Gel Growth of K2PbCu(NO2)6-Elpasolite Single ... - ACS Publications

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Gel Growth of K2PbCu(NO2)6‑Elpasolite Single Crystals Lianyang Dong,†,‡ Tiglet Besara,‡ Alyssa Henderson,‡ and Theo Siegrist*,†,‡ †

Department of Chemical and Biomedical Engineering, FAMU-FSU College of Engineering, Tallahassee, Florida 32310, United States ‡ National High Magnetic Field Laboratory, Tallahassee, Florida 32310, United States ABSTRACT: Single crystals of K2PbCu(NO2)6 have been grown from aqueous solution; however, convection effects often introduce defects in the crystals and contribute to a high nucleation rate. Here, we show that K2PbCu(NO2)6 single crystals can be grown in silica gels where convection is largely eliminated, resulting in diffusion controlled crystal growth. The diffusion coefficients of the Cu2+- and Pb2+- acetates in potassium silica gel were determined using a classical onedimensional (1D) diffusion model. The influence of the reactant concentrations on the nucleation location and growth was investigated. The growth rate was analyzed using Frank’s model for diffusion-controlled growth rates for a radially symmetric crystal. The magnetic susceptibility of K2PbCu(NO2)6 shows evidence of 1D antiferromagnetic behavior based on a Bonner− Fisher fit.

1. INTRODUCTION

In particular, K2PbCu(NO2)6 undergoes two successive structural transitions at 281 and 273 K which were previously investigated using electron paramagnetic resonance (EPR) techniques.6 On the basis of precise X-ray and neutron measurements, Noda et al. found that the K2PbCu(NO2)6 elpasolite phase transitions proceed from the face centered cubic phase 1 to the tetragonal phase 2 at 281 K with the c-axis shortened, and to phase 3 with a symmetry lower than orthorhombic, due to the cooperative Jahn−Teller effect.1 It was further noticed that K2PbCu(NO2)6 is also a potential candidate hosting one-dimensional (1D) antiferromagnetic chains with S = 1/2 per Cu2+ due to the Cu orbital order induced by the Jahn−Teller distortions.7 The Cu spins are weakly interacting due to the long contact distance of 7.55 Å between Cu atoms, and even weaker interactions between the resulting chains. This leads to interesting low temperature magnetic effects, with three-dimensional long-range antiferromagnetic order well below 1 K. Heisenberg antiferromagnetic linear chain systems are hosts to a number of interesting magnetic properties, such as quantum critical behavior in a magnetic field,8 formation of a Tomonaga-Lüttinger liquid,9 and gapless spin excitations.10 Further investigations require high quality single crystals of sufficient size (mass) for measurements of the spin dynamics using a number of different techniques, such as neutron scattering, and thermodynamic measurements such as magnetic field dependent specific heat measurements, solid state NMR studies, to mention just a few.8,11,12

K2PbCu(NO2)6 belongs to the crystal system of the elpasolite family, with composition A2BM(NO2)6, where A = K, Rb, Cs, Tl; B = Ca, Sr, Ba, Pb; and M = Fe, Co, Ni, Cu. These systems have been studied extensively, and especially the copper elpasolite K2PbCu(NO2)6 has attracted special interest as it exhibits successive phase transitions due to the cooperative Jahn−Teller effect.1−5 The structure of copper elpasolite K2PbCu(NO2)6 at room temperature is face centered cubic (space group Fm3̅) with lattice parameter a = 10.677(5) Å (see Figure 1): The Cu atoms are at the centers of CuN6 octahedra with the triangular NO2 groups having their oxygen atoms pointing outward and positioned in the xy, yz, and zx planes. Six NO2 groups from neighboring octahedra ordering in planes envelop the Pb atoms. The K atoms are in large interstitial sites between the CuN6 octahedra.

Received: May 5, 2017 Revised: August 12, 2017 Published: August 22, 2017

Figure 1. Crystal structure of K2PbCu(NO2)6. © 2017 American Chemical Society

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DOI: 10.1021/acs.cgd.7b00646 Cryst. Growth Des. 2017, 17, 5170−5177

Crystal Growth & Design

Article

Figure 2. Four different combinations to form K2PbCu(NO2)6 in U-tube gel. acetate in the right arm, and KNO2 in both arms. The four different setups are placed on a shelf in a temperature-controlled environment for observation of the growth process. 2.4. Determination of Diffusion Coefficients. The mass transport through the gel is via diffusion, and it is the only mechanism available to supply the solute to the growing crystal. Thus, the diffusion coefficients of the Pb- and Cu-acetates were determined. This was facilitated since both acetates are clearly distinguishable from the gel. The diffusion coefficients at room temperature are determined by visual inspection of the diffusion front over the duration of several weeks. A simple linear geometry was implemented (see Figure 3), with

To investigate the low temperature physical properties of this compound, high quality single crystal samples are required. Single crystals of K2PbCu(NO2)6 had been grown previously from aqueous solutions using the temperature reduction method:2,4 an almost saturated warm aqueous solution (40 °C) of reactants was cooled slowly to room temperature. However, the crystals grown above room temperature frequently show defects and irregular shapes. Here, we describe a novel way to grow K2PbCu(NO2)6 single crystals at room temperature by crystallization in a gel. Using a gel and counter-diffusing solutions, it is possible to control the degree of supersaturation, and therefore, the nucleation rate. Additionally, gels support a growing crystal, but also provide physical isolation from the container walls, thus reducing heterogeneous nucleation. Gels further eliminate convective mass transport, giving an exclusively diffusioncontrolled environment for crystal growth.13

2. EXPERIMENTAL SECTION 2.1. Preparation of Stock Solution. To avoid impurity inclusion in the elpasolite K2PbM(NO2)6, a potassium metasilicate based gel was chosen. First, a potassium metasilicate stock solution was prepared by combining 50 g of potassium metasilicate with 125 mL of DI water, stirring vigorously for 24 h to obtain a uniform mixture, followed by filtering. The resulting mixture was stored in an airtight container to reduce unintended oxidation. 2.2. Preparation of Gels. Gels were produced by adding 0.5 M acetic acid to a 5:1 water to potassium metasilicate stock solution. After the solution was stirred, the gels were allowed to set, with gelling action normally occurring within a few minutes, resulting in slightly basic gels. 2.3. Setup of K2PbCu(NO2)6 Single Crystals Growth in Gel. Typically, the container used for the gel method is a test tube or a Utube. In a test tube, at least one of the reactants is dispersed in the gel , and the other reactant solutions are layered slowly over the gel, providing a solution-to-gel interface. However, no large single crystals were obtained in our trials. Subsequently, a U-tube setup with a neutral gel that provides two independent interfaces was used to grow K2PbCu(NO2)6 crystals. The reactants used for the growth of K2PbCu(NO2)6 crystals are KNO2, and the acetates Pb(CH3COO)2 and Cu(CH3COO)2. The solubility of the reactants at room temperature is about 3220 g/L for KNO2, 80 g/L for Cu(CH3COO)2, and 443 g/L for Pb(CH3COO)2.14 Upon direct mixing of the solutions in any combination that was used for the gel growth, precipitation of polycrystalline K2PbCu(NO2)6 occurred immediately, indicating low solubility of the product. After gelation, the reactant solutions were carefully layered over the gel so that the solution-to-gel interface remained undisturbed. Three components are needed to form the elpasolite phase, and KNO2 plays an important role as discussed below. Four different combinations are explored as shown in Figure 2: (1) KNO2 in the left arm, both Pb- and Cu-acetates in the right arm; (2) KNO2 and Pb-acetate in the left arm, Cu-acetate in the right arm; (3) KNO2 and Cu-acetate in the left arm, Pb-acetate in the right arm; and (4) Pb-acetate in the left arm, Cu-

Figure 3. Model of 1D diffusion in a gel 8 mL of neutral gel prepared in a test tube, connected to a large reservoir containing 80 mL of reactant solution to ensure that the concentration at the gel−solution interface remains constant. In addition, the diffusion coefficients of the reactants are assumed to be concentration independent. Upon contact with the gel, the reactant solution starts to diffuse into the gel: the diffusion front location was recorded at different times until the reactant reached the end of the gel. This simple setup is described by the 1D diffusion equation given as D

∂ 2C ∂C = ∂t ∂x 2

t>0

(1)

where D is the diffusion coefficient, C is the reactant concentration, x is the distance, and t is the time. This specific problem can be solved using a similarity method by introducing a single independent dimensionless variable η, which is a function of diffusion distance and time: C(η) = 1 −

2 π

where η is given as x η= 2(Dt )1/2

∫

0

η

e −s

2

dS ≡ erfc(η) (2)

(3)

and erfc(η) is the complementary error function15 as plotted in Figure 4. It is seen that erfc(2.0) is essentially 0, indicating that the concentration change between η = 0 and η = 2 is close to 100%. Therefore, we define the diffusion front for η: C(η) = 0, and η ≈ 2.0, giving the diffusion coefficient (D) as 5171



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gel, and similarly, the diffusion coefficient of Cu2+ in aqueous solution is about 13−17 × 10−6 cm2/s,19 about 10 times of that in the gel. Therefore, the gel not only eliminates convection, it also decreases the diffusion coefficients of the ions in comparison to water, which is beneficial for controlling the nucleation and growth rate.

3. RESULTS AND DISCUSSION 3.1. Crystal Formation and Location of Nucleation. The concentration and the initial physical location of the reactants are considered in more detail. In the following, the influence of the concentrations will be discussed: The chemical reaction for the formation of K2PbCu(NO2)6 is given as 2K+ + 6NO2− + Pb2 + + Cu 2 + = K 2PbCu(NO2 )6 ↓

(5)

For our experiments, the reactants have concentrations [K+], [NO2−], [Pb2+], [Cu2+], and the product has a concentration of 1 for a solid so that the total change in Gibbs free energy from equilibrium for the reaction is

Figure 4. Plot of complementary error function.

D = x 2/16t

ΔG = RT ln Keq {[K+]2 [NO2−]6 [Pb2 +][Cu 2 +]}

(4)

It is therefore clear that the concentration of KNO2 will dominate the thermodynamic equilibrium conditions for crystal growth. To map out the conditions for nucleation and growth, 4 different ion combinations and 3 different concentrations resulting in 12 setups, are listed in Table 2. After approximately two months, images of the different setups were acquired and are presented in Figure 6. As can be seen, higher concentrations increase the nucleation rate, especially for the case with a higher [NO2−] concentration. In addition, smaller crystals are located closer to high [NO2−] concentration, demonstrating that the [NO2−] concentration plays a dominant role in the nucleation and growth processes as it enters with the power of 6 in eq 6. It is also noteworthy that the color of the reactant solution and gel changes gradually to green due to the diffusion and consumption of the reactants. Furthermore, crystal growth is expected to slow as the reactant concentration continually decreases. 3.2. Crystal Size Distribution and Crystal Location. To evaluate the 12 experiments, a simple statistic is given in Figures 7 and 8, where the location and sizes of the crystals are recorded. In general, large crystals tend to form at the center of the U-tube. Specifically, setups 1 and 4 have most of their crystal formation in the middle of the gel, indicating that the presence of KNO2 does not influence the diffusion rates of Pb(CH3COO)2 and Cu(CH3COO)2. The off stoichiometric ratio in setup number 7, with excess [NO2−], pushed the crystal formation position further to the right, with the higher [NO2−] concentration (left arm) not only producing many small crystals, but also larger crystals primarily located at positions further from the [NO2−] side. This indicates that as the concentration of [NO2−] decreases, the nucleation rate is reduced, forming fewer nuclei, but resulting in larger crystals. In setups 2 and 5, the number of nuclei is and the crystal sizes are reasonable, but not as large as expected due to the reduced concentration. Setup 8, with higher [NO2−], yields a larger number of small crystals due to the increased nucleation rate. It is not clear if the formation of Cu[NO2−]64−ion facilitates the growth of K2PbCu(NO2)6 crystals, but setups of 3, 6, and 9 have a distinctly higher numbers of nuclei, and thus smaller crystal sizes located close to the Cu2+ and [NO2−] side. In contrast, setups 2 and 5 yielded fewer nuclei and therefore

Therefore, the diffusion coefficients are easily determined by measuring both diffusion time and diffusion distance. Figure 5 shows the setups for determining the diffusion coefficients of the Cu2+- and Pb2+-acetates (0.3 M), and results are tabulated in

Figure 5. Setup for measurements of the diffusion coefficients of Cu2+and Pb2+-acetates. Table 1. Unfortunately, no diffusion coefficients could be obtained for K+ and NO2− as they are not easily observed in the gel.

Table 1. Diffusion Distance of Pb(CH3COO)2 and Cu(CH3COO)2 Increases with Time time (h)

diffusion distance (mm) of Pb(CH3COO)2

diffusion distance (mm) of Cu(CH3COO)2

0 0.5 1 1.5 2 3

0 2.0 2.9 3.5 4.1 5.1

0 2.0 3.0 3.6 4.2 5.2

(6)

Each experiment lasted 3 h, using a concentration of the reactant solutions of 0.3 M. The resulting diffusion distance versus diffusion time is shown in Table 1. According to eq 4, the diffusion coefficients for Pb(CH3COO)2 and Cu(CH3COO)2 are 1.50 × 10−6 cm2/s and 1.57 × 10−6 cm2/s, respectively, typical for a number of different ions in silica gel.16,17 Hence, for setups with counter-diffusing Cu2+- and Pb2+-acetates, the reactants are expected to reach each other in the middle of the gel. Typically, the diffusion coefficients of ions in water are about 5−10 times larger than in the gel. For instance, the diffusion coefficient of Pb2+ in water is 9.39 × 10−6 cm2/s,18 about 6 times higher than in the 5172



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Table 2. Combinations and Concentrations for All Twelve Different Setups setup no.

left side

1

1.8 M KNO2

2

1.8 0.3 1.8 0.3 2.4

M M M M M

KNO2 Pb(CH3COO)2 KNO2 Cu(CH3COO)2 KNO2

2.4 0.4 2.4 0.4

M M M M

KNO2 Pb(CH3COO)2 KNO2 Cu(CH3COO)2

3 4 5 6

right side

setup no.

0.3 M Pb(CH3COO)2 0.3 M Cu(CH3COO)2 0.3 M Cu(CH3COO)2

7

3.0 M KNO2

8

0.3 M Pb(CH3COO)2

9

0.4 M Pb(CH3COO)2 0.4 M Cu(CH3COO)2 0.4 M Cu(CH3COO)2

10

0.4 M Pb(CH3COO)2

12

3.0 0.3 3.0 0.3 1.2 0.2 1.8 0.3 2.4 0.4

11

left side

M M M M M M M M M M

KNO2 Pb(CH3COO)2 KNO2 Cu(CH3COO)2 KNO2 Pb(CH3COO)2 KNO2 Pb(CH3COO)2 KNO2 Pb(CH3COO)2

right side 0.3 M Pb(CH3COO)2 0.3 M Cu(CH3COO)2 0.3 M Cu(CH3COO)2 0.3 M Pb(CH3COO)2 1.2 0.2 1.8 0.3 2.4 0.4

M M M M M M

KNO2 Cu(CH3COO)2 KNO2 Cu(CH3COO)2 KNO2 Cu(CH3COO)2

Figure 6. Results of all 12 different setups after a two-month growth run.

mm to 4.5 mm on edge, is shown in Figure 9, on a 1 mm × 1 mm grid. Suspended in the soft gel, the crystals grow close to thermal equilibrium and develop {1 0 0} face bounded cubes, reflecting the cubic symmetry. With increasing size, {1 1 1} and {1 1 0} faces develop to reduce the surface area, thereby lowering the free energy. To remove gel from the crystal surfaces, the crystals were “etched” in water, giving a rough surface morphology, as can be seen in the two crystals to the right in Figure 9. In contrast, the inset figure at upper left of Figure 9 shows the crystals obtained by the temperature reduction method. Optical inspection shows that the number of defects and intergrowths are higher than in the gel grown crystals, and the largest crystal is about half the size and 1/10th in mass as compared to the gel grown crystal. Using the temperature reduction method, crystals tend to nucleate heterogeneously at the container walls and show defects likely caused by fast growth due to convection at elevated temperatures.

larger crystals. The largest crystal was obtained in setup 7, on the opposite side of [NO2−], confirming that the [NO2−] concentration determines the nucleation rate. For setups 10, 11 and 12, nucleation increases with increasing concentration, crystals with size up to 3.5 mm, 3 mm, and 2.5 mm grew at the center of the gels. A small number of crystals form at the Pb2+ and [NO2−] side, but a higher number of nuclei and thus smaller crystals form close to the Cu2+ and [NO2−] side, similar to what was observed previously. In Figure 8 bottom right, only the result of setup 10 is shown because three combinations of KNO2 (1.2 M), Pb(CH3COO)2 (0.2 M), and Cu(CH3COO)2 (0.2 M) with one reactant solution in one end, and two in the other, did not produce any crystals. In setup 10, KNO2 (1.2 M) was added at both sides and again, cube shaped crystals were obtained. The addition of KNO2 in both sides increased the concentration product of the participant reactant ions. 3.3. Crystal Morphology and Growth Rate. A row of crystals grown from the gels, with crystal sizes varying from 1 5173



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Figure 7. Comparison of concentration under same combination (interval 0.5 cm, origin at left interface, gel length 10 cm).

⎡ 2 ⎛ s ⎞⎞⎤ π⎛ ⎜1 − erf⎜ ⎟⎟⎥ = AF(s) Co − C∞ = A⎢s−1e−s /4 − ⎝ 2 ⎠⎠⎦ ⎣ 2 ⎝

To estimate the growth rate of K2PbCu(NO2)6 single crystals in the gel, the reactant concentration C∞ (supersaturated) at large distances from the crystal is considered constant. This assumption is valid if the total mass of the crystal is small compared to the mass of the reactants in the gel, and it is especially appropriate for the initial stages of crystal growth. At the crystal surface, the concentration will initially have the value of C∞ but will adjust to a lower value Co in response to the growth process, defining the crystal growth domain as the space where the concentration is affected by the growing crystal. The concentration profiles may be predicted based on Fick’s diffusion laws; however, the concentration profiles change with time and adjust to the growing crystal surface. The concentration profiles become even more complicated when multiple crystals form close to each other and their growth domains overlap.20 Frank had given a description of diffusion-controlled growth rates for a radially symmetric crystal shape.21 A “reduced radius” s was defined as r 1/2 for a spherical system, where r is

(8)

where A is a constant and F is a function of s. If the difference Co − C∞ is constant, the value of s also remains constant, so that the crystal radius r is expected to be proportional to the square root of time t. In our experiments, crystals start out as cubes and become polyhedral instead of spherical. However, the length a of the cube edge can serve as the relevant dimension for the analysis of the growth rate. A plot of a2 vs t for a crystal well isolated from all other crystals during its growth period is shown in Figure 10. As can be seen, linear behavior (along blue line) is observed until the crystal edge reaches about 2 mm for this specific crystal. However, the saturation difference cannot remain constant as the reactants are exhausted and the stress on the crystal surfaces from the deformation of the gel increases. Furthermore, the shape of the crystal is not spherical but polyhedral (cubic) in our case, with {100} faces. The development of different faces such as {111} and {110} is expected to affect the overall growth rate, as different faces are expected to have different growth velocities. All these factors give rise to the observed nonlinearity. A second order polynomial fit of the square of the cubic edge length a (red

(Dt )

the crystal radius, t is time, and D is the diffusion coefficient of the reaction product. The solution to the diffusion equation

∂C = D∇2 C ∂t

(7)

is given as 5174



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Figure 8. Comparison of combination under the same concentration (interval 0.5 cm, origin at left interface, gel length 10 cm)

Figure 10. Crystals size as a function of time.

Figure 9. Examples of crystals harvested from the gel method (inset figure: examples of crystals harvested from the temperature reduction method).

growth by adding reactants may maintain the linear behavior to larger crystal sizes. 3.4. Magnetic Susceptibility. The magnetic properties of the of K2PbCu(NO2)6 at low temperatures is shown in Figure 11, where 1D magnetic behavior is observed. The magnetic susceptibility of K2PbCu(NO2)6, measured from room temperature down to T = 1.7 K exhibits a broad peak at 3.6 K, indicating short-range intrachain order due to the

curve) gives a2 = 0.343t − 0.001t2. The deviation of the growth rate is mainly observed for large crystals, where the above assumptions no longer hold. However, the overall deviations are considered minor, and increasing the concentrations during 5175



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Lianyang Dong: 0000-0003-2729-8137 Tiglet Besara: 0000-0002-2143-2254 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Science Foundation, under Award NSF DMREF-DMR 1534818. Part of the work was carried out at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation under Award NSF-DMR-1157490, the State of Florida and Florida State University.

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Figure 11. Temperature dependent magnetic susceptibility of K2PbCu(NO2)6 under 0.1 T.

REFERENCES

(1) Noda, Y.; Mori, M.; Yamada, Y. Successive Jahn-Teller PhaseTransitions in K2PbCu(NO2)6. J. Phys. Soc. Jpn. 1978, 45, 954−966. (2) Kashida, S. Ultrasonic Velocities of K2PbCu(NO2)6 Near the Cooperative Jahn-Teller Phase-Transitions. J. Phys. Soc. Jpn. 1978, 45, 1874−1879. (3) Kashida, S. Effect of Uniaxial-Stress upon the Successive PhaseTransitions in K2PbCu(NO2)6. J. Phys. Soc. Jpn. 1979, 47, 1134−1140. (4) Kashida, S.; Kaga, H. Effect of Hydrostatic-Pressure on Successive Phase-Transitions in K2PbCu(NO2)6. J. Phys. Soc. Jpn. 1978, 44, 930− 932. (5) Kashida, S. Successive Jahn-Teller Phase-Transitions in K2PbCu(NO2)6. J. Phys. Soc. Jpn. 1978, 45, 414−421. (6) Harrowfield, B.; Pilbrow, J. Jahn-Teller Effect in K2PbCu(NO2)6. J. Phys. C: Solid State Phys. 1973, 6, 755−766. (7) Okuda, K.; Mollymoto, H.; Miyako, Y.; Mori, M.; Date, M. OneDimensional Antiferromagnetism in Rb2PbCu(NO2)6. J. Phys. Soc. Jpn. 1984, 53, 3616−3623. (8) Blanc, N.; Trinh, J.; Dong, L.; Bai, X.; Aczel, A.; Mourigal, M.; Balents, L.; Siegrist, T.; Ramirez, A. P. Quantum Criticality Among Entangled Spin Chains, to be submitted, 2017. (9) Yan, X.; Li, W.; Zhao, Y.; Ran, S.; Su, G. Phase diagrams, distinct conformal anomalies, and thermodynamics of spin-1 bond-alternating Heisenberg antiferromagnetic chain in magnetic fields. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 134425. (10) Yoshioka, D.; Arakawa, G.; Ichinose, I.; Matsui, T. Néel-dimer transition in an antiferromagnetic Heisenberg model and deconfinement of spinons at the critical point. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 70, 174407. (11) Mourigal, M.; Enderle, M.; Klopperpieper, A.; Caux, J.; Stunault, A.; Ronnow, H. Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain. Nat. Phys. 2013, 9, 435−441. (12) Kuhne, H.; Klauss, H. H.; Grossjohann, S.; Brenig, W.; Litterst, F. J.; Reyes, A. P.; Kuhns, P. L.; Turnbull, M. M.; Landee, C. P. Quantum critical dynamics of an S = 1/2 antiferromagnetic Heisenberg chain studied by C-13 NMR spectroscopy. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 045110. (13) Dahanaraj, J.; Byrappa, K.; Prasad, A.; Dudley, M. In Special Topics of Crystal Growth; Springer Handbook of Crystal Growth; Springer-Verlag: Berlin, 2010; Chapter 48, pp 1607−1636. (14) Dean, J. A. Inorganic Compounds. In Lange’s Handbook of Chemistry; McGraw-Hill Companies, 1998; Section 3, pp 13−60. (15) Andrews, L. C. In Other Functions Defined by Integrals; Special Functions of Mathematics for Engineers; SPIE Publications: 1997; Chapter 3, pp 110−112. (16) Patrick, W.; Allan, B. The diffusion of electrolytes in silica gel. J. Phys. Chem. 1934, 38, 771−778. (17) Dennis, J. Diffusion Studies in Silica Gels. J. Colloid Interface Sci. 1968, 28, 32−35.

emergence of quasi-1D spin correlations caused by the Jahn− Teller effect driven orbital order. A fit to the magnetic susceptibility using the Bonner−Fisher result for isolated 1D spin chains22,23 yields an excellent agreement with an in-chain exchange energy J/kB = 5.4 K. Physical property measurements are known to be affected by crystal quality. In our case, the study of the 1-D magnetic behavior of K2PbCu(NO2)6 clearly depends on the length of the Cu chains that form due to orbital order. Therefore, a low number of defects is desired to ensure that physical properties approach intrinsic behavior. Further studies investigating the quantum critical phenomena that are observed at low temperatures and high magnetic fields will be reported elsewhere.8

4. CONCLUSION The gel growth method has been demonstrated to be a successful way to grow high quality K2PbCu(NO2)6 single crystals. For a given small domain in the gel, the crystal nucleation and growth depend on the concentration product of the participant reactant ions inside the domain, as well as diffusion rates of reactants at the domain boundary. These factors evidently change with time due to the reactant diffusion from the high to the low concentration end, and the growth of the crystal. The concentration profile can be easily determined in the case of diffusion only. However, when crystal formation occurs, the growing crystals affect the concentration profiles. When more crystals form and crystal growth domains interact, the concentration profiles become exceedingly complex, rendering a detailed analysis of them difficult. Knowledge of the concentration profile of each reactant ion along the U-tube containing the gel helps to estimate the overall mass transport. Our simple model of the crystal growth gives a good understanding of the different effects that control nucleation and growth. Further optimization of the growth of K2PbCu(NO2)6 single crystals may include variations of the initial concentrations, the volume of the solution to the gel ratio, concentration replenishing during growth, as well as changing the geometry of the container. 5176



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(18) Sato, H.; Yui, M.; Yoshikawa, H. Ionic Diffusion Coefficients of Cs+, Pb2+, Sm3+, Ni2+, SeO42− and TcO4− in Free Water Determined from Conductivity Measurements. J. Nucl. Sci. Technol. 1996, 33, 950− 955. (19) Ribeiro, A. C. F.; Esteso, M. A.; Lobo, V. M. M.; Valente, A. J. M.; Simoes, S. M. N.; Sobral, A. F. N.; Burrows, H. D. Diffusion Coefficients of Copper Chloride in Aqueous Solutions at 298.15 and 310.15 K. J. Chem. Eng. Data 2005, 50, 1986−1990. (20) Henisch, H. K. Growth Mechanisms. In Crystals in Gels and Liesegang Rings; Cambridge University Press: New York, 1988; pp 48− 88. (21) Frank, F. Radially Symmetric Phase Growth Controlled by Diffusion. Proc. R. Soc. London, Ser. A 1950, 201, 586−599. (22) Bonner, J.; Fisher, M. Linear Magnetic Chains with Anisotropic Coupling. Phys. Rev. 1964, 135, A640−A658. (23) Hatfield, W. New Magnetic and Structural Results for Uniformly Spaced, Alternately Spaced, and Ladder-Like Copper (ii) Linear-Chain Compounds. J. Appl. Phys. 1981, 52, 1985−1990.

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