presented in Figure 3. Adapted from ref 1. hydrodynamic diameter, dH, from the translation diffusion coefficient Dt (Stokes-Einstein equation):

dH ) kBT/3πη0Dt

(9)

where kB is the Boltzmann constant, T is the absolute temperature in kelvin, η0 is the viscosity of the solvent, and dH is the diameter of spheres or longer semiaxes of rods, respectively. The fractal dimension, Df, of these particles has been determined from the SLS data.30,31 The total Rayleigh ratios were plotted as a function of q using the known relation

R(q) ∝ q-Df

(10)

as was reported by us earlier.2 The q values in our case are between 5.09 × 10-3 and 3.32 × 10-2 nm-1 (θ ) 17-150°), and linear log R(θ) used for Df determination was up to 2 orders of magnitude. The linear part of log R(θ) against q was measured over the range equivalent to approximately 30-333 nm in real space. Radii of gyration, Rg, were determined from the SLS measurements using the known Zimm plot.

Results and Discussion A. Spontaneous Precipitation (Quasi Thermodynamic Equilibrium). In Figure 1 is presented the precipitation diagram of the LaF3 solid phase prepared by mixing the precipitation component KF-La(NO3)3 solutions, at 298.15 K and 60 min after mixing (time taken as approaching the equilibrium state); 1 day aged systems were also checked. Circles denote the lowest pair of concentrations with solid phase detected by LS and the corresponding pH of these systems. (The diagram is very informative for further explanation of

Figure 2. Measured and calculated pH values (Mineql+ program) of the systems at the solubility boundary (shown in Figure 1) as a function of La(NO3)3 concentration (KF was varied for each run). The solid line is the sigmoidal Boltzmann fitting curve calculated.

the results and hence it is redrawn from ref 1.) All the experimental data (circles) may be well fitted to a Gaussian curve. The left side of the Gauss curve corresponds to the systems with the ratio a(La)/a(F) . 1 and the pH in the acidic region of about 4, and the right side corresponds to the systems with a(La)/a(F) , 1 and the pH of about 6 (calculated using the Mineql+ program). This suggests on first sight that two different precipitation processes are taking place, which are as follows: In the first concentration region (region I), systems at the left side of the Gauss curve, the solubility line is parallel to the equivalence line which means generally neutralization of charges. Due to the charge suppression (finally neutralization) of the positively charged stable sol particles (stabilized by the potential determining La3+ ions which prevail in this region), the repulsion between charged particles decreases, because of the increase in ionic strength along the equivalence line toward the left. This is the main mechanism of aggregation and growth of submicrocrystals, and microcrystals spontaneously precipitated. In the second case (region II), the right side of the Gauss curve, the fitting line of the experimental data coincides with the calculated solubility line for 1:3 salts; hence, the ionic solubility process is dominant in this region. It is the pH region where La-hydrolyzable species prevail in solution (the La-ion is coordinated by 9 water molecules); hence the amorphous gel, polymer lanthanum hydroxide, was formed first as a precursor. It is the less soluble phase (the solubility product is more than 2 orders of magnitude lower as compared to that of LaF3). The amorphous phase (the precursor being formed instantly) undergoes crystalline phase transformation (as we will confess and corroborate more in part 2 of this work). Calculation of the activity ratios of La- to F-ions of systems at the solubility line shows that a(La)/ a(F) . 1 for the systems at the left side (region I) and opposite for the systems at the right side (region II) of the Gauss curve, respectively. Figure 2 shows the pH values measured and calculated for the systems at the solid/solution boundary as a function of the solid-phase concentration, i.e., the


concentration of lanthanum nitrate. It determines the solid-phase concentration, because the La(NO3)3 concentration is either in equivalent ratio or in shortage toward KF. The pH curve looks like a two-step titration curve; that supports the above-mentioned mechanism of precipitation in the concentration region I. Few of the particle hydrodynamic diameters (median weight average) determined by the DLS are marked at the curve. The first inflection in the S-shaped pH curve is at -log c(La(NO3)3 ) 2.01 ( 0.05 and pH ) 4.1 (in the pH region 3.9-4.3), and the second very steep one is at 3.00 ( 0.02 and pH ) 5.6 (in the region 4.6-5.9), determined by the regression of a sigmoidal fit. Below pH ) 4.1, the La3+ species prevails and particles are strongly, positively charged;1,32 that is the reason for their high colloidal stability. At pH ∼ 4.5 species LaF2+ and at pH ∼ 5.1 species LaF2+, respectively, prevail. At pH g 6 neutral mono- and polynuclear La(OH)30 (aq) species or La(H2O)x species dominate, as was calculated. The pHexp and pHcalc agree pretty well but better in the acidic region (as is obvious in Figure 2). B. Kinetics of Crystal Growth by CCM (pF-Stat Growth). The results and analysis of the pF-stat rate (eq 3a) of crystal growth of LaF3 particles on crystal seeds in metastable, supersaturated solution measured potentiometrically at 298.15 K are presented in Figure 3a-c. The ln vg data in the whole pH region (ln vg vs pH) may be well fitted to a sigmoidal equation, as is common procedure for a simple, sigmoidal, titration curve (where cationic and anionic species prevail below and above an equivalence point, respectively), Figure 3a. In the restricted region however, related to the growth only from the acidic supersaturated solution at pH 3.95-4.4 (start of region I in Figure 1 and Figure 2), the rates are slow (stable blue sol formation) and the data may be fitted to a linear function with the slope equal to 1 (1.1 ( 0.1), corresponding to the first-order kinetics. In this case, the main mechanism of growth is assumed to be diffusion of ions through solution toward the crystal surface, their adsorption on the crystal surface, dehydration, and incorporation into the crystal lattice.5 Further a layer-by-layer formation2,26,33 was observed on micrographs (see later). When crystal growth is performed from the solutions at pH around 5.6 (steep part of the pH curve, Figure 2), then the rate starts to be about 10 times greater and the size of particles is 10-30 times larger (micrographs of the samples). The order of reaction in the vicinity of pH 5.6 was calculated to be 4.1 ( 0.3, which assumes the fourth-order kinetics of growth. The mechanism that governs growth in the last case may be polynuclear growth on the crystal surface.5,33 The fourth-order dependence of crystal growth rate on supersaturation at high supersaturations means that it is always limited by either surface reaction or the integration into the surface step. The pH-stat (pF-stat) rate as a function of the pH in supersaturated solution is shown in Figure 3a, where ln vg vs pH data are well fitted to a sigmoidal Boltzmann equation (R2 ) 0.99) (as was told at the beginning). We calculated the center of the S-curve at pH(0) ) 5.0 ( 0.1 with the greatest change in rate in the pH region 5.0 ( 0.3; the rate increases from (1.4 ( 0.6) × 10-10 (calculated from fitting parameter A1) to maximum value (7.6 ( 2.0) × 10-9 m s-1 or to about


Figure 3. (a) The pF-stat kinetics of crystal growth of lanthanum fluoride as a function of pH in the supersaturated solution used for LaF3 crystal growth. Data were fitted to the sigmoidal Boltzmann equation (R2 ) 0.99149). From the parameters ln A1 and ln A2, we get vg(min) ) (1.1 ( 0.8) × 10-10 and vg(max) ) (6.2 ( 1.1) × 10-9 m s-1; the center (xo) of the S curve is at pH ) 5.0 ( 0.1 and the pH interval of the greatest change in vg is at pH ) 5.0 ( 0.3, by means of dx ) 0.32 ( 0.06, where the maximum change in vg ) 1 × 10-9 m s-1 (from (A1 + A2)/2). (b) The pF-stat kinetics of crystal growth of lanthanum fluoride at given constant pH values from 4 to 6 as a function of supersaturation, according to eq 3a together with the linear fit of the data (R ) 0.9917), from which the order of growth and the polynuclear rate constant were calculated. Error bars correspond to 2% of the linear rate of growth at low supersaturations and 5% at high supersaturations, respectively. (c) The pF-stat growth rate: (ln vg) as a function of (1/ln s) data fitted to an exponential decay function of the first order, from which a pre-exponential factor A was calculated, according to an Arrhenius type of equation for nucleation theory (refs 25 and 27).

118


7.6 nm s-1 (calculated from parameter A2). At about IEP pH ) 5.5, the linear rate of growth is about 5-6 nm s-1. However, all the data from pH 4 to 6 are excellently fitted to the linear function of the ln form of eq 3b, yielding the overall order of growth p ) 3.3 ( 0.4. Taking 2% error at smaller s and 5% error in vg calculated for higher s-values, we see that both p ) 3 and 4 are in the limits of error. The overall rate constant for polynuclear crystal growth kp is calculated to be (7.1 ( 0.7) × 10-12 m s-1 (ln kp ) ln A, intercept). This is a very low value and very sensitive to the changes in slope; hence fitting the data to p ) 3 (third-order reaction) we calculated kp ) 1 × 10-10 m s-1, that is, growth of 0.1 nm s-1. The fitting line with slope 4 (fourth-order reaction) is also in the limits of experimental error. From the fit of the data (ln vg vs 1/ln s) with an exponential decay function of the first order, shown in Figure 3c, we obtained the pre-exponential factor A ) (2.7 ( 0.4) × 109 m s-1 (∼3 × 1012 m-3 s-1). This value is in the range of approximately 106-1013 m-3 s-1, as was given and discussed in ref 25, for some sparingly soluble salts, although the theoretical value for the factor A should be approximately 1033(3 m-3 s-1, according to the Arrhenius type equation for the rate of surface critical nucleus formation in nucleation theory.25,27 The pF-stat kinetics of growth from the solution at pH 6 and above has not been studied in detail yet. This rate of transformation in the pH region above 6 may be studied and published elsewhere. C. Characterization of Particles by the DLS and SLS Method. The DLS data of the seeds used for the kinetic measurements described above are shown in Figure 4. The second-order autocorrelation function, Γ(2), obtained by the homodyne method together with the fitting line (upper part of the figure), as well as the histogram of the mass (weight) average hydrodynamic diameters converted from the Γ(2) distribution (lower part) show that seed particles are of submicrometer size and rather monodisperse, with the polydispersity index dw/dn ) 1.01, measured at θ ) 90°. The other systems prepared beyond the same pH and at the 1/3 concentration ratios are more or less similar (bimodal with smaller frequency of smaller sizes), only the average size increases linearly with the concentration of the components. The supersaturated solutions (ws) (more or less clear to the naked eye) used for crystal growth of the seeds were also heterogeneous systems although with nanoand submicrosized particles from about 10 nm above (with increasing concentration of LaF3 and simultaneously with increasing the pH of solution), as was determined by SLS and DLS. Radii of gyration (measured at θ ) 20-150°, 10° step), i.e., diameters calculated from them, and median weight (mass) average hydrodynamic diameters (measured at θ ) 20°) are presented in Figure 5, as a function of LaF3(s) concentration. Both sets of data show a linear dependence on concentration. Extrapolations to zero concentration give diameter values of about 8 nm for primary particles, which is about the lower limit of the size determination with our instrument.

Stubicˇar

Figure 4. Dynamic light scattering measurements of the LaF3 seeds used for kinetic measurements. (a) Correlation curve. (b) Histogram of the weight (mass) average hydrodynamic diameters together with the cumulative frequency curve.

Fractal dimensions of particles in the ws, deduced from the negative slope of the double logarithmic plots of the SLS data (eq 10) reported by us,2 increased from 1.36 for the diffusion-limited aggregation (solution at pH 4) and reached the final value of polydisperse aggregates Df ) 2.10((0.05) for the reaction-limited aggregation (solution at pH 5) and imply also that they are fairly loose aggregates. This will be corroborated also by the observation on the photomicrographs described below in paragraph D. So, the universality of the aggregation process regardless of the particular system is here once again justified.30,31 Titration of KF with La(NO3)3 proceeded at the beginning with the appearance of a glassy, gel phase


Figure 5. Radii of gyration, Rg, and the median hydrodynamic diameters, dH, of the particles in supersaturated solutions used for growth experiments as a function of LaF3 (s) concentration.


Figure 7. Hydrodynamic diameters (weight average) as a function of scattering vector, q, measured on the system obtained by titration of KF with La(NO3)3 from pH ) 6.6 up to the concentration ratio 1:1 and pH ) 4.5.

Figure 8. Hydrodynamic diameters (weight average) as a function of scattering vector, measured 8 years after preparation by the pF-stat method (CCM) up to 600% of growth from the solution with the concentration ratio 1:3 and at constant pH ) 3.95 during the growth. This blue colloid system (sol) is rather stable even after such a long time.

Figure 6. Panels a (upper part) and b (lower part): Some of the correlation functions at different angles obtained on the system presented in Figure 7 and Figure 8, respectively.

(the precursor appeared instantly) at c(La)/c(F) ) 1/6 and pH ) 6.6 (details in part 2), followed by blue/white sol precipitation at c(La)/c(F) ) 1/2.3 and at pH ) 5.95 containing the particles mainly of dH (wa) ) (129 ( 6)

nm (at θ ) 20°), and further by changing the ratio to c(La)/c(F) ) 1/1.4 and dropping the pH to 4.5. The correlation curves measured at certain angles of the last mentioned system (pH ) 4.5) are presented in Figure 6a, and the corresponding dependence of dH on the scattering vector is shown in Figure 7. The other system presented is also in the acidic pH region, but at lower pH ) 3.95, which was kept constant during the kinetic constant composition experiment (CCM) up to 600% of growth at the ratio of the titrants c(La)/c(F) ) 1/3. The system is a colloidal, stable blue sol even 8 years after preparation, Figure 6b and Figure 8. In that case, the particles are much smaller (because of lower pH), we may say in the nanoscale region. Both mentioned systems (Figures 6-8) show the same pattern of two distinct relaxation modes: a slow mode at smaller angles, at around θ ) 35°, and a fast one at larger angles of the correlation curves, at θ ) 90°. The q position of the maxima at ∼35° is rather close to each other

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Stubicˇar

Figure 9. Optical microscopic pictures of the heterogeneous systems prepared from La(NO3)3-KF acidic and about neutral aqueous solutions, respectively. (f-h) Anisotropic dendrites, with various interesting features, and stable sols prepared by the pF-stat method at pH ) 4.0 and with an excess of La(NO3)3, grown up to 600% of growth (m/m0) ) 7, using the pF-stat method. The magnification is 230× and 150×, respectively. (o,p) Large crystals with rhomboidal and hexagonal structure (as was determined by the XRD method, see part 2) several hundreds of microns large, prepared by the pF-stat method at moderate supersaturation, concentration ratio 1/3 and pH ) 5.85, grown also up to 600% of growth (m/m0 ) 7). The magnification is 150×. For other details about the preparation procedure, see the experimental part.

(probably the same) for both systems prepared differently (that is, Figure 7 by titration and Figure 8 by CCM). The qmax corresponds to the real space of 108.3 nm for the former case and 98.3 nm for the latter case, respectively. Also about the same maximum value in dH was measured at θ ) 120° (backward direction of the scattered light). Two distinct slopes (relaxation processes) indicate two different sets of spherical particles: primary particles accommodated inside the larger vesicles. The innermost parts of the correlation curves (Figure 6b) correspond to the diffusion coefficient D ) 7 × 10-7 cm2/s, dH ) 4.6 ( 0.4 nm (the polydispersity index is dw/dn ) 1.01, indicating the monodisperse system). The slopes at larger decay times correspond to the much smaller number of micron-size particles (trimodal size distribution, one set has dH (wa) ) (5660 ( 420) nm)); averaged it is dH ) (1266 ( 2732) nm (dw/

dn ) 11.54) and the corresponding D is equal to 1.92 × 10-8 cm2/s (measured at θ ) 30°). The correlation curves of the system prepared by the pF-stat method and measured repeatedly after such long time show not so distinct intersections between fast and slow relaxation (instead the slope has changed continuously with decay time), i.e., exponential curves at smaller angles up to θ ) 60°. Besides, the dH values (median, weight average) measured at θ ) 35°, 120°, and 90° are much smaller, as follows: (24 ( 3), (26 ( 1), and (4.1 ( 0.4) nm, respectively. If we suppose spherical shape of the vesicles (determined by the size at 35° and 120°) and of the primary LaF3 nanoparticles (determined at 90°), then we calculate from their volumes that vesicles accommodate ∼6 primary particles in this case. According to Figure 5, the size of primary particles obtained by extrapolation to zero concentration (i.e., independent


measurements) gave dH ) 8 nm. That would be about 3 primary particles inside the vesicle. Meanwhile, we may assume also the ellipsoidal shape of the particles grown inside the ribbons, which coexist in the same system. That was confirmed by microscopic examination (part 2) although on the higher length scale (micronsize scale) and assuming the self-similarity of the shape and the structure (as in a vast number of cases in the literature). Both shape assumptions are evidenced and corroborated below in paragraph D. In this case, biaxial ellipsoids with semiaxes b and a, where (a < b) and with axial ratio F ) b/a, give F > 1 for an oblate ellipsoid and F < 1 for a prolate ellipsoid. Then our data presented in Figure 8 can be interpreted as follows: the fast relaxation mode at 90° (and around it) may be attributed to the cross-section of the vertical stacks, and the slow mode at lower angles is related to the horizontally stacked longer dimension of the rod-shaped particles. We calculate that F ) 3.0 ( 0.2, and oblate ellipsoids have to be distinguished. Since qL < 1 in our case, the contribution of the rotation of rods to the decay rate is negligible.15,29 D. Characterization of Particles by POM. Although in the micron-size scale range microscopic examination is the most direct method for the determination of the particle shape (ex situ), it is a very useful complementary method. Hence, shape should not be a priori assumed when analyzing the scattering data, and that makes the analysis more straightforward. Using polarized light with a λ plate, the birefringence of particles can be confessed too. Figure 9f-h shows some interesting features of the anisotropic dendrites and stable sol mentioned above, at pH ) 4, grown by a firstorder kinetics (large excess of one component); further, Figure 9o,p shows large crystals grown via an exponential growth rate, at pH ) 5.85. The stoichiometric concentration ratio was 1/3 in both cases, only the supersaturation of the working solution was much higher in the last case; also the samples were observed microscopically after 600% of the pF-stat seeded growth. So, we compared different systems at the same stage of growth, i.e., at m/m0 ) 7. Conclusions The main idea of these investigations (described in this paper and part 2) was to prove the crucial thing in the preparation of new materials for their special applications, which is to confirm the self-similarity of particle shape (morphology control) and axial ratio of the anisotropic particles (in this case rod-shaped particles), in the micron- and submicron-scale range (this paper) as well as in the nanoscale range (which is corroborated in part 2). New materials with their enriched properties and wide application are based on their augmented structural, optical, and electrical properties. The potentiometric pF-stat method is shown to be appropriate for the preparation of new metal fluoride particles using specific chemical control, such as the state of ions in solution at the attained pH and ionic strength (without addition of acid/base or neutral electrolyte). In particular, several main conclusions can be made on the basis of the results presented in part 1. (1) A thermodynamic equilibrium approach to the spontaneous precipitation of solid phases from La(NO3)3


and KF solutions in the broad range of several orders of magnitude in concentration, at 25 °C, was used, and a precipitation diagram was determined. The solubility data at the solid/solution boundary (based on the LS data) were fitted to a Gaussian curve, and two concentration regions with quite different ionic species in solutions can be distinguished (from the calculation by the Mineql+ program), which includes different mechanisms of growth from these particular solutions. (2) The pF-stat and pH-stat potentiometric rates of LaF3 growth on well-defined crystal seeds from the metastable supersaturated solution at pH from 4 to 6 were measured and analyzed. A polynuclear growth with an order of crystallization of 3-4 was determined by the analysis of the growth rate as a function of supersaturation (bilogarithmic plot), and the polynuclear rate constant kp ) (7.1 ( 0.7) × 10-12 m s-1 was calculated. (3) The systems investigated are pure inorganic heterogeneous systems (without any additives, inhibitors, or promoters) having narrowly distributed physical characteristics as particle shape, particle size distribution, and axial ratio (F) all calculated (here in part 1) on the basis of the DLS measurements and analysis for submicrometer particles. Acknowledgment. Financial support from the former Ministry of Science and Technology, now the Ministry of Science, Education and Sport of the Republic of Croatia, is acknowledged (Projects No. 119 495 and 0119 622). The author is grateful to Prof. Dr. Gary W. Poehlein for editing the manuscript. References (1) Stubicˇar, N. In Trends in Colloid and Interface Science XIII; Tezˇak, \., Martinis, M., Eds.; Progress in Colloid and Polymer Science, Vol. 112; Springer: New York, 1999; pp 200-205. (2) Stubicˇar, N. In Trends in Colloid and Interface Science XV; Koutsoukos, P. G., Ed.; Progress in Colloid and Polymer Science, Vol. 118; Springer: New York, 2001; pp 119-122. (3) Davies, C. W.; Jones, A. L. Trans. Faraday Soc. 1955, 51, 812-829. (4) Stubicˇar, N. M.Sc. Thesis, University of Zagreb, Zagreb, Croatia, 1973. Tezˇak, B.; Tezˇak, \.; Stubicˇar, N. Croat. Chem. Acta 1973, 45, 275-295. (5) Nielsen, A. E.; Toft, J. M. J. Cryst. Growth 1984, 67, 278288. (6) Nielsen, A. E. J. Cryst. Growth 1984, 67, 289-310. (7) Stubicˇar, N.; Tezˇak, \. Kem. Ind. (Zagreb) 1978, 28, 205210. (8) Stubicˇar, N.; C Ä avar, M.; Sˇ krtic´, D. Prog. Colloid Polym. Sci. 1988, 77, 201-206. (9) Stubicˇar, N.; Sˇ cˇrbak, M.; Stubicˇar, M. J. Cryst. Growth 1990, 100, 261-267. (10) Stubicˇar, N.; Markovic´, B.; Tonejc, A.; Stubicˇar, M. J. Cryst. Growth 1993, 130, 300-304. (11) Tomson, M.; Nancollas, G. H. Science 1978, 200, 1059. (12) Zieba, A.; Nancollas, G. H. J. Cryst. Growth 1994, 144, 311319. (13) Muir, J. D. Microscopy: Polarizing Microscope. In Physical Methods in Determinative Mineralogy; Zussman, J., Ed.; Academic Press: London, 1977; p 38. (14) Keller, A. Crystallinity and Kinetics of Crystallization. In Polymers, Liquid Crystals and Low Dimensional Solids; March, N., Tosi, M., Eds.; Plenum Press: New York, 1984. (15) Chu, B. Laser Light Scattering; Academic Press: New York, 1974; p 221.

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(16) Chen, S. H.; Chu, B.; Nossal, R. Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems; Plenum Press: New York, 1981. (17) DLS-700S, Photal Otsuka Electronics Co., Japan, Operation Manual 22.01.1991. (18) Kratky, O. Nova Acta Leopoldina 1983, NF 55 (256), 1-103. The World of Neglecting Dimensions: Small-Angle Scattering of X-rays and Neutrons of Biological Macromolecules; Deutsche Akademie der Naturforscher Leopoldina: DDR-4010 Halle. (19) Stubicˇar, N. Langmuir 1998, 14, 4322-4330. (20) Program Mineql+; Environmental Research Software: Hallowell, ME (21) Ho¨gfeld, E. Stability Constants of Metal-Ion Complexes, Part A: Inorganic Ligands; IUPAC Chem. Data Series, No. 21; Pergamon Press: Oxford, 1980. (22) Lingane, J. J. Anal. Chem. 1967, 39, 881; 1968, 40, 935. (23) Manon, M. P.; James, J. J. Chem. Soc., Faraday Trans. 1 1989, 85, 2683. (24) Christoffersen, J.; Christoffersen, M. R. J. Cryst. Growth 1990, 100, 203. (25) Wu, W.; Nancollas, G. H. Adv. Colloid Interface Sci. 1999, 79, 229-279.

Stubicˇar (26) OHara, M.; Reid, R. C. Modelling of Crystal Growth Rates from Solution; Prentice Hall: U.K., 1973. (27) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes; Academic Press: New York, 1971; pp 84 and 104. (28) Mohan, R.; Myerson, A. S. Chem. Eng. Sci. 2002, 57, 42774285. (29) Lehner, D.; Lindner, H.; Glatter, O. Langmuir 2000, 16, 1689-1695. (30) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Klein, R.; Ball, R. C.; Meakin, P. J. Phys.: Condens. Matter 1990, 2, 30933113. (31) Asnaghi, D.; Carpineti, M.; Giglio, M.; Vailati, A. Physica A 1995, 213, 148-158. (32) Tani, Y.; Umezava, Z.; Chikama, K.; Hemmi, A.; Soma, M. J. Electroanal. Chem. 1994, 378, 205-213. (33) Mann, S. Biomineralization, Principles and Concepts in Bioinorganic Materials Chemistry; Oxford University Press: Oxford, 2001.

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