Investigating Hydrolytic Polymerization of Aqueous Zirconium Ions

Monika Dominska , Maciej Mazur , Kelly P. Greenough , Monique M. Koan , Paweł G. Krysiński , G.J. Blanchard. Bioelectrochemistry 2007 70 (2), 421-43...
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3568

J. Phys. Chem. B 2002, 106, 3568-3575

Investigating Hydrolytic Polymerization of Aqueous Zirconium Ions Using the Fluorescent Probe Pyrenecarboxylic Acid J. J. Tulock and G. J. Blanchard* Department of Chemistry, Michigan State UniVersity, East Lansing, Michigan 48824-1322 ReceiVed: July 26, 2001; In Final Form: January 10, 2002

We report on the hydrolytic polymerization of zirconium in aqueous solution using the transient optical response of the fluorescent molecule 1-pyrenecarboxlyic acid (PCA) as a probe. We have measured the fluorescence lifetimes and rotational diffusion dynamics of PCA in “aged” aqueous solutions of tetravalent zirconium to understand the ability of fluorescent probes to report on the structural aspects of hydrous metal oxide selfassembly. The degree of polymerization, and therefore the extent of solute organization, is controlled by allowing the Zr4+ solutions to polymerize (age) for selected amounts of time at 85 °C. Fluorescence lifetimes and reorientation times measured for PCA in zirconium solutions increase rapidly, approaching a constant value after 30 h of aging. The data point to growth of polymers that achieve a maximum volume. The reorientation data place limits on the size and shape of polymers and provide insight into the mechanism for their growth.

Introduction Much research has focused on understanding the aqueous chemistry of zirconium because of its importance in a variety of applications, including ceramics,1 sol-gel chemistry,2 and nuclear fuel processing.3 In acidic solution, it has long been accepted that zirconium exists as the tetrameric species [Zr(OH)2‚4H2O]48+ where four metal ions are arranged nearly at right angles joined by double bridging hydroxyl bonds.4 Four water molecules then occupy the remaining positions of each metal coordination sphere. If solutions of zirconium are allowed to stand, these tetramer units undergo hydrolysis according to the following equilibrium:

[Zr(OH)2‚4H2O]48+ a [Zr(OH)(2+x)‚(4 - x)H2O]4(8-4x)+ + 4xH+ Heating or the addition of base accelerates hydrolysis, giving rise to gelatinous precipitates which, when refluxed for extended periods, produce either cubic or monoclinic hydrous zirconias.5 Clearfield proposed a mechanism to explain these observations by assuming precipitates formed from the joining of tetrameric units through hydroxyl bridging bonds either in an ordered fashion, producing two-dimensional sheets, or randomly, resulting in three-dimensional growth.5-7 Heating of solutions was thought to allow slow growth of tetramers in an ordered fashion, while addition of base was believed to cause rapid, disordered growth. Production of cubic and monoclinic phases under different experimental conditions was thought to result from ordered and random growth, respectively. While there has been experimental evidence to support Clearfield’s theory for the formation of two-dimensional sheets,8 some recent evidence suggests that tetramers can aggregate to form rodlike particles upon heating of Zr4+ solutions.9 Recent work in this area has attempted to distinguish between these several growth mechanisms. * To whom correspondence should be addressed. E-mail: blanchard@ photon.cem.msu.edu.

Our interest in this chemical system stems from previous work in our laboratory that has explored the use of fluorescent probe molecules to sense solute self-assembly in crystallizing systems.10-13 This strategy, termed the “lock-and-key” approach, relies on the use of probe molecules possessing functional groups that promote their association with the crystallizing solute. By ensuring close proximity between the probe and solute, when solute self-organization occurs, the probe steady-state and timeresolved emission responses report on the relevant chemical processes. Typically, the probe is introduced as a trace impurity (∼10-6 M) into solutions containing the solute. The probe chromophore emission response and rotational diffusion behavior are studied as the solute concentration is varied from below saturation through supersaturation. We have applied this strategy to the study of self-assembly of adipic acid in aqueous solution and have found that, while fluorescence lifetimes and steady-state emission spectra report the onset of solute organization and the average environments sensed by probes, information concerning the range and structure of solute organization is derived most efficiently from rotational diffusion times. The information content of each measurement can be optimized by careful choice of the probe molecule. The level of detail that can be extracted from reorientation measurements depends, in part, on the solute identity because the exact relationship between the persistence time of transient solute organization and the time constant for reorientation of aggregates is not known. Because of this limitation, only the average size of adipic acid aggregates could be estimated. We have chosen aqueous solutions of zirconium as a model system for study because organization of this metal ion is known to proceed through hydrolytic polymerization. Once formed, solute moieties persist over long times, enabling a correlation to be made between probe molecule dynamics and aggregate structure. The spectroscopy of pyrenecarboxylic acid (PCA; Figure 1) is well characterized11 and was chosen as a probe for this system because of the propensity of carboxylic acids to complex with Zr4+.7 One goal of the work we present here is to extend the lock-and-key strategy beyond the organic systems we have

10.1021/jp012901f CCC: $22.00 © 2002 American Chemical Society Published on Web 03/14/2002

Polymerization of Aqueous Zirconium Ions

J. Phys. Chem. B, Vol. 106, No. 14, 2002 3569

Figure 1. Structure of the probe molecule PCA.

studied previously10-13 to include inorganic systems. Understanding the aqueous chemistry of zirconium is also a matter of practical significance in the construction of robust multilayer assemblies.14-28 We have measured fluorescence lifetimes and rotational diffusion dynamics of the probe molecule PCA in solutions of Zr4+ “aged” between 0 and 110 h to understand the limit of structural information that can be obtained using the lock-and-key strategy. Lifetimes were found to decrease with solution age, and the rotational diffusion behavior of PCA in these solutions reveals that growth of zirconium hydrous polymers occurs until they approach an upper size limit. This result is consistent with the mechanism for the precipitation of hydrous zirconias proposed by Singhal et al., indicating that polymers grow to achieve a constant size and subsequently aggregate to form crystal nuclei.9 We find the rotational diffusion of PCA in solutions of Zr4+(aq) senses not only the overall motion of probe/solute aggregates, but also the rotation of PCA tethered to zirconium hydrous polymers. These data, when interpreted within the model for reorientation in macromolecular and membrane systems developed by Szabo,29 place firm boundaries on the size and shape of polymers and provide evidence to indicate the most likely mechanisms for their formation. Experimental Section Chemicals. 1-Pyrenecarboxylic acid (>98%) was purchased from Fluka Chemical Co. and recrystallized twice from methanol. The purified probe molecule exhibited a single-exponential fluorescence lifetime decay in aqueous solution, as measured by single-photon counting, and was judged to be pure on this basis. ZrOCl2‚8H2O (>99.5%) was purchased from Riedel de Hae¨n and used as received. Stock solutions of 0.2 M zirconium and 1.8 M HCl were prepared using deionized water (18 MΩ cm; Millipore Corp.). For aging experiments, zirconium solutions were prepared by adding 1 equiv of acid to an appropriate amount of the zirconium stock solution to produce a final zirconium concentration of 0.05 M. Solutions were then aged at 85 ( 2 °C using a temperature-controlled silicone oil bath, then removed, cooled to room temperature, and allowed stand for at least 2 h. The probe molecule was then introduced at a concentration of 5 × 10-7 M just prior to measurement. Time-Correlated Single-Photon-Counting (TCSPC) Spectrometer. Fluorescence lifetimes and rotational diffusion dynamics of PCA in aged zirconium solutions were measured using a time-correlated single-photon-counting spectrometer. A detailed description of this instrument has been given previously, and only the experimental parameters important for this work are described here.30 The second harmonic of the output of a mode-locked (100 ps pulses, 80 MHz repetition rate) Nd:YAG laser (Quantronix 416) was used to excite a synchronously pumped, cavity-dumped dye laser (Coherent 702-2) with 532 nm light. Samples were excited by pulses of light produced by the dye laser at 646 nm (Kiton Red dye) and frequency-doubled to 323 nm (Type I SHG with LiIO3) at a repetition rate of 8 MHz. Temperature control was achieved by placing the sample cuvette in a brass block holder maintained at 20.0 ( 0.1 °C with a Neslab (model EX-221) circulating bath. Fluorescence

Figure 2. Stern-Volmer plot of fluorescence lifetime with proton concentration. Uncertainties for each data point are shown at the 95% confidence limit.

lifetimes were recorded with emitted light collected at a polarization of 54.7° with respect to the excitation polarization to eliminate motional contributions to the decay. Emission was collected at 425 nm with a 10 nm band-pass using a subtractive double monochromator (American Holographic B-10), and detection was accomplished using a Hamamatsu R3809U-50 microchannel plate PMT. The typical instrument response function for this system is 35 ps fwhm. Results and Discussion We have obtained complementary bodies of data to characterize organization in solutions of Zr4+ at various stages of hydrous polymer formation. At room temperature, organization proceeds very slowly;31 the extent of organization can be controlled by heating solutions for selected amounts of time and then cooling them to room temperature. The emission response and rotational diffusion dynamics of PCA in these aged solutions have been studied to evaluate the ability of fluorescent probes to report on solute self-assembly. We first consider fluorescence lifetime measurements and discuss the pH dependence of this response in terms of the probe’s ability to sense the extent of Zr4+ hydrolysis. Next, rotational diffusion data are studied to elucidate contributions to the experimental anisotropy decays from overall and internal motions of probe/ solute complexes. With this information in hand, the rotational diffusion behavior of zirconium polymers is used to place boundaries on their size and shape. We consider the lifetime and reorientation data separately. Lifetime Measurements. As mentioned previously, polymerization of the hydrous oxide of Zr4+ results in a decrease in solution pH due to hydrolysis of the tetrameric species. It is important to consider that, for complex systems, fluorescent molecules can often distribute themselves in environments with properties that do not reflect the bulk composition and, when this situation arises, the emission properties of probes can reveal important clues about molecular-scale processes. We have measured fluorescence lifetimes of PCA in buffered solutions and compared these data with those measured in aged solutions of zirconium to understand the ability of this probe to sense changes in the local environment associated with Zr4+ hydrous oxide polymer formation. As can be seen from Figure 2, the lifetimes of PCA in buffered solutions exhibit a measurable pH dependence, varying

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Tulock and Blanchard

Figure 3. Behavior of fluorescence lifetimes with increased age of zirconium solutions.

from 5.4 ns at pH 1.7 to 5.0 ns at pH 1.1. The acid and base forms exhibit lifetimes of 5.6 and 40 ns, respectively,11 and this result is not consistent with the simultaneous existence of the two probe species in solution. If this were the case, we would expect, in the limit that each form persisted significantly longer, on average, than their respective fluorescence lifetimes, to observe two time constants, with the contribution from each being proportional to the concentration of each species. Conversely, if proton exchange were very fast compared to the fluorescence decay time constant, we would expect the observed lifetime to represent a weighted average of fluorescence lifetimes of the protonated and deprotonated forms of the probe. The pKa of this probe is 4.8, and it seems unlikely that this dependence is the result of a shift in equilibrium between protonated and deprotonated forms of the probe, given the fraction protonated changes by less than 0.06% over the pH range of 1.1-1.7.11 We assert that these changes result from quenching of PCA by protons. For fluorophores undergoing collisional quenching, their lifetime will depend on the quencher concentration according to the Stern-Volmer equation32

τ-1 ) τ0-1 + kq[Q]

(1)

where kq is the bimolecular quenching constant, τ is the observed lifetime, and τ0 is the lifetime in the absence of quencher. Figure 2 shows a Stern-Volmer plot of the lifetimes with increasing proton concentration. We observe a linear relationship between τ-1and [H+], consistent with collisional quenching of PCA by protons, and we recover the quenching constant Ksv ) 1.3 ( 0.2 M-1 and bimolecular quenching constant kq ) (2.4 ( 0.2) × 108 M-1 s-1. With this information in hand we consider lifetimes measured in aged Zr4+ solutions. The data in Figure 3 show the dependence of PCA lifetime on the age of Zr4+ solutions. Each point in the figure is the result of six determinations with uncertainties reported to the 95% confidence limit. We note two important features in these data; the first is that lifetimes decrease with increasing solution age. Because the pH is known to decrease with the hydrolysis of zirconium, this finding, when compared to the data for buffered solutions, indicates that the probe molecule is sensing the pH change of the solution associated with the degree of polymerization. While other solution properties such as ionic strength or dielectric response may contribute to the lifetime data, it is clear that the probe molecule is sensitive to the progress of this reaction. Examining the data further reveals

that lifetimes approach a constant value, indicating polymerization to be complete to within our ability to discern after ∼30 h. The lifetime measurements we report indicate that PCA is sensitive to the growth of zirconium polymers, most likely by sensing the solution pH, but because changes in pH may result from a variety of growth mechanisms, lifetime measurements do not contain detailed information on the structure of the hydrous metal oxide polymer. In contrast, rotational diffusion measurements are sensitive to the shape of the reorienting species. The structure of polymers depends on the mechanism by which tetramer units add during polymerization, and the reorientation dynamics of probes associated with these polymers should contain important structural information. The lifetime data are important, however, in providing a reference point for interpreting the reorientation data discussed below. Rotational Diffusion Dynamics. The reorientation dynamics of the system we report on here are more complex than those covered by the traditional models of rotational diffusion. We discuss these simpler models very briefly to provide a frame of reference for our data. The induced orientational anisotropy function, R(t), is constructed from the experimental data:

R(t) )

I|(t) - I⊥(t) I|(t) + 2I⊥(t)

(2)

where I|(t) and I⊥(t) are the experimental signal intensities. Expressions relating the functionality of R(t) to chemical properties are well developed.33-35 R(t) is predicted to decay as a sum of exponentials, with the number of components depending on the orientations of the transition dipole moments and effective rotor shape of the reorienting moiety. The modified Debye-Stokes-Einstein (DSE) equation36 is a common starting point in the interpretation of experimental data. This equation relates molecular reorientation times to the bulk system parameters solution viscosity, η, solute hydrodynamic volume, V,37 and temperature, T.

τOR )

1 ηVf ) 6D kBTS

(3)

The term f accounts for frictional contributions to the solventsolute boundary condition. The value of f can vary between the so-called “slip” (0 < f < 1) and “stick” (f ) 1) limits, depending on the specific system.38,39 S is a shape factor to account for nonspherical reorienting moieties.40 For rigid systems with reorientation dynamics characterized by more that one decay time constant, Chuang and Eisenthal’s equations relate the experimental data to molecular properties.35 These equations have proven exceptionally useful in cases where solventdependent changes in the functionality of the chromophore anisotropy decay are seen.41,42 For labile systems, such as the ones we report on here, the presence of multiple components in the anisotropy decay cannot, in general, be interpreted using Chuang and Eisenthal’s formulation. The reorientation dynamics of PCA in the presence of zirconium are different from the above descriptions because complexation of this probe produces a labile system where no fixed relationship between the transition moments and diffusion axes can be assigned, a priori. The rotational diffusion behavior of PCA in the presence of Zr4+ (Figure 5) is very different from that in pure water (Figure 4). The interesting features to note from these data are the decay time constants and functionality exhibited by R(t) in each system. For pure water, R(t) decays as a single exponential with τOR ≈ 90 ps (Figure 4b), but in freshly prepared solutions of Zr4+, R(t) exhibits two decays, τ1

Polymerization of Aqueous Zirconium Ions

J. Phys. Chem. B, Vol. 106, No. 14, 2002 3571

Figure 4. Rotational diffusion behavior of PCA in water: (a) raw fluorescence intensity signals I|(t) and I⊥(t), (b) anisotropy function constructed from the experimental data shown in (a). The best fit regression function for these data is R(t) ) 0.23 exp(-t/(90 ps)).

Figure 6. Behavior of the anisotropy decay constants τ1 (9) and τ2 (O) as a function of Zr4+ solution age.

TABLE 1: Anisotropy Data for 1-Pyrenecarboxylic Acid in Aged Solutions of Zr4+ age (h)

τ1 (ns)

τ2 (ns)

R1

0 0.75 ( 0.01 0.047 ( 0.003 0.167 ( 0.005 1 1.4 ( 0.3 0.17 ( 0.07 0.18 ( 0.04 2 1.8 ( 0.1 0.24 ( 0.07 0.18 ( 0.01 3 2.8 ( 0.2 0.25 ( 0.11 0.16 ( 0.05 4 3.0 ( 0.2 0.19 ( 0.06 0.20 ( 0.01 5 5.0 ( 0.9 1.4 ( 0.3 0.16 ( 0.02 15 6.3 ( 0.9 1.4 ( 0.3 0.16 ( 0.02 20 6.4 ( 1.0 1.4 ( 0.4 0.16 ( 0.02 30 6.6 ( 1.3 1.4 ( 0.4 0.17 ( 0.02 40 6.4 ( 0.3 1.3 ( 0.2 0.16 ( 0.01 51 6.7 ( 1.2 1.9 ( 0.4 0.15 ( 0.02 60 6.9 ( 1.4 1.4 ( 0.4 0.16 ( 0.02 70 5.8 ( 0.6 1.5 ( 0.3 0.17 ( 0.01 91 5.7 ( 1.4 1.4 ( 0.5 0.15 ( 0.05 110 6.7 ( 0.4 1.0 ( 0.1 0.16 ( 0.01

Zr4+

Figure 5. Rotational diffusion behavior of PCA in 0.05 M aged 0 h at 85 °C: (a) raw fluorescence intensity signals I|(t) and I⊥(t), (b) anisotropy function constructed from the experimental data shown in (a). The best fit regression function for these data is R(t) ) 0.17 exp(-t/(50 ps)) + 0.06 exp(- t/(750 ps)).

≈ 50 ps and τ2 ≈ 750 ps (Figure 5b). This change in R(t) can result only from changes in the effective rotor shape of the reorienting moiety, the angle between the chromophore transition moments, δ, or the orientation of the chromophore with respect to the diffusion axes. The quantity δ can be evaluated from the value of R(t) at time zero according to the relationship

R(0) ) 0.4P2(cos δ)

(4)

where P2 is the second-order Legendre polynomial.43 As can be seen from eq 4, R(0) depends only on δ and is independent of both the form of R(t) and the rotor shape exhibited by the probe. Table 1 shows the behavior of R(0) to be constant for all solutions measured, and we are left to consider a change in the shape of the reorienting species to explain the behavior of R(t). We assert that the most likely explanation for the observed behavior of R(t) is that association of PCA with zirconium hydrous polymers produces a change in the effective rotor shape of the chromophore. To this point we have not addressed the origin of τ1 and τ2, and making the correspondence between the time constants and the shape and size of the reorienting moiety is central to this

R2

R(0)

0.17 ( 0.03 0.07 ( 0.03 0.05 ( 0.01 0.05 ( 0.02 0.05 ( 0.01 0.09 ( 0.02 0.09 ( 0.02 0.08 ( 0.01 0.08 ( 0.02 0.06 ( 0.01 0.07 ( 0.02 0.06 ( 0.02 0.05 ( 0.01 0.08 ( 0.05 0.06 ( 0.01

0.34 ( 0.03 0.25 ( 0.07 0.24 ( 0.01 0.22 ( 0.07 0.24 ( 0.02 0.25 ( 0.04 0.24 ( 0.03 0.24 ( 0.03 0.25 ( 0.04 0.21 ( 0.01 0.22 ( 0.04 0.23 ( 0.04 0.21 ( 0.02 0.23 ( 0.09 0.22 ( 0.01

work. While it may be tempting to attribute these decay constants to reorientation of PCA that is free in solution and complexed with the solute, τ1 ) 50 ( 3 ps (Figure 5) is measurably different from τOR ) 95 ( 18 ps, which we have measured for PCA in water (Figure 4). We need to consider these data in the context of a model that takes into account the spatial and dynamical relationship between PCA and the Zr4+ hydrous oxide polymer. The rotational diffusion behavior of PCA depends on the solution “age” as shown in Figure 6. Each point reported is the result of at least six individual measurements, and the uncertainties are reported at the 95% confidence level. A minimum of 1000 counts were acquired in the region used for normalization of the experimental signals I|(t) and I⊥(t). All decay curves were fit to both single- and double-exponential models and were found to fit only when two decay components were used. The quality of the fit was judged on the basis that residuals were centered about zero only for fits of doubleexponential decay functions. As can be seen from Figure 6, pronounced changes in τ1 and τ2 occur with increased solution age, with each time constant reaching a limiting value after ∼30 h at 85 °C. The behavior of these time constants with solution age and their magnitude suggest that they do not originate from a mixture of free and complexed PCA. The time constants of τ1 ≈ 1.4 ns and τ2 ≈ 6.4 ns at long age times are not consistent with reorientation of the free probe. For either of these values to be consistent with the modified DSE model (eq 3), solution viscosity would have to increase 10-fold from that of water, and we observe no such change in solution bulk viscosity with solution age. We consider

3572 J. Phys. Chem. B, Vol. 106, No. 14, 2002

Tulock and Blanchard of the (polymer) cylindrical rotor, ZM. Figure 7b illustrates the coordinate system relating the chromophore to the polymer. The Szabo model differs from Chuang and Eisenthal’s treatment in that it considers the general case of fluorescence depolarization where intramolecular motions contribute to anisotropy decays. We assume that the probe transition moments lie within the π-system plane of the pyrene chromophore, making φAE ) 0 or π, depending on the choice of coordinates (vide infra).

2 (P (cos β))2P2(cos θA)P2(cos θE) + 5 2 9 sin2 2β cos φAE sin 2θA sin 2θE + 32 9 sin4 β cos 2φAE sin2 θA sin2 θE exp(-6Dxt) + 32 2 3P (cos θA)P2(cos θE)(sin2 β cos2 β) + 5 2 2 3 1 cos φAE sin 2θA sin 2θE cos2 β - (1 - cos β) + 4 2 2 1 2 (1 + cos β) - cos β + 2 2 3 1 cos 2φAE sin2 θA sin2 θE (sin β)(cos β - 1) + 4 2 2 1 - (1 + cos β) sin β exp(-(5Dx + Dz + DP)t) + 2 23 P (cos θA)P2(cos θE)(sin4 β) + 54 2 2 3 1 cos φAE sin 2θA sin 2θE - (1 + cos β) sin β + 4 2 2 1 3 (cos β - 1) sin β + cos 2φAE sin2 θA sin2 θE × 2 4

R(t) )

[

]

[

((

(

)

))

((

Figure 7. Coordinate systems showing the relationship between (a) the transition moments and the chromophore local z-axis and (b) the chromophore and macroscopic (polymer) coordinate systems. The quantities θA and θE describe the orientations of the absorption, µA, and emission, µE, transition dipole moments with respect to the chromophore axis of rotation, ZP. δAE is the angle between the transition moments, and φ is the difference between their dihedral angles with respect to the chromophore coordinate system. The angle βMP describes the orientation of the chromophore and macroscopic polymer coordinate systems.

)

) )]

(

[

((

(

)

))

]

(cos8(β/2) + sin8(β/2)) exp(-(2Dx + 4Dz + DP)t) (5) the fact that both time constants increase with solution age to be consistent with the reorientation of a single moiety, PCA complexed with an oligomer, and the age-dependent increases in reorientation times reflect the size of the solute species as polymerization proceeds. We find the reorientation times to be correlated with fluorescence lifetimes in the sense that both bodies of data indicate growth of polymers to be complete after ∼30 h. While the reorientation data are sensitive to the growth of polymers, the relationship between τ1 and τ2 and the physical dimensions of these structures is not unambiguous. In the general case, which almost certainly applies to this system, these time constants contain contributions from two types of motion: (1) overall motion of PCA/solute aggregates and (2) rotation of the fluorescent moiety relative to the zirconium hydrous polymer. To understand the role of both probe and polymer motions in determining τ1 and τ2, we interpret our data using the model developed by Szabo.29 This model considers the case of reorientation of chromophores attached to macroscopic particles, where the chromophore is free to rotate about its axis of attachment to the particle. To relate reorientation of the transition moments to the motion of the macroscopic particles, a coordinate system is needed to describe the relationship between the chromophore and the macroscopic particle. Figure 7a shows the angles, θA and θE, relating the orientation of the chromophore transition moments, µA and µE, to the probe z-axis, ZP, and φAE, the dihedral angle between µA and µE. The probe and macroscopic coordinate systems are related through the angle β, which is relevant only for nonspherical (cylindrical) particles. β is the angle between the chromophore z-axis and the C∞ axis

where Dz and Dx are the Cartesian components of the macroscopic diffusion constant DM and DP is the diffusion constant for rotation of the probe molecule about ZP (Figure 7b). It can be seen that eq 5 predicts R(t) can exhibit three time constants, with the first term sensing only rotation of the macroscopic particle. The second and third terms depend on the motion of the macroscopic particle and of the probe about its fixed axis. In addition, because diffusion of the probe about ZP should be much faster than the motion of the macroscopic particle, the longest decay time constant will be associated with the first term in all cases. When either µA or µE is coincident with ZP, β ) 0, and DP ) 0, eq 5 reduces to Chuang and Eisenthal’s formulation for a long axis polarized prolate rotor. Although eq 5 predicts three time constants, only two are resolved within our experimental uncertainty. This result can obtain only for a limited number of conditions, and the sign and magnitude of the exponential prefactors place significant constraints on the orientation of the chromophore relative to the macroscopic particle. Our experimental data are of the form

R(t) ) 0.17 exp(-t/τ1) + 0.06 exp(-t/τ2) τ1 ) (6Dx)-1, τ2 ) (5Dx + Dz + DP)-1

(6)

and there are two sets of angles θ, β, and φAE that are consistent with experiment. These two sets of angles correspond to the same physical situation, differing only by the (arbitrary) assignment of the probe and particle Cartesian axes. If we set β ) 0, and use the solution-phase result that δ ) 32° (φAE )

Polymerization of Aqueous Zirconium Ions

J. Phys. Chem. B, Vol. 106, No. 14, 2002 3573 TABLE 2: Calculated Reorientation Times and Corresponding Structural Parameters for the Possible Polymer Shapesa oblate prolate sphere (S ) 0.19) (S ) 0.11) (S ) 1) τOR (ns) volume (Å3) no. of tetramer units X (Å) Z (Å)

6.1 4700 15 110 9

6.1 2900 9 9 69

6.4 24 700 77 18 18

no chromophore intramolecular motion 1.4, 6.4 1866 9 6 66

a The oblate, prolate, and sphere results were determined assuming chromophore motion about its tether to the polymer. If this assumption is lifted, the results in column 4 obtain.

between DM and the observed decay constants are more complicated for the cylindrical geometry, where τ1 is related only to the component Dx (eq 6). To calculate DM, Dz must be known and substituted into

DM )

Figure 8. (a, top) Coordinate systems and angle definitions consistent with experimental data. Left panel: schematic for {θA, θE, φAE, β} ) {(6°, (38°, 0°, 0°}. Right panel: schematic for {θA, θE, φAE, β} ) {(16°, -16°, 180°, 23°}. (b, bottom) Dependence of the diffusion constant ratio Dz/Dx on the cylinder aspect ratio, p ) L/2r. Open circles are based on calculations adapted from ref 44. The solid point is experimental data from this work.

0 based on the structure of PCA), the only angles θ that yield prefactors consistent with the experiment are {θA, θE} ) {6°, 38°} or {-6°, -38°}. It is important to note that the assignment of θA ) -θE ) 16° and φAE ) π (symmetric displacement of µA and µE about the ZP axis) produces prefactors in eq 5 inconsistent with the functionality of the experimental data. For these angle assignments, the prefactor A1 ≈ 0.31 and the prefactor A2 ≈ -0.08. Because it is the orientations of the transition dipole moments that are sensed in these measurements, the symmetric assignment of the angles θ about ZP is appropriate, and if this assignment of θA, θE, and φAE is enforced, β must be set to 23° to bring eq 5 into agreement with the experimental results. In fact, the two sets of angles {θA, θE, φAE, β} ) {(6°, (38°, 0°, 0°} and {(16°, -16°, 180°, 23°} are two different descriptors of a single chromophore orientation relative to the macroscopic particle (Figure 8a), with the only difference between the two being in the assignment of the chromophore Cartesian axes relative to the transition moments. We consider next the information contained in the decay time constants τ1 and τ2. The largest contribution to R(t) is predicted using eq 5 to be associated with the longer decay constant. This result is in agreement with the experimental values recorded in Table 1, which shows two time constants at increased solution age (>30 h). The longer decay constant has an amplitude of ∼0.17, while the shorter constant has an amplitude of ∼0.06. The importance of eq 6 is that it relates the values of τ1 and τ2 to the macroscopic diffusion constants Dx and Dz (Figure 7b). By extracting values of Dx and Dz, the DSE model can be used to estimate the volume of polymers. For the limiting case of a spherical rotor, τ1 is related directly to DM ()Dx) and thus equal to τOR in eq 3. The relationships

Dz + 2Dx 3

(7)

Dz is contained in τ2 (eq 6) and cannot be determined independently unless DP is known. To first order, DP can be estimated by subtracting the reciprocal of τ2 from that of τ1 to give the quantity DP + Dz - Dx. This is valid because diffusion of macroscopic particles will be much slower than rotation of the probe about ZP. Estimates of DM can be substituted into eq 3 to recover the volumes for cylindrical and spherical geometries if the shape factor of each is known. These results are summarized in Table 2. Calculating a volume for the spherical case is straightforward. For a spherical rotor, the time constant of 6.4 ns corresponds to a volume of 24 700 Å3. For a cylindrical particle, the relationship between τ and V depends on the aspect ratio of the cylinder. For example, a rod-shaped particle with an aspect ratio of 5:1 corresponds to S ) 0.22, and for 10:1, S ) 0.075. These values of S yield V ) 5400 Å3 and V ) 1850 Å3, respectively, for a reorientation time of 6.1 ns. Therefore, the volume of 24 700 Å3 calculated for the spherical case must be considered the upper limit for these polymers. Ambiguity arises in calculating volumes for cylindrical particles because we must consider that cylinders are described in terms of their aspect ratio, either rod-shaped or disk-shaped. This choice corresponds to rotors behaving as prolate or oblate ellipsoids, respectively, yielding different values of S for a given aspect ratio. Since the dimensions of polymers should be related to those of zirconium tetramers, not all values of S produce volumes that are reasonable. We can place a lower limit on the volume by varying S until the corresponding dimensions of each ellipsoid approach those of the tetramer (8.9 Å × 8.9 Å × 5.8 Å).8 This treatment reveals that the volume of polymers may be as small as 4700 Å3 (prolate) or 2900 Å3 (oblate). The root of our inability to describe the shape of the polymer more precisely lies in the fact that we attempt to treat the motion of the probe chromophore about its bond to the polymer. If we make the assumption that this motion is unimportant, we can use the same model (eq 5), taking DP ) 0. Ignoring probe intramolecular motion can be justified on two grounds: (1) the probe motion is fast relative to that of the polymer and is thus averaged out over the duration of the measurement and (2) the location of the chromophore on the polymer is not well understood and samples a variety of environments. Setting DP ) 0 in eq 6, we recover τ1 ) 6Dx-1 ) 6.4 ( 0.4 ns and τ2 ) (5Dx + Dz)-1 ) 1.4 ( 0.2 ns, corresponding to Dx ) (2.60 (

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Tulock and Blanchard

0.18) × 107 Hz and Dz ) (5.84 ( 0.84) × 108 Hz. Our data predict a cylindrical rotor with a ratio Dz/Dx ) 22.5 ( 5.0, assuming DP ) 0. With the ratio Dz/Dx evaluated, we consider the aspect ratio and volume of the cylinder consistent with our data. The relationship between cylinder size and aspect ratio and diffusion constants has been treated previously.44 The diffusion constants Dz and Dx are given by

Dz )

kBT

( )

4p2 A0πηL3 1 + δz

Dx )

3kBT

(ln p + δx)

πηL3

(8)

where L is the length of the cylinder, p ) L/2r is the cylinder aspect ratio, η is the solution viscosity, A0 ) 3.841 is an empirical fitting parameter, and the terms δz and δx are endeffect corrections for the cylinder. The dependence of the terms δ on p has been tabulated by Tirado and Garcia de la Torre.44 The ratio Dz/Dx depends on the cylinder aspect ratio, p, and we present this dependence in Figure 8b. We infer that, for our system, p ) 11.0 ( 1.3. From this ratio and the experimental values of Dx and Dz, we use eqs 8 to determine L ) 66 Å and d ) 6 Å for our system, yielding a cylinder volume of 1866 Å3. Thus, the simplified model leads to a cylindrical polymer with an aspect ratio of 11. We have stated earlier that one goal of this work is to evaluate the ability of fluorescent probes to report on the structural aspects of solute organization. With this in mind it is useful to compare the values in Table 2 to existing information on the size of zirconium hydrous polymers formed in solution. Clearfield has found the size of crystallites formed from zirconium solutions range in size between 48 and 96 tetramer units, representing the upper limit of size polymers attain in solution.6 The actual value must be lower than this since growth of crystallites can occur after precipitation. Using an interpretation where we allow for intramolecular probe motion, we recover a polymer size range between 9 and 77 tetramer units, an estimate that overlaps substantially with that made by Clearfield. We note that the simplified interpretation leads to a cylindrical polymer with nine tetramer units. We can compare our data to the result of Singhal that polymers grow initially to form rod-shaped particles with a cross sectional radius of 4.3 Å.9 This cross section corresponds to the lower bound we establish for rod-shaped particles. According to the data in Table 2, rod-shaped species of cross section 4.3 Å would have an aspect ratio of 5 and be comprised of nine tetramer units. When we assume DP ) 0, we extract a cylindrical particle with a cross section of 6 Å and an aspect ratio of 11. It is clear that the reorientation dynamics of fluorescent probes can be used to provide insight into the dimensions of oligomeric species in solution. Conclusion We have measured the fluorescence lifetime and rotational diffusion dynamics of PCA in aged solutions of Zr4+(aq). The lifetime data in buffered solutions reveal that this probe molecule undergoes quenching by protons at low pH. This pH-dependent response enables the probe molecule to sense hydrolysis of zirconium tetramers and the completion of their growth to form polymers. The rotational diffusion data also sense the growth of polymers and reveal that, for chemical systems where organization persists for times greater than the reorientation time of solute aggregates, detailed structural information can be obtained. Zirconium polymers were found to attain a constant size between 9 and 15 monomer units, if of cylindrical

symmetry, and a size of 77 tetramer units if growth produces spherical particles. However, because spherical growth would produce polymers potentially larger that what could be supported in solution, this mechanism seems least likely. We believe that solubility limits the ultimate oligomer size and consequent saturation of the dynamics we measure. The central issue in understanding the relationship of polymer solubility to our data lies in whether these hydrous oxide polymers form a small number of long polymer chains or whether shorter individual chains tend to aggregate, and this is a matter of kinetic competition. Future work will center on better understanding the factors that determine the apparent size limitation for these materials. Acknowledgment. We are grateful to the National Science Foundation for support of this work through Grant CHE 0090864. We are deeply indebted to “reviewer 66” for a highly insightful review of this work. References and Notes (1) Dominguez-Rodriguez, A.; Gutierrez-Mora, F.; Jimenez-Melendo, M.; Routbort, J. L.; Chaim, R. Current understanding of superplastic deformation of Y-TZP and its application to joining. Mater. Sci. Eng., A 2001, A302, 154-161. (2) Woolfrey, J. L.; Bartlett, J. R. Processing colloidal powders, sols and gels. Ceram. Trans. 1998, 81, 3-20. (3) Franklin, D. G.; Lang, P. M. Zirconium-alloy corrosion: a review based on an International Atomic Energy Agency (IAEA) meeting. ASTM Spec. Tech. Publ. 1991, 1132, 3-32. (4) Mukherji, A. K. Analytical Chemistry of Zirconium and Hafnium; Pergamon Press: New York, 1970. (5) Clearfield, A. J. Mater. Res. 1990, 5, 161-162. (6) Clearfield, A. Inorg. Chem. 1964, 3, 146-148. (7) Clearfield, A. ReV. Pure Appl. Chem. 1964, 14, 91-108. (8) Fryer, J. R.; Hutchison, J. L.; Paterson, R. J. Colloid Interface Sci. 1970, 34, 238. (9) Singhal, A.; Toth, L. M.; Beaucage, J.-S. L.; Peterson, J. J. Colloid Interface Sci. 1997, 194, 470-481. (10) Tulock, J. J.; Blanchard, G. J. J. Phys. Chem. A 2000, 104, 83408345. (11) Tulock, J. J.; Blanchard, G. J. J. Phys. Chem. B 1998, 102, 71487155. (12) Rasimas, J. P.; Berglund, K. A.; Blanchard, G. J. J. Phys. Chem. 1996, 100, 7220-7229. (13) Rasimas, J. P.; Berglund, K. A.; Blanchard, G. J. J. Phys. Chem. 1996, 100, 17034-17040. (14) Lee, H.; Kepley, L. J.; Hong, H. G.; Akhter, S.; Mallouk, T. E. J. Phys. Chem. 1988, 92, 2597-2601. (15) Putvinski, T. M.; Schilling, M. L.; Katz, H. E.; Chidsey, C. E. D.; Mujsce, A. M.; Emerson, A. B. Langmuir 1990, 6, 1567-1571. (16) Katz, H. E.; Schilling, M. L.; Chidsey, C. E. D.; Putvinski, T. M.; Hutton, R. S. Chem. Mater. 1991, 3, 699-703. (17) Hong, H. G.; Sackett, D. D.; Mallouk, T. E. Chem. Mater. 1991, 3, 521-527. (18) Vermeulen, L. A.; Thompson, M. E. Nature 1992, 358, 656-658. (19) Frey, B. L.; Hanken, D. G.; Corn, R. M. Langmuir 1993, 9, 18151820. (20) Schilling, M. L.; Katz, H. E.; Stein, S. M.; Shane, S. F.; Wilson, W. L.; Buratto, S.; Ungashe, S. B.; Taylor, G. N.; Putvinski, T. M.; Chidsey, C. E. D. Langmuir 1993, 9, 2156-2160. (21) Vermeulen, L. A.; Snover, J. L.; Sapochak, L. S.; Thompson, M. E. J. Am. Chem. Soc. 1993, 115, 11767-11774. (22) Katz, H. E. Chem. Mater. 1994, 6, 2227-2232. (23) Thompson, M. E. Chem. Mater. 1994, 6, 1168-1175. (24) Vermeulen, L. A.; Thompson, M. E. Chem. Mater. 1994, 6, 7781. (25) Hanken, H. G.; Corn, R. M. Anal. Chem. 1995, 67, 3767-3774. (26) Horne, J. C.; Blanchard, G. J. J. Am. Chem. Soc. 1996, 118, 1278812795. (27) Neff, G. A.; Mahon, T. M.; Abshere, T. A.; Page, C. J. Mater. Res. Soc. Symp. Proc. 1996, 435, 661-666. (28) Horne, J. C.; Blanchard, G. J. J. Am. Chem. Soc. 1998, 120, 63366344. (29) Szabo, A. J. Chem. Phys. 1984, 81, 150-167. (30) Dewitt, L.; Blanchard, G. J.; Legoff, E.; Benz, M. E.; Liao, J. H.; Kanatzidis, M. G. J. Am. Chem. Soc. 1993, 115, 12158-12164.

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