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Impurity Partitioning During Colloidal Crystallization Jun Nozawa, Satoshi Uda, Yuhei Naradate, Haruhiko Koizumi, Kozo Fujiwara, Akiko Toyotama, and Junpei Yamanaka J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp309550y • Publication Date (Web): 01 Apr 2013 Downloaded from http://pubs.acs.org on April 12, 2013

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The Journal of Physical Chemistry

Impurity Partitioning During Colloidal Crystallization Jun Nozawa, † , * Satoshi Uda, † Yuhei Naradate, † Haruhiko Koizumi, † Kozo Fujiwara, † Akiko Toyotama, ‡ Junpei Yamanaka, ‡ †

Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

‡

Graduate School of Pharmaceutical Sciences, Nagoya City University, 3-1 Tanabe, Mizuho, Nagoya, Aichi 467-8603, Japan

Abstract We have found that an impurity partitioning takes place during growth of colloidal crystals, which was recognized by the fact that the impurity concentration in the solid (C S ) was different from that in the initial solution (C 0 ). The effective partition coefficient k eff (= C S /C 0 ) was investigated for pure polystyrene and polystyrene dyed with fluorescent particles by changing the ratio of particle diameters d i mp /d cryst and growth rate V. At each size ratio for the polystyrene impurity, k eff was less than unity and increased to unity with increasing V, whereas at a given growth rate, k eff increased to unity as d imp /d cr ys t approached unity. These results were consistent with the solute behavior analyzed using the Burton, Prim, and Slichter (BPS) model. The obtained k 0 , equilibrium partition coefficient, from a BPS plot increased as d i mp /d crys t approached unity. In contrast, while the fluorescent particles also followed the BPS model, they showed higher k 0 values than those of the same size of polystyrene particles. A k 0 value greater than unity was obtained for impurities that were similar in size to the host particle. This behavior is attributed to the positive free energy of fusion associated with the incorporation of the fluorescent particles into the host 1 ACS Paragon Plus Environment


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matrix. Such positive free energy of fusion implies the presence of the enthalpy associated with interaction between particles.

Keywords: colloidal crystal, impurity partitioning, melt growth, crystal growth

I. INTRODUCTION The control of chemical composition during crystal growth is one of the most important issues for bulk crystals, because the physical properties, such as the electric and optical properties of materials, are quite sensitive to chemical composition. During crystal growth, the partitioning of the impurities and the solute plays a pivotal role in determining the compositional homogeneity of the crystals. The partitioning phenomenon is the result of equilibrium and dynamic processes, including transport of elements to the moving interface and incorporation of an element from the melt to the crystal phase, and hence it can be regarded as the most essential process in melt growth. One method for controlling the partitioning is to apply an external electric field to the liquid–solid interface. 1 –3 Pfann and Wagner discussed the transportation and partitioning of a charged solute under an electric field. 4 This model, which uses k 0 for the partitioning of the charged solute, has been regarded as a standard treatment for field-modified partitioning. 5 ,6 Tiller and Sekerka, however, proposed another model, which introduced the field-modified equilibrium partition coefficient k E0 in place of the conventional k 0 . 7 Uda et al. experimentally proved that k E0 should be used for the partitioning of ionic species under an electric field. 8 –11 Detailed investigation of the partitioning phenomenon is still required, however, not only for 2 ACS Paragon Plus Environment

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industrial applications, but also for the development of a greater understanding of its fundamental mechanisms. Interfacial phenomena during melt growth, such as impurity partitioning, are less understood compared to those involved in vapor and solution growth. For vapor or solution growth, some direct in-situ observations have been carried out using an electric scattering technique, such as reflection high-energy electron diffraction (RHEED) analysis, and microscopy with molecular-scale resolution, including scanning tunneling microscopy (STM) and atomic force microscopy (AFM). These in-situ observations have enabled great progress in the understanding of elemental growth processes. 12– 1 4 Direct observation of the liquid–solid interface during melt growth is difficult, however, due to the high temperature environment and very rich solute concentration in the melt. Molecular scale understanding is also needed to depict the detailed interfacial phenomena of melt growth, and for this purpose, we investigated the growth of a colloidal crystal in order to learn more about the solute partitioning at the solid–liquid interface during melt growth. Colloidal crystals, periodic arrays of colloidal particles, have attracted great interest in a large number of fields because of their potential application in functional devices, such as photonic crystals. 15 In addition, colloidal crystals are also useful for modeling diverse physical phenomena, e.g., glass transition, 16 nucleation, 17

and sublimation. 1 8,19 Although the interactions between atoms and between

colloidal particles are different, it has been recognized that some of the behaviors of colloidal systems follow the classical theory of atomic systems,

20,21

and hence, the

study of colloidal systems provides us with precious information about those processes that are difficult to experimentally detect for atomic systems due to the 3 ACS Paragon Plus Environment


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atomic spatial/time scale. For the impurity partitioning study, differently sized particles from those of a host were employed as an impurity in a colloidal dispersion. Some studies have been reported on the doping of differently sized particles into colloidal crystals. Geerts et al. 22 confirmed that two solid phases were crystallized from a binary mixed liquid as a result of size fractionation, which was also expected theoretically. 23 Recently, Yoshizawa et al. carried out in-situ observation of the exclusion of impurity particles during grain growth. 24 Fabrication of colloidal crystals from a polydisperse dispersion was also studied both theoretically and experimentally. van Megen and co-workers investigated the influence of particle size polydispersity on the nucleation and growth rate of colloidal crystals, and they suggested that particle size fractionation occurred at the crystal-fluid interface. 25 ,26 Although there are a number of studies discussing the fractionation of different particles, to our knowledge, the investigation of the partitioning of impurities during colloidal crystallization has never been applied as a model for understanding an ordinary melt growth system. In this study, we fabricated colloidal crystals containing differently sized particles as doped impurities and determined how much of the impurities were incorporated into the crystal under different growth conditions.

II. EXPERIMENTAL The colloidal particles used in this study were polystyrene spheres purchased from Duke Scientific. Pure polystyrene (PS) particles (500 nm) served as the host. Two types of particles were used as impurities: pure polystyrene like that of the host, 4 ACS Paragon Plus Environment

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and polystyrene dyed with a fluorescent agent (w/fluor). Different sizes of pure polystyrene particles (300, 400, 600, and 700 nm) and fluorescent bearing particles (300, 400, 520, 600, and 700 nm) were doped as impurity particles. All of the experiments were carried out using a 2% initial impurity concentration in the dispersion (C 0 = 2%); namely, there were two impurity particles in 98 constituent particles. A higher impurity concentration was difficult to obtain, because it leads to insufficient crystallinity of the grown colloidal crystals. Convective assembly 27,28 was employed to grow the colloidal crystals. A schematic illustration of colloidal crystal growth is shown in Fig. 1a. The colloidal dispersion was held in an ~80 µl cell made of a glass plate. A thin glass plate (~0.12 mm thickness) as a substrate was inserted into the dispersion and then pulled out with precise control using a step motor. Colloidal particles gathered at the meniscus due to solution flow and formed a colloidal crystal. Growth occurred at the meniscus when the volume fraction, φ , reached approximately 0.49, and the growth rate changed from 0.36 to 3.6 mm/h. The growth direction was the opposite to the pull out direction, which is similar to the growth mode in conventional unidirectional solidification. More than several tens of layers of colloidal crystals were formed on the glass plate (Fig. 1b). The cleavage surface was used to evaluate the bulk impurity composition. That is, the glass plates were previously scratched with a glasscutter, and then cracked after growth of the crystal so as to expose the cleavage surface, which was observed by scanning electron microscopy (SEM). The number of impurities was counted in a given 120 µm 2 area of the SEM images in order to determine the impurity concentration of the crystal (C S ). Each area contained approximately 500 particles, and several tens of areas were examined for one growth. 5 ACS Paragon Plus Environment


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The effective partition coefficient, k eff , was obtained by dividing C S by C 0 .

Fig. 1 (a) Schematic illustration of the growth cell of a colloidal crystal. (b) Cleavage surface of a resultant colloidal crystal with a few tens of layers.

III. RESULTS AND DISCUSSION Typical SEM images of cleavage surfaces of the colloidal crystals are presented in Fig. 2. They show an ordered, close packed, {111} face of a face centered cubic structure. Although {100} and {110} faces were also occasionally observed, only the {111} face was selected in the present study. In addition, the colloidal crystals were not a complete single crystal, but contained a partially polycrystalline region. Impurity concentrations were only determined in the areas of the single crystal with high crystallinity.

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Fig. 2 FE-SEM images of impurity particles in the colloidal crystal. (a), (b), (c), and (d) show substitution impurities of 300, 400, 600, and 700 nm, respectively.

Using the SEM images, the numbers of impurities in the steady-state growth regions were counted. Most of the impurity particles substituted constituent particles, as shown in Fig. 2. These impurities were regarded as substitute-type impurities similar to those observed in an ordinary crystal. The k eff (= C S /C 0 ) values obtained at various growth rates for each impurity size are shown in Fig. 3. For each sample with different impurities, k eff approached unity as the growth rate increased. With respect to the impurity size, the closer the size of the impurity was to that of the host particle, the easier it became for k eff to approach unity. In addition, for each impurity size, the k eff of the fluorescent polystyrene particle was larger than that of the pure polystyrene particle. It should also be noted 7 ACS Paragon Plus Environment


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that the k eff of the 520 nm fluorescent particles surpassed unity.

Fig. 3 Plot of k eff versus growth rate for the crystals. Solid lines refer to pure polystyrene impurities (PS) and dotted lines refer to fluorescent polystyrene (w/fluor).

The dependency between the growth rate and k eff follows the Burton, Prim, and Slichter (BPS) model 29 as described by eq. (1), which is often employed to describe impurity partitioning during melt growth:

keff =

k0

 Vδ  k0 + (1− k0 ) exp  − c   DL 

,

(1)

where k 0 is the equilibrium partition coefficient, V is the growth rate of the crystal, D L is the diffusion coefficient of the impurity in dispersion, and δ c is the thickness of the diffusion boundary layer. When k 0 is less than unity, the impurity is rejected by 8 ACS Paragon Plus Environment

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the interface and forms an impurity boundary layer. The shape of boundary layer is determined by both diffusion and convection. The fluid dynamics is critical for the magnitude of the thickness of boundary layer, δ c . Here, based on the BPS model, distribution of the impurity is obtained from four parameters, k 0 , V, D L and δ c . The k eff is determined by using these parameters as eq (1), and eventually, impurity concentration in boundary layer depends on k eff . Rearranging eq. (1), we obtain the following equation for a BPS plot:

 1  1  δ ln  −1 = − c V + ln  −1 DL  keff   k0 

(2)

The k eff values are replotted in Fig. 4 as a function of the growth rate V according to eq. (2). It should be noted that ln[(k eff ) – 1 –1] has a linear relationship with the growth rate, indicating that the partitioning behavior of the colloidal crystal can be evaluated using the BPS model. The slope of the fitted line gives δ c /D L , while the intersection at V = 0 gives k 0 (Table 1). In order to obtain δ c , the diffusion coefficient of each particle was determined using the Stokes–Einstein equation, and the results were found to be similar for the different particles and on the order of 10 -12 m 2 /s. In contrast, the experimental δ c values in the present study were at most 10 times larger than the particle sizes. The diffusion layer length of an ordinary melt growth is often on the order of ~100 µ m, which is about 10 6 times larger than the size of atomic or molecular impurities.



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Fig. 4 Plot of ln(1/k eff – 1) versus growth rate for the crystals.

Table 1. Values of k 0 and δ c for the polystyrene (PS) and fluorescent impurity (w/fluor) particles obtained from the BPS plot.

The obtained k 0 values are summarized in Fig. 5. The blue solid lines represent the polystyrene impurities, while the red dashed lines indicate the polystyrene impurities 10 ACS Paragon Plus Environment

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with the fluorescent agent. The k 0 of the 500 nm polystyrene impurity is also plotted in the figure at unity without experimental data, because it is the same material and the same size as the crystal. The value of k 0 for the polystyrene impurities other than those with a diameter of 500 nm was always less than unity, and decreased as the difference between the size of the impurity and the 500 nm host particle increased. In contrast, the value of k 0 for the 520 nm fluorescent particle surpassed unity. Moreover, the k 0 of each fluorescent polystyrene impurity was larger than that of the corresponding pure polystyrene particle.

Fig. 5 Summary of the k 0 values for each impurity.

Thurmond and Struthers 30 formulated an equation for the equilibrium partition coefficient of a solute in an alloy based on the van’t Hoff equation. The simple form of k 0 in eq. (3) from Thurmond and Struthers includes the transition free energy difference, ∆GTrj , between the solid and liquid phases of an impurity ‘j’ at the 11 ACS Paragon Plus Environment


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transition temperature, T, at which the transition from the solid to the liquid phase of the host material occurs, and the excess enthalpy of solute, ∆H S , which is caused by incorporation of the impurity into the host material as follows:

k0 =

 ∆G j − ∆H  S   Tr = exp RT xLj   xSj

(3)

where xSj is the mole fraction of the impurity (‘j’) in the solid, xLj is the mole fraction of the impurity (‘j’) in the liquid, and R is the gas constant. This model was applied in order to discuss the difference in the equilibrium partition coefficient between the pure polystyrene and the polystyrene with fluorescent particles. During typical crystal growth, ∆GTrj is obtained at the transition temperature of the host material. On the other hand, the volume fraction, φ , which is the total particle volume divided by that of the system volume, is used for colloidal crystal growth as a parameter for the phase transition of the colloidal dispersion to a colloidal crystal. Investigation of the volume fraction of the phase transition, φ Tr , for two different types of impurity is necessary to evaluate ∆GTrj and k 0 for both. Comparing the φ Tr of the impurity to that of host particles, whether contribution of

∆GTrj in eq.(3) is positive or not is obtained. Therefore, φ Tr was evaluated by means of the effective volume fraction, φ eff , which is defined as:

d  4π  eff   2  φeff = n 3

3

(4)

where n is the number density of the particles and d eff is the effective diameter of the particles. The effective diameter, d eff , is the sum of the real radius of the particle, d p , and the additional portion due to the thickness of the electrical double layer, d c : 12 ACS Paragon Plus Environment

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deff = dp + dc

(5)

The number density, n, was measured by counting the number of particles in liquid phase. Fig. 6(a) shows a typical colloidal crystal image observed by optical microscopy using an oil immersion lens. When the colloidal dispersion is sealed in a cell, a small amount of overflowed dispersion is trapped in the small gaps between the silicone sheet and the glass plate, and colloidal crystals grow at these sites (Fig. 6(b)). The number density of each particle type or size was calculated from the images. Observations were carried out for liquid coexisting with the solid phase, for which the phase transition volume fraction of the liquid could be obtained.

(a)

(b)

1 µm

Fig. 6 (a) Optical microscopy image of a 2-D colloidal crystal for number density measurement. (b) Schematic illustration of a growing cell of 2-D colloidal crystals.

The d eff was calculated from the Zeta potential value that was obtained via electrophoretic mobility measurements (Table 2-1). On the basis of Barker and Henderson perturbation theory, 31 d eff is obtained as:

deff = d p + ∫

∞ r=d p

1− exp {−UY ( r ) kBT }dr.  

(6)

where r is the center-to-center distance between the particles and k B is Boltzmann’s constant. The term U r is the Yukawa-type interaction pair potential given by:

 U Y r = A Ze 

()

( )

2

 4πε  exp −κ r r , 

( )

(7) 13

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where Z is the number of the charge that is deduced from the Zeta potential measurement, e is the elementary charge, ε is the permittivity of water, and 1/ κ is the

 Debye length. The term A  = exp κ a 



( ) (1+ κ a)  is a geometric factor that shows the 2

effect of the particle size. The obtained values for d eff for each particle are listed in Table 2-1.

Table 2-1 Zeta potential ( ζ ) , effective diameter (d eff ), number density (n), and effective volume fraction ( φ eff ) of the polystyrene (PS) and polystyrene with fluorescent particles (w/fluor.).

Particle

ζ (mV)

deff (nm)

n (number/µm 2 )

φ eff

PS-300

-73.8

371

8.85

0.489

PS-400 PS-500 PS-600 PS-700 w/fluor.-300

-74.6 -73.9 -79.5 -75.0 -84.7

476 632 679 826 378

5.52 3.13 2.53 1.75 8.57

0.484 0.490 0.494 0.494 0.498

w/fluor.-400 w/fluor.-500 w/fluor.-600 w/fluor.-700

-49.3 -49.5 -61.1 -58.0

455 589 665 791

5.63 3.22 2.68 1.90

0.461 0.435 0.484 0.470

Table 2-2 Ratio of effective volume fraction ( φ eff ) between PS and w/fluor. particles.

Particle

φ Fleff /φ PSeff

300 400

1.018 0.953

500 600 700

0.888 0.981 0.950

Combining the number density, n, with the effective diameter, d eff , gives φ eff, as 14 ACS Paragon Plus Environment

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shown in eq. (4). The values of d eff for the pure and fluorescent polystyrene particles with the same size were then compared. The ratios for φeffw/fluor. φeffPS are also listed in PS Table 2-2. For each particle size, it was found that φeffw/fluor. is less than φeff . In the hard

sphere model, the phase transition begins at φ ∼ 0.49, as discussed in terms of only the entropy.

However,

because

charged

colloidal

particles

experience

electrical

interactions with other particles, the interaction for different materials would give rise to different levels of enthalpy contribution to the free energy, and hence they would have values for φ Tr that could be different from 0.49. Based on the fact that

φeffw/fluor. φeffPS 0 , 2

(17)

the mole fraction, x, monotonically increase with the φ , and hence the experimental w/fluor. PS w/fluor. PS result of φTr(imp) leads to xTr(imp) . Moreover, when we assume that unit < φTr(host) < xTr(host)

volume of vacancy, v Va , is the same as that of the colloidal particle, v col , the mole 16 ACS Paragon Plus Environment

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fraction x is equivalent to the volume fraction φ .

Fig. 7 Molar Gibbs free energy with respect to mole fraction of colloid for the (a) host polystyrene crystal, (b) polystyrene impurity, and (c) fluorescent impurity.

Here, we will discuss the reason why the k 0 of the fluorescent impurity can surpass unity. The molar free energy curve for the polystyrene impurity is the same as that of the host because the materials are the same (Fig. 7a and b). Since the phase transition volume fraction between polystyrene and that of host is the same, which PS PS corresponds to the same mole fraction, xTr(imp) , the free energy change of the = xTr(host)

polystyrene, ∆GTrPS (= µ LPS − µSPS ), is zero (Fig. 7b). This is because the common tangent lines are the same for host and impurity. In contrast, because the fluorescent particle is a different material from the host, w/fluor. PS w/fluor. PS they have values for φTr(imp) that are smaller than φTr(host) , namely xTr(imp) . The free < xTr(host)

energy change of the fluorescent particles, ∆GTrw/fluor. , for the case of solidification 17 ACS Paragon Plus Environment


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PS from the solution of xTr(host) is shown in Fig. 7c as the difference between µ L and µ S , PS obtained from the tangent line of G L at xTr(host) and the parallel tangent line to G S ,

which indicates that ∆GTrw/fluor. is positive. The values for ∆GTrj (j=PS, w/fluor.) for the pure polystyrene and fluorescent PS particles at xTr(host) are shown in Fig. 8a assuming that it is constant regardless of

particle size. The value of ∆H S is related to the strain or bonding energy change due to impurity partitioning, 36 and is assumed to be the same for both types of impurities. The value of –∆H S increases as the difference in the sizes of the impurity particle and host crystal increases (Fig. 8b). The values for ∆GTrj − ∆H S are plotted in Fig. 8c. Because ∆GTrPS = 0 and then ∆GTrPS − ∆H S is negative for the polystyrene impurity, k 0 is always

less

than

unity

(Fig.

8d).

For

the

fluorescent

particles,

however,

∆GTrw/fluor. − ∆H S is larger than the pure polystyrene, because ∆GTrw/fluor.> 0, which makes k 0 of the fluorescent particles slightly larger than that of the pure polystyrene. When ∆H S becomes small enough as the impurity particle size reaches that of the host, it is possible that ∆GTrj − ∆H S > 0, and thus the k 0 of the fluorescent impurity can surpass unity.

Fig. 8 (a) Variation of the free energy of transition from solid to liquid for an impurity, and (b) excess enthalpy due to partitioning for the polystyrene and 18 ACS Paragon Plus Environment

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fluorescent impurities. Vertical dashed line corresponds to the same impurity particle size as the crystals. (c) and (d) show ∆GTrj − ∆H S and k 0 for the polystyrene and fluorescent impurities, respectively.

IV. CONCLUSION We have experimentally and analytically confirmed that solute partitioning occurs during colloidal crystallization. The partitioning behavior of the colloidal crystal demonstrates that k eff approaches unity with increasing growth rate, and k 0 of the impurity approaches unity when its size becomes close to that of the host. That is, the impurity partitioning of colloidal crystals basically follows the BPS model. Impurity partitioning also changed depending on the type of impurity particles. The k 0 was different for pure polystyrene and polystyrene with fluorescent particles, which was due to the effective volume fraction, φ eff . This difference implies a contribution of the enthalpy to the free energy of the colloidal system in addition to the entropy effect that has been thought of as the only cause of colloidal crystallization.

ACKNOWLEDGEMENTS This work was supported by Grants-in-Aid for Scientific Research ( 22656141 ) from the Ministry of Education, Science, Sports and Culture of Japan. The authors would like to thank the cooperative program of the Advanced Research Center of Metallic Glasses, Institute for Materials Research, Tohoku University, for assistance 19 ACS Paragon Plus Environment


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with the FE-SEM observations.


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Tiller, W.

A.;

Sekerka, R.

F.

Redistribution

of

Solute

During Phase

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Impurity Partitioning During Colloidal Crystallization - American

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