Predicting the Size Distribution in Crystallization of TSPP:TMPyP

Article pubs.acs.org/crystal

Predicting the Size Distribution in Crystallization of TSPP:TMPyP Binary Porphyrin Nanostructures in a Batch Desupersaturation Experiment Morteza Adinehnia, Ursula Mazur,* and K. W. Hipps* Department of Chemistry and Materials Science and Engineering Program, Washington State University, Pullman, Washington 99164-4630, United States S Supporting Information *

ABSTRACT: Crystallization of a binary porphyrin nanostructure (BPN) of TSPP (meso-tetra(4-sulfonatophenyl)porphyrin) and TMPyP (meso-tetra(N-methyl-4-pyridyl)porphyrin) was studied. The morphology and crystallinity of the BPN was investigated using transmission electron (TEM) and atomic force microscopies (AFM). The composition of the BPN was analyzed using X-ray photoelectron spectroscopy (XPS), elemental analysis, and UV−visible spectroscopy. These techniques revealed a 1:1 composition of anionic to cationic porphyrins in the structure. Our initial studies on the synthesis of these materials revealed that the average size of these crystals increases monotonically with synthesis temperature and decreasing monotonically with initial concentration (supersaturation) of the mother solution. In this work we have developed a model to simulate the growth of these organic monocrystalline materials for the first time. This model encompasses all the major kinetic and thermodynamic steps of crystallization including homogeneous nucleation, growth, and Ostwald ripening. The model is then validated by comparing the simulation results with experimental crystallization histograms. The unknown parameters are extracted by fitting the simulation to the experimental data. This investigation will help in better understanding of crystallization and size control in this class of photoactive organic materials. The integration rate constant preexponential is found to be (2.9 ± 1.3) × 106 m4/(mol s), and the activation energy for the integration rate is determined as 44 ± 2 kJ/mol.

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INTRODUCTION Binary porphyrin nanostructures made by self-assembly of anionic and cationic porphyrins provide us with a new class of materials that have numerous applications in catalysis,1 gas storage,2 field-effect devices,3 optical devices,4,5 solar cells,6 and photovoltaics.7,8 Porphyrin nanostructures are particularly promising as light-harvesting and electron-transfer components in solar energy utilizing devices. Self-assembled porphyrin nanostructures also serve as useful models of biological lightharvesting structures. Well-defined nanostructures such as porphyrin nanorods, nanotubes, nanofibers, nanosheets, nanospheres, and other morphologies have recently been reported.9 Porphyrin-based nanomaterials can be produced by several methods including (1) ionic self-assembly (ISA),10,11 which is generally used in making binary structures; (2) the reprecipitation method,10,12,13 which is probably the most common method for making single porphyrin structures; or (3) coordination polymerization.14,15 Ionic self-assembly is the most facile and promising method for synthesis of binary porphyrin nanostructures (BPNs) made of oppositely charged water-soluble porphyrin species.12 ISA is basically mixing the solutions of two oppositely charged water-soluble porphyrins so that the oppositely charged ionic blocks (tectons) can pair © 2014 American Chemical Society

together and produce new binary porphyrin material with novel properties. Many groups including ours have performed quality research on electronic16,17 and mechanical18,19 properties of these materials; yet, the area of size control has not been addressed. For commercial applications, it will be necessary to know how to make BPN crystals in a reproducible way with a given morphology and size distribution. A model is developed here that provides the needed guidance. This model simulates crystallization in BPNs, and particularly 1:1 TSPP:TMPyP crystals, in a batch desupersaturation experiment (experiments where a supersaturated mixture is created and left to reach saturation). The model is established from material balance and simultaneous numerical solution of nucleation and growth formulas. This model includes the three main processes of crystallization, namely, nucleation, growth, and Ostwald ripening. Almost none of the physical and chemical properties for this pair of porphyrins are known and also kinetic and thermodynamic parameters of the crystallization have not Received: October 9, 2014 Revised: November 7, 2014 Published: November 11, 2014 6599

dx.doi.org/10.1021/cg501506s | Cryst. Growth Des. 2014, 14, 6599−6606

Crystal Growth & Design

Article

TSPP−4 (aq) + TMPyP+4 (aq) → TSPP:TMPyP(s)

been investigated for this general class of materials. During this research we measure, calculate, or estimate numerous properties and then extract the crystallization parameters by fitting the simulation to the experimental data.

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(1)

After completion of crystallization, a small amount of suspension was withdrawn. AFM and TEM images were obtained from each sample. Crystal size distributions were plotted by measurement of dimensions of several hundred crystals from each sample. Eight experiments were performed corresponding to different values for temperature, concentration, and mixing time. Thirteen seconds was used for the majority of mixing times because it was easy to do uniformly by hand. The times used in E1−E5 allowed uniform addition of reagents by hand. E6 was done using matched syringes.

EXPERIMENTAL SECTION

Materi al s. Starting materials include m eso -tetra(4sulfonatophenyl)porphyrin dihydrochloride ([H2TSPP]·2HCl, 99%, Frontier Scientific), meso-tetra(N-methyl-4-pyridyl)porphyrin tetrachloride ([H2TMPyP]·Cl4, 99%, Frontier Scientific), and sodium hydroxide (NaOH, 30% w/w, J. T. Baker). These were used without further purification. Stock solutions (500 μM) were prepared by dissolution of porphyrin crystals in Millipore water (18 MΩ), stored, and used within 2 weeks. NaOH stock solution (0.002 M) was prepared from dilution of NaOH (14.7 M) with Millipore water (18 MΩ) and was used to adjust and maintain the pH of the solutions to neutrality. Characterization. The atomic force microscopy (AFM) images were obtained with a Bruker Multimode atomic force microscope using RTESPA tips (silicon cantilever with driving frequency of ∼300 MHz and force constant of ∼40 N/m) in tapping mode at 0.5 Hz scan rate. AFM samples were prepared by depositing 2 drops of crystal solution on a freshly cleaved mica surface (SPI Supplies, Inc.) mounted on a metallic disk. After 3 min the metallic disk was affixed to a spin dryer using double sided tape and spin-dried for 15 s at 1000 rpm to dispose of excess solution and to prevent clustering of aggregates. Transmission electron microscopy (TEM) was used to image the structure of the BPNs and also to acquire selected area diffraction (SAED) patterns. TEM micrographs (images) provide us with information about the general morphology of the crystals. They are also our primary source for measurement of crystal dimensions. SAED patterns give us information about crystal structure (lattice spacing of the unit cell and space group) of our BPNs. The transmission electron micrographs in this work are obtained either with a Philips CM200 UT Intermediate Voltage or High Resolution Transmission Electron Microscope from FEI Company at 200 keV electron beam energy. TEM samples were prepared by depositing 10 μL of the TSPP:TMPyP crystal solution onto a 100 mesh Formvar coated, nickel grid (TED PELLA, Inc.). After a few minutes, the extra moisture was absorbed from the grid using a piece of Kimwipe, to touch the edge of the grid. The X-ray photoelectron spectroscopy (XPS) analysis was performed with a Kratos Analytical Axis 165 ESCA spectrometer equipped with a monochromatic Al Kα source (1486.5 eV.), the binding energy analysis was referenced to C 1s signal at 285 eV. Sample charging was neutralized using an electron flood gun. Quantitative analysis of the ESCA peaks was performed by CasaXPS (Casa Software, Ltd.). The XPS samples were prepared by embedding the crystalline powder in a thin clean high purity indium layer pressed on the XPS stub. The elemental analysis of the nanocrystals was obtained from ALS Environmental (Tucson, AZ.). Note that these data are consistent with a 1:1 salt with eight molecules of solvent inclusion. Anal. Calcd for C88H68N12O12S4·8H2O: C, 60.05; H, 4.93; N, 9.55; O, 18.18; S, 7.25. Found: C, 60.36; H, 4.93; N, 9.55; O, 18.57; S, 6.80. The uncertainty in S based on multiple analysis is ±0.40. UV−vis absorbance spectra of porphyrin monomers at pH 7 were collected using a PerkinElmer 330 UV−vis-NIR spectrophotometer at 0.5 nm resolution. The absorption spectra and spectroscopic proof of relative composition are provided in the Supporting Information in Figures S1−S4. Synthesis of BPN. The desupersaturation experiments were performed in a batch reactor in a temperature-controlled Brinkmann (Lauda-Brinkmnn LP.) bath. The single porphyrin solutions were made at the required concentration for each particular experiment, preheated to the required temperature, and mixed afterward. Each experiment lasted about 2 h. The precipitation reaction is

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MODELING Details of modeling are described in this section. Our modeling method relies on these points: 1) The nucleation rate is a function of supersaturation as descried by Volmer and Weber.21 2) The growth path is controlled both by the diffusion of nutrients from the bulk of the solution to the crystal interface and by the integration of the species in the crystal lattice. 3) The solubility is a function of size according to the Gibbs−Thomson relationship.22 4) The needle-shaped crystal morphology is taken into account through surface and volume shape factors. 5) The contents of the reaction vessel is assumed to be well macro-mixed and micromixed. 6) The required time for mixing the two solutions is taken into account in the model. Nucleation. We assume homogeneous nucleation in crystallization of this binary porphyrin nanostructure. This is justified by the Nielsen criterion, 23 which states that homogeneous nucleation is dominant where there are more than 1012 nuclei/m3 in suspension. Our nucleation rates are in the order of 1013−1014 nuclei/m3 (as predicted by eq 2). According to classical nucleation theory, the rate of nucleation J (the number of nuclei per unit time per unit volume) is expressed in the form of an Arrhenius equation: ⎛ −4β 3γ 3V 2 ⎞ CL m ⎟ J = A exp⎜⎜ 2 3 3 2⎟ ⎝ 27α k T (ln S) ⎠

(2)

where γCL is the interfacial tension between crystal and solution (N/m); k is the Boltzmann constant (J/K); T is the temperature in Kelvin; Vm is the volume of the growth unit (m3); α is the volume shape factor; β is the surface shape factor, and S is the relative supersaturation, defined as follows:

S=

C Ceq

(3) 3

C is the concentration of the solution (mol/m ) and Ceq is the large crystal equilibrium concentration. Kind and Mersmann24 derived an expression for the pre-exponential factor A by combining an impact coefficient and a Zeldovic factor25 (eq 4); DAB is the diffusivity of solute species (m2/s); NA is Avogadro’s number (mol−1); and Cc is the molar density of a large crystal (mol/m3). ⎛ 1 ⎞ γCL A = 1.5DAB(CNA )7/3 ⎜ ⎟ ⎝ CCNA ⎠ kT

(4)

The global interfacial tension between crystal and solution, γCL, is important in crystallization since it is one of the key 6600



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dm = k rA(C i n − Ceq n)r dt

parameters in determining the primary nucleation rate. Nielsen and Söhnel26−28 measured γCL for 58 crystal systems and Mersmann derived an equation for predicting interfacial tension in aqueous systems.29 γCL =

KkT ⎛⎜ Cc ⎞⎟ ln⎜ ⎟ dm 2 ⎝ Ceq ⎠

Integration of growth units on the surface of a crystal is a rather complex process. According to Kossel’s model, the flat looking surface of a crystal is in fact made of steps (layers) of molecular height that incorporate kinks31−33 (see Figure 1).

(5)

where dm is the size of cubic growth unit (dm = (CcNA)−1/3) and K is a constant between 0.31 and 0.414. A value of K = 0.333 is suggested from comparison with experimental data, while Mersmann derived K = 0.414 from an estimate of interfacial geometry.29 We will treat K as a fitting parameter. At constant temperature, there is always a rapid increase in nucleation rate when supersaturation goes up. The relationship between nucleation rate and temperature is unfortunately not that simple. Temperature appears in cubic form in the denominator of eq 2 but also affects the values of A, γCL, and S. In our case the nucleation rate decreases with increasing temperature. Ostwald Ripening. Ostwald ripening is described by the Gibbs−Thomson equation,22 which states that the solubility of a compound is a function of particle size. For a spherical particle of size L (see proof in ref 30), the equilibrium solubility is given by eq 6. Therefore, small particles have higher solubility. This effect becomes particularly important for small particles below 1 μm, and especially in the case of reactive crystallization where high levels of supersaturation produce extremely small particles. ⎛ γ Vm ⎞ Ceq = C∞ exp⎜ CL ⎟ ⎝ kTL ⎠

Figure 1. Kossel’s model of a crystal surface.

When a growth unit arrives at the crystal surface it starts to move on the surface. The initial bonds between the growth unit and the plane are enough to keep it on the surface for a while but not strong enough to inhibit surface diffusion. Thus, the growth unit must find an energetically favorable site or desorb back in the solution. If the crystal surface is rough on the molecular scale (high density of kinks and steps), growth is accelerated. In practice, application of eqs 6 and 7 is difficult because they include the interfacial concentration term, which is hard to measure. It is normal to eliminate it and to consider an overall concentration diving force (C − Ceq). For our growth model we did not restrict ourselves to the simple models of growth, but instead did a rather extensive review of numerous models presented in the literature for different systems29,34−39 and selected the model, which closely matches the identity of our system. The selected model (eq 9) is that which Chiang and Donohue developed for multicomponent precipitation crystallization systems,39 where kr is modeled by an Arrhenius equation (eq 10) and Er is the integration activation energy (kJ/mol). In our analysis we treat k0r and Er as fitting parameters.

(6)

Ostwald ripening is considered in the derivation of the nucleation rate formula, critical nuclei radius, and in the growth formulation. Growth. As soon as stable nuclei (particles larger than critical size) in a supersaturated solution are formed they begin to grow into crystals of visible size. This is done by the addition of growth units to the crystal. While growth parameters are well-known for many inorganic crystals, very little is known in the case of organic materials in general and ionic organic compounds in particular. We model the growth of our crystals by diffusion-integration theory. Diffusion-integration theory assumes that there are two separate steps involved in the growth of the crystal. First the solute will diffuse through the bulk of the solution to get to the interface of crystal/solution (modeled by a diffusion equation), and there it must incorporate (integrate) itself into the crystal lattice (modeled by a reaction equation). The diffusion process is described by eq 7, where m is the mass of crystal; A is the surface area of the crystal; kd is the mass transfer coefficient; C is the concentration of the solute in the bulk of the solution; and Ci is the solute concentration at the crystal/solution interface. The diffusion step is linearly dependent on the supersaturation. The integration rate is given by eq 8, where kr is the rate constant and n is the number of components that make the fundamental crystal composition unit; n = 2 in the case of a binary compound. dm = kdA(C − C i) dt

(8)

RG

(1 + C) − = 2k r kd

1+

4k r kd

(C +

)

kr C 2 kd eq

2k r kd 2

k r = k 0r exp( −Er /RT )

(9) (10)

The mass transfer coefficient (kd) depends on temperature and hydrodynamic conditions. It can be calculated to a good approximation from Armenante and Kirwan correlation23,40 given by eq 11 where η is the viscosity of the solution (Pa s) and ds and dv are the diameters of stirrer and growth vessel (m). ⎛ ρ L4/3 ⎞0.52 ⎛ d ⎞0.17 ⎛ η ⎞0.33 kdL ⎟ ⎜ s ⎟ ⎜⎜ ⎟⎟ = 2 + 0.52⎜⎜ s ⎟ DAB ⎝ η ⎠ ⎝ d v ⎠ ⎝ ρs DAB ⎠

(11)

According to eq 9, growth rate is a function of kd and kr, which are both increasing with temperature. Hence one expects

(7) 6601



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and the new concentration and supersaturation are calculated to be used in the next time interval. There are some considerations in this simulation to make it imitate the real experiment. For example, a mixing time is considered. Our method neglects breakage and agglomeration since we have seen little breakage or agglomeration phenomena at the temperature and concentration range under study in the TEM and AFM images. Also dissolution rate is negligible for this class of barely soluble crystals. These simplifications enable us to calculate size distributions without using population balance solution methods. Further details concerning the program are available as Supporting Information.

the growth rate to be increasing monotonically with temperature. The physical properties for organic ionic crystals are not widely investigated and for this work they had to be measured or estimated. Density was determined by flotation of crystals in a mixture of heptane ρ = 0.684 g/cm3 and carbon tetrachloride ρ = 1.594 g/cm3. The proportion of these two was adjusted until the crystals remained suspended in the mixture. That density was ρ = 1.46 ± 0.08 g/mL. Solubility was measured by dissolution of crystals at various temperatures and calculating the concentration of the equilibrated solution from the UV−vis spectrum. The results show that the solubility is increasing from almost insoluble (3.1 × 10−9 mol/L) at 293 K to barely soluble (6 × 10−8 mol/L) at 343 K. The solubility versus temperature is plotted in Figure S9 in the Supporting Information. The diffusion coefficient was estimated by the method proposed by da Costa et al.41 This method is derived for porphyrin molecules and calculates diffusion coefficients from molecular dimensions. Density functional theory (DFT) was used to determine the molecule dimensions. Further information about diffusion coefficient calculations and a sample of the Gaussian DFT optimization file is provided in the Supporting Information and in Figures S5 and S8. Mersmann’s method was used for prediction of interfacial tension in this aqueous system.29 The formulation of diffusion-integration theory is based on spherical crystals; therefore, a modification is required to make it applicable to crystals with nonspherical shape. This is done by inclusion of shape factors in the growth formulation.42 The volume shape factor (α = Vc/L3) and surface shape factor (β = Vc/L3) of the crystals were determined by first obtaining the actual area, volume, and length by statistical analysis of AFM and TEM data. The values of α and β were then calculated and plotted as a function of the characteristic dimension, L (for needle-shaped crystals it is common to take length as the characteristic dimension). Figures S6 and S7 in the Supporting Information show volume shape factor and surface shape factor graphs versus length for the particles. We simultaneously and numerically solve the equations for nucleation and growth along with mass balance in the reactor. The mass balance equation is given by eq 12 where the first term is the sum over all crystals in the solution (n) having a particular growth rate (Rn,i) and particular surface area(An,i). The second term is the amount of mass spent in making new nuclei at time-step i. ⎛ dm ⎞ ⎜ ⎟ = ⎝ dt ⎠ i

∑ (R n,iA n,i)Ni + Nnucl,iVnucl,iρc n

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RESULTS AND DISCUSSION We determined the composition of our TSPP:TMPyP BPNs at pH 7 through UV−vis, XPS, and elemental analysis tests. In UV−vis experiments we mixed porphyrin solution with excess amounts of TSPP or TMPyP. After crystallization and filtration of crystals, we measured the concentration of the unreacted compounds in the remaining solution. The results show a 1:1 composition of TSPP:TMPyP. Further discussion about UV− vis results and the corresponding graphs are provided in the Supporting Information. Elemental analysis values are provided in the experimental section. In a 1:1 ratio of TSPP:TMPyP the theoretical molar ratios should be C/N = 7.33 and S/N = 0.33. In our crystals, molar ratios of C/N = 7.36 and S/N = 0.31 were observed. XPS. The XPS measurements also suggest a 1:1 structure. A high-resolution scan of the nitrogen 1s region is depicted in Figure 2. To understand this spectrum one should note that

(12)

Figure 2. High-resolution XPS spectrum of nitrogen 1S peak, showing equal number of pyrrole NH, imine N, and pyridinium N atoms. This implies a 1:1 ratio of +4 TMPyP and −4 TSPP. Freebase TMPyP ([H2TMPyP]+4) and freebase TSPP ([H2TSPP]−4) are also shown with their respective N atoms.

We divide the time into small discrete intervals (order of 10 ms) and then calculate the concentration, supersaturation, stable nuclei size, and nucleation rate at the beginning of each time interval. Then the particles that are created at that particular interval, or are already present in the solution, are grown with respect to the growth rate that is calculated as a function of size for each crystal in the solution. The number and size of all crystals that lie at size interval (L, L + ΔL) is stored at the end of each iteration. This histogram is modified in the next iteration according to the change in the total number of crystals (nucleation) and growth that moves crystals to larger size intervals. At the end of each time interval the amount of nutrients consumed by the crystals are calculated

there are three types of nitrogen in TMPyP and TSPP. Two of these three types are present in the TSPP molecule. The imine (−CN−) nitrogen with binding energy near 398 eV and the pyrrole (−NH−) nitrogen at 400 eV.43,44 The TMPyP molecule has both imine and pyrrole nitrogens, but it also has four nitrogens on the meso-substituted groups. These Nmethyl-pyridinium nitrogens and their corresponding peak appear near 402 eV.45,46 The area under the peaks in Figure 2 6602



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Figure 3. (a) AFM image of TSPP:TMPyP nanorods revealing layer growth. The longer edges of layers are oriented perpendicular to the length of the rod (fastest growing direction, shown by the arrow). The average step height is approximately 2 nm. (b) TEM micrograph showing layer growth in BPN nanorods. (c) HRTEM micrograph of TSPP:TMPyP crystals, showing lattice fringes and the lattice spacing of 1.25 nm. The lattice lines are oriented along the length of the crystal.

noticeable point in this figure is that the aspect ratio of the crystals also changes with synthesis temperature. At lower temperature they are more grain-like, while at higher temperature they are more needle-shaped. This is because the linear growth rate is different along the three axes of the crystal. To address this properly we determined the volume and surface shape factors as a function of size of the crystals (Figures S6 and S7 in the Supporting Information). Figure 5a shows SAED patterns at four different spots on a crystal. The diffraction patterns are all parallel to each other,

indicates that there is a 1:1:1 ratio between these different types of nitrogen in the crystal structure. This is further corroboration of a 1:1 crystal composition. TEM and AFM. The first evidence of growth at the molecular scale was revealed in AFM images. In Figure 3a,b, steps are clearly visible and are oriented parallel to the fastest growing direction (length) of the rod. This growth pattern is consistent with Kossel’s model.31 According to surface energy models this means that the face that is normal to the width of the crystal has lower energy than the face that is normal to the length of the crystal. We also note that the average step height is 2 nm, which is close to the height and width of one dimeric growth unit. A HRTEM image (Figure 3c) reveals a lattice spacing of 1.25 nm along the width of the crystal. Our starting point for quantitative growth studies is the effect of temperature and initial concentration on the final size of the crystals. A series of desupersaturation experiments were performed to investigate this effect. Figure 4 shows AFM and TEM images of crystals made at different temperatures. From the images it is obvious that the size of crystals increases with the synthesis temperature. An increase in crystal size with temperature has also been reported for metalated porphyrins47 as well as several other crystal systems.48,49 The other

Figure 5. (a) SAED patterns obtained at different spots along the length of a crystal. (b) TEM micrograph of the same crystal. (c) TEM micrograph of a larger crystal. Diffraction spots overlaid on the image of a single rod section showing interplanar spacing of 1.3 ± 0.1 nm between planes oriented along the length and 0.75 ± 0.05 nm between planes oriented along the width of the crystal.

which indicate that the molecular arrangement and orientation are the same along the length of crystal and that the crystal is in fact a single crystal and not polycrystalline or the result of attachment of several layers of material in different orientations. It is also satisfying to note that the ticker regions of the nanorod give more defined spots than the thinner regions. In Figure 5c, SAED pattern reveals the interplanar spacings to be 1.3 ± 0.1 and 0.75 ± 0.05 nm. The 1.3 nm spaced planes are parallel to the preferential growth direction of the rods. These

Figure 4. AFM images and TEM micrographs of crystals made at different temperatures. Crystals clearly grow bigger with temperature. 6603



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This was done at the small particle end of the distribution in order to not weight small contaminant particle or the occasional rod fragment. At the large particle end, very large particles were seen so seldom that we had no statistical confidence in the histogram values. Modeling Results. Using the model explained above, we simulated the nucleation and growth of crystals. The parameters k0r, Er, and K were found and optimized by means of nonlinear least-squares fitting (trust-region-reflective algorithm). The optimized function was the simultaneous fitting of E1−E5 conditions with initial guess values for the three parameters (Table 2).

are the same planes that are illustrated in Figure 3c. The distance observed in Figure 3c (1.3 nm) is quite consistent with this value obtained from SAED (1.3 ± 0.1 nm). Crystal Size Distribution Histograms. Eight experiments were performed according to Table 1. After 2 h reaction time, Table 1. Conditions for Crystal Precipitation experiment reference

temperature (K)

concentration (μM)

mixing time (s)

E1 E2 E3 E4 E5 E6 E7 E8

293 303 313 323 313 313 313 313

5 5 5 5 20 5 5 5

13 13 13 13 13 1.5 16 63

Table 2. Parameters Used in Computer Simulation parameter interfacial tension constant (K) integration pre-exponential term (k0r) integration activation energy (Er)

small amounts of suspension were sampled and deposited on nickel grids or a mica surface for TEM or AFM measurements, respectively. For each experiment several hundred crystal dimensions were measured. The results are presented for experiments E1, E2, E3, and E4 in Figure 6 and for E3 and E5

values eq 5 eq 8

0.340 ± 0.007 (2.9 ± 1.3) × 106 m4/(mol s)

eq 8

44 ± 2 kJ/mol

Figures 6 and 7 show the simulation results (colored solid bars) at different temperatures and concentrations and compares them to experimental histograms (hollow bars). The horizontal axis is the characteristic length of crystals and the vertical axis is the number of crystals at each bin divided by total number of counted crystals divided by bin-width (N/ (NtotΔL)). This is required to be able to compare simulation with experiment. Since experimental histograms are based upon counting a few hundred particles in each case, simulation histograms are based on records of millions of particles in the solution. Dividing by ΔL is necessary to eliminate the effect of different bin-widths in simulation and in experiment. Looking at the comparison we see that the simulation does a good job of predicting experimental results. This indicates that the thermodynamic parameters used in running the program are reliable. Comparing these values obtained by us to similar values reported for other crystals reveals that our values are close to the values that other groups calculated or measured for inorganic crystals (Tables 3 and 4). The value for the interfacial tension, K = 0.340 ± 0.007 is in quite good agreement with the value of K = 0.333 as suggested from a general survey of experimental data.29

Figure 6. Comparison of experimental histograms (hollow bars) and computer simulation (solid bars) of crystals made at 293, 303, 313, and 323 K (all at 5 μM).

Table 3. Comparison of Integration Energy Barrier (Er) Value in Different Crystals Er

compound iron fluoride trihydrate (β-FeF3.3H2O) potassium sulfate (K2SO4)50 zeolite (TPA-silicalite-1)51 calcium carbonate (CaCO3)52 TSPP:TMPyP crystals

36

Figure 7. Comparison of experimental histograms (hollow bars) and theoretical simulation (solid bars) of TSPP:TMPyP crystals made at 5 and 20 μM (all at 313 K).

61 kJ/mol 42 kJ/mol 42 kJ/mol 39.2 kJ/mol 44 kJ/mol

Effect of Mixing Time. The crystal size distributions (CSD) in Figures 6 and 7 are not Gaussian. The deviation from a Gaussian is greater in cases where nucleation rate is lower (higher temperature and lower concentration). We assign these shape variations as a consequence of a finite mixing time. To support this, we show the results (Figure 8) of three experiments for which the initial concentration and temperature were the same, but the porphyrin solutions were mixed with different mixing times. The shortest mixing time (1.5 s)

in Figure 7 (hollow bars). It is clear that the average size of the crystals is increasing monotonically with synthesis temperature and decreasing monotonically with initial concentration (supersaturation) of the solution. Also the distribution is wider at higher temperatures. In preparing the histograms, we report and use only sizes greater than or equal to 1/6 th of the distribution maximum. 6604



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porphyrin molecules and structures. This, however, is the first time that kinetics of nucleation and growth has been investigated for porphyrin crystals This research links the science of porphyrin structures to the technology of making catalysts, sensors and transistors, and solar cells by providing the engineer (designer) the means to predict and control the size of these materials appropriate to a particular application. Work is underway to test this model on porphyrins crystals having other growth shapes.

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ASSOCIATED CONTENT

S Supporting Information *

UV−vis absorptions, calculation of extinction coefficients, proof of 1:1 structure based on UV−vis technique, DFT calculations method, basis set, and a sample of Gaussian input file, simulation program, volume and surface shape factor plots, diffusion coefficient measurement for TSPP and TMPyP, and solubility as a function of temperature. This material is available free of charge via the Internet at http://pubs.acs.org.

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Figure 8. Comparison of experimental (hollow bars) and model histograms (colored solid bars) with mixing times of 1.5 s (E6), 16 s (E7), and 63 s (E8). The simulation results are also shown with hollow bars, and the parameters used were those from Table 2.

*E-mail: [email protected]. Tel: +1-509-335-5822. Fax: +1509-335-8867. *E-mail: [email protected]. Tel: +1-509-335-3033. Fax: +1-509335-8867.

belongs to E6 where the two porphyrin solutions are mixed rapidly and all the nutrients are quickly available for nucleation; hence, we have a higher number of nuclei and smaller crystals with a narrower distribution. On the other extreme (E8) where we mix the two solutions for a longer period of time (63 s), some nuclei are formed in the very beginning of the mixing in the reactor, and as the gradual feeding is continued, the added nutrients are spent on both growing the existing nuclei and making new nuclei. The total number of crystals will be less in this case but the distribution will be wider. The relatively larger deviations between calculated and observed histograms for the very long mixing time experiment is attributed to short time variations in the addition rate within the 63 s overall interval.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the US National Science Foundation for their support in the form of grant CHE-1152951. We also thank the Francheschi Microscopy and Imaging Center at Washington State University for the use of their transmission electron microscopes.

Table 4. Comparison of Integration Constant Values at 70°C in Different Crystals

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integration constant (kr) @ 40 °C

compound 23

silver bromide (AgBr) sodium chloride TSPP:TMPyP crystals

AUTHOR INFORMATION

Corresponding Authors

REFERENCES

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0.136 m4/(mol s) 0.91 m4/(mol s) 0.13 m4/(mol s)

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CONCLUSIONS In this work a binary porphyrin nanostructure of anionic TSPP and cationic TMPyP was synthesized for the first time and characterized by AFM and TEM imaging. The molecular ratio in this structure has been investigated through UV−vis, XPS, and elemental analysis techniques and found to be 1:1. The crystallinity of these needle-shaped structures has been verified by selected area diffraction studies. A crystal size distribution prediction model was developed based on mass balance and thermodynamic models for nucleation and growth. This model accurately predicts the size distribution of the materials in the studied temperature and concentration ranges. From this model, integration constants were extracted (Table 2). Many groups including ours are developing the science that describes the chemical, physical, mechanical, and electronic behavior of 6605



Article

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Predicting the Size Distribution in Crystallization of TSPP:TMPyP

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