Elasticity of Prussian-Blue-Analogue Nanoparticles

Graphical Abstract

The elastic behavior of Prussian-blue-analogue nanoparticles has been investigated, using high-pressure synchrotron X-ray diffraction and nuclear-inelastic-scattering techniques. We have extracted all of the elastic moduli for different sizes of Prussian-blue-analogue nanoparticles. This result is the basis for understanding electron–lattice coupling phase transitions at the nanoscale.

Abstract

We report on the elastic properties of Ni/[Fe(CN)₆] Prussian-blue-analogue nanoparticles investigated by high-pressure synchrotron X-ray diffraction and nuclear inelastic scattering. For 3 nm and 115 nm particles, we have obtained bulk moduli of (30.3 ± 3.8) GPa and (24.5 ± 3.2) GPa, with Debye sound velocities of (2496 ± 46) m s^–1 and (2407 ± 38) m s^–1, respectively. Combining these results, Poisson's ratio, Young's modulus, the shear modulus, and the transversal/longitudinal sound velocities have been calculated for each particle size. All of these physical quantities suggest a stiffening of the lattice when the particle size decreases, which is mainly attributed to the reduction of iron ions on the particle surface.

Introduction

Prussian-blue analogues (PBA) are a broad family of face-centered cubic coordination networks, composed of a pair of transition-metal ions connected by cyanido bridges of the general formula A_1–xN^II[M^III(CN)₆]_1–x/3 (A is an alkaline ion; N and M are transition-metal ions). The different combinations of metallic ions (and electronic configurations) allow different physicochemical properties to be obtained, while keeping the same framework structure. Notably, certain PBA materials can display interesting ferromagnetic behavior1-4 or electronic phase-change phenomena, such as spin-state switching5, 6 and Jahn–Teller switching,7, 8 which are – in most cases – induced by an intervalence charge transfer between the metal ions. These electronic phenomena have received much attention, as they are accompanied by a spectacular change of magnetic and optical properties.9, 10

Due to the strong electron–lattice coupling, electronic phase transitions in PBA solids are governed, to a large extent, by the lattice properties. Of particular interest is the substantial change of lattice volume between the two states, which has been extensively investigated by powder X-ray diffraction methods under different external stimuli (temperature, pressure, light irradiation, etc.).5, 6, 11, 12 On the other hand, the elastic properties of PBA materials remain much less explored: only a few bulk modulus measurements have been reported in the literature. By means of high-pressure crystallographic methods, Bleuzen et al. have measured a bulk modulus of (43 ± 2) GPa for photomagnetic CoFe PBA materials,11 while Pajerowski et al. obtained ca. 31 GPa for ferromagnetic NiCr PBA materials.13 On the other hand, Boudkeddaden et al. have estimated the bulk moduli B_LT = (30 ± 3) GPa and B_HT = (23 ± 2) GPa for the switchable PBA compound RbMn[Fe(CN)₆] in the low- and high-temperature phases, respectively.14 These data are particularly important for the better understanding of the effect of external pressure on the phase stability and also to rationalize the elastic-energy changes, which accompany the phase transitions in PBA materials.15 The in-depth knowledge of the elastic properties of PBA compounds is also an asset for their possible future applications in mechanical actuators (using the volume change).16

Besides bulk PBA compounds, more recently, the development of PBA nano-objects has led to the observation of new physical phenomena in this class of nanomaterials, such as superparamagnetism.17, 18 Core@shell particles, such as PBA@PBA and Au@PBA heterostructures, with original magnetic19-22 and optical23 properties, have also been synthesized. These heterostructures exhibit, in many cases, mechanical strain-mediated coupling between the electronic and magnetic properties of the core and the shell. When one of the components undergoes an electronic phase transition, this coupling can even allow the switching of the properties of the second component.22, 24 Theoretical analysis has highlighted that the pressure the core (shell) can apply to the shell (core) upon the phase transition can be estimated if Young's moduli and Poisson's ratios of both components are known.25 Then, combining the theoretical prediction with synthetic control of the core and shell thicknesses, the applied pressure on the core and the shell can be controlled, opening the way to monitoring the synergy between magnetism, optical, and electronic phenomena in PBA heterostructures. However, the elastic properties of nanoscale PBA materials remain completely unknown. To further understand the mechanical properties of these materials (both bulk and nanoscale), the determination of elastic coefficients thus becomes indispensable.

In a previous work, using Mössbauer spectroscopy, we analyzed the evolution of lattice stiffness as a function of the particle size in Ni^II/[Fe^III(CN)₆] nanoparticles and revealed a stiffening of the lattice for ultrasmall particles (ca. 3 nm), reflected by the increase of their Debye temperature Θ_D.26 Based on Raman and FTIR spectroscopy data, this stiffening was attributed to a reduction of the ferric ions on the surface of the particles. In the present work, we combined high-pressure synchrotron X-ray diffraction (XRD) and nuclear-inelastic-scattering (NIS) methods to determine the bulk modulus B and the Debye sound velocity v_D for 3 nm and 115 nm Ni/[Fe(CN)₆] nanoparticles. Using these data, we were able to calculate Young's modulus Y, shear modulus G, Poisson's ratio ν, and transversal/longitudinal sound velocities (v_t, v_l). Additionally, we have also confirmed surface reduction, in the case of 3 nm nanoparticles.

Results and Discussion

PBA nanoparticles were synthesized using previously reported methods. The smallest nanoparticles (3 nm) were obtained in the presence of a stabilizing agent [bis(3-aminopropyl)poly(ethylene glycol), PEG-NH₂] during the synthesis, while the larger nanoparticles were obtained from the controlled precipitation of Ni²⁺ and [Fe(CN)₆]^3– moieties, wherein the increasing concentration and addition rate leads to an increase of the nanoparticle size. The resulting nanoparticles were post-synthetically functionalized by the PEG-NH₂ stabilizer.26

Figure 1a shows synchrotron X-ray diffraction patterns for different sizes of Ni/[Fe(CN)₆] nanoparticles and for two bulk materials, Ni₃[Fe^III(CN)₆]₂ and Ni₂[Fe^II(CN)₆]. The intense peaks at 5.2° and 7.4° correspond to the (200) and (220) reflections, respectively, of the well-known face-centered cubic structure27, 28 (see inset of Figure 1b). Interestingly, for the largest nanoparticles (115 nm, 70 nm, 37 nm), the width of the diffraction peaks is narrower than in the bulk material – due, most likely, to some disorder in the latter. Usually, larger peaks are expected when decreasing the particle size (cf. Scherrer equation). This phenomenon is indeed observed here for the nanoparticles. We explain the unusual behavior of bulk materials by the difference between the bulk and the nanoparticles in synthetic procedures, which can introduce different kinds of defects into the crystal structures.

Figure 1

(a) Synchrotron X-ray diffraction patterns of Ni/[Fe(CN)₆] measured at room temperature in the 2θ interval 4–14° (λ = 0.477 Å) for different sizes of nanoparticles. (b) Lattice parameter of Ni/[Fe(CN)₆] particles as a function of their size (the cubic structure is shown in the inset). The dashed lines represent the lattice parameters of the bulk Ni₃[Fe^III(CN)₆]₂ and Ni₂[Fe^II(CN)₆] materials.

The cell parameter a of the Prussian-blue cubic lattice was then calculated using the position of the (200) reflections. Figure 1b shows the evolution of a as a function of the particle size. The cell parameters of the two bulk compounds {Ni₃[Fe^III(CN)₆]₂ and Ni₂[Fe^II(CN)₆]} are also shown. As expected, the density of the bulk Ni₂[Fe^II(CN)₆] material is substantially higher than that of the bulk Ni₃[Fe^III(CN)₆]₂. The cell parameters of the 115 nm and 70 nm particles are similar (slightly higher) when compared with those of bulk Ni₃[Fe^III(CN)₆]₂. When decreasing the size, the cell parameter also decreases. Using the Mössbauer spectrum of the 3 nm [37 nm]26 particles, we obtain a percentage of (85 ± 7) % [(10 ± 17) %] and (15 ± 7) % [(90 ± 20) %] for the Fe^II and Fe^III species, respectively. Then, at 3 nm, the influence of the reduced species Fe^II–CN–Ni^II on the surface is dominant, which leads to a cell parameter approaching that of the bulk Ni₂[Fe^II(CN)₆] material.

To gain further insights into their elastic properties, the 3 nm and 115 nm samples were further investigated using high-pressure XRD. It is important to note that the diffractograms in Figure 1a were recorded in a liquid (methanol/ethanol/water) medium. However, when applying pressure to this mixture, the lattice parameters of the PBA samples evolved in an “anomalous” manner, probably due to a structural phase transition, mediated by the strong physicochemical interactions between the particles and the liquid. We have not investigated this phenomenon in detail, but we changed to a more inert pressure-transmitting medium (neon), which was then successfully used to apply hydrostatic pressures of up to 35 GPa (at room temperature) to our samples. The evolution of the cell parameters, as a function of the pressure, is displayed in Figure 2a and b for the 3 nm and 115 nm samples, respectively. The unit-cell parameter a was calculated, in both cases, using the (200) reflection. (Note: the differences in the unit-cell parameters between the results in Figures 1 and 2 are most likely due to the different sample environments.) From these measurements, we have extracted the bulk modulus of the particles using the second-order Murnaghan approximation for the bulk modulus [Equation 1]:29

(1)

where B₀ is the bulk modulus at zero pressure, and B₀′ and B₀′′ are the first- and second-order constants of the bulk modulus, respectively. The second-order approximation was used, because a simple approximation of the linear evolution of the bulk modulus as a function of the pressure was not satisfactory (B₀′′ is not negligible). Taking into account the general definition of the bulk modulus (inverse of compressibility χ_T) [Equation 2]:

(2)

and the evolution of the unit-cell parameter of a cubic lattice as a function of the pressure can be written as follows [Equation 3]:

(3)

where a₀ is the cell parameter at zero pressure. Using Equations (1) and (3), we found a bulk modulus of (30.3 ± 3.8) GPa and (24.4 ± 3.2) GPa for the 3 nm and 115 nm particles, respectively. This result is in good agreement with our previous Mössbauer spectroscopic investigation,26 which indicated that the stiffness of the NiFe PBA nanoparticles increases with decreasing size.

Figure 2

Evolution of the unit-cell parameter as a function of the pressure for the (a) 3 nm and (b) 115 nm particles. The squares and lines represent the experimental data and the fits, respectively, using Equations (1) and (3). The fitted parameters are displayed in the inset of each graph.

To fully characterize the elastic properties of the 3 nm and 115 nm samples, we have determined further lattice dynamical parameters using the nuclear-inelastic-scattering technique. NIS spectra were collected for 3 nm and 115 nm particle sizes and were treated as described in ref.30 to finally obtain the iron vibrational density of states (DOS) of Ni/[Fe(CN)₆] nanoparticles (Figure 3a). In the present work, the powder form of the samples implies a partial DOS for iron, averaged over all directions of phonon polarizations (i.e., over all directions of atomic displacements).

Figure 3

(a) Iron DOS g̃(E) of Ni/[Fe(CN)₆] for the 3 nm and 115 nm nanoparticle samples. (b) Plots of g̃/E² in the low-energy range (acoustic modes) for the two samples.

Several modes can be identified by means of the composition factor,31, 32 which is derived from the selection rules of the NIS. In a randomly oriented powder, the composition factor e_j² (with e_i being the polarization vector) is directly related to the fraction, g̃, of DOS, such as [Equation 4]:

(4)

By assuming a total decoupling of inter- and intramolecular vibrations, the upper limit of the composition factor for each acoustic branch can be written as [Equation 5]:

(5)

where m̃ is the mass of the resonant atoms in the cell and M_Σ is the mass of the primitive cell (1/4 of the mass of the chosen conventional face-centered cubic cell). Considering the three acoustic modes, the limit of the acoustic part E_ac can be estimated as [Equation 6]:

(6)

The estimated upper acoustic limit E_ac ≈ 18 meV is roughly the same for both the 3 nm and 115 nm particles. The first peak in the DOS, ca. 8–12 meV, is thus composed of acoustic modes, while the broad peaks at ca. 23 meV and ca. 33 meV are due to optical branches.

In the same way, using the composition factor, it is possible to identify stretching modes of the Fe coordination octahedron. In the case of the 115 nm particles, the well-defined optical mode at 66 meV probably corresponds to the threefold degenerated F_1u stretching modes of the perfect octahedron. In this case, by assuming the stretching of two rigid fragments, with respective masses of M₁ and M₂, and assuming that the resonant atom belongs to the fragment with mass M₁, the composition factor for each mode can be written as [Equation 7]:

(7)

The composition factor has been determined by fitting the peak with a pseudo-Voigt function and by integrating it as described by Equation (4). By using Equation (7) and taking into account the threefold degeneracy of the mode, the fragment mass of the resonant atom is estimated to be M₁ = 107 a.u., which is in good agreement with the expected fragment mass associated with one iron atom and two CN groups (109 a.u.). Therefore, this mode can be clearly identified as the threefold degenerated stretching of the iron ion, with two CN groups in the local environment. In the case of the 3 nm particles, the same peak is located around 73 meV, due to the reduction of iron ions on the surface. Indeed, this reduction leads to a decrease of the Fe–C bond length, which is associated with a stiffening of the bond, leading to the upshift of the peak. This result confirms our previous observations of a chemical reduction Ni^II/[Fe^III(CN)₆] → Ni^II/[Fe^II(CN)₆] at the surface of the particles, as evidenced by Raman spectroscopy, IR spectroscopy, and Mössbauer spectroscopy.26 By determining the composition factor, the fragment mass is calculated as M₁ = 130 a.u., which is larger than for the 115 nm particles. This can be explained by: (i) an increase of the fragment mass M₁; (ii) a decrease of the primitive-cell mass, which leads to an overestimation of M_Σ; or (iii) the presence of an additional vibrational mode in the peak area. The first explanation is unlikely and can be excluded. The second is possible if a decrease of the averaged mass of the primitive cell, due to the strong contribution of the surface (incomplete cell on the surfaces, due to the breaking of the periodicity), is considered. The third explanation is the most likely, due to the fact that each optical mode does not present a similar shift due to the surface reduction. It can lead to the superimposition of several peaks, and thus, to an increase of the composition factor, which is proportional to the area under the curve.

Table 1 summarizes some selected lattice dynamical parameters, extracted from the DOS, according to the methods described in ref.33 The Lamb–Mössbauer factor, the vibrational amplitude, and the mean force constant show the same trend: an increase of lattice stiffness upon size diminution. The vibrational internal energy tends to be higher for stronger bonds, while a decrease of the vibrational entropy is observed.

Table 1. Selected lattice dynamical parameters extracted from the NIS spectra

Ni3[Fe(CN)6]2		3 nm	115 nm
Lamb–Mössbauer factor	f LM	0.58(1)	0.555(1)
Vibrational amplitude [Å]	√<x^2Fe>	0.176(8)	0.184(2)
Mean force constant [N/m]	<CFe>	587(15)	502(10)
Internal energy per atom [meV]	u vib,N	100(1)	97.0(1)
Entropy per atom (kB)	s vib,N	2.57(8)	2.65(2)
Specific heat (kB)	C V	2.26(8)	2.33(2)

The Debye sound velocity v_D can be also extracted from the extrapolation of the DOS at E = 0 meV, using the Debye model, such as [Equation 8]:34, 35

(8)

where g̃_D, m̃, ϱ, and ħ are the DOS in the Debye model, the mass of the resonant atom (⁵⁷Fe), the volumetric mass density of Ni₃[Fe(CN)₆]₂, and the reduced Planck constant, respectively. In practice, v_D was extracted using Figure 3b. The sound velocity for the 115 and 3 nm particles are ν_D(115 nm) = (2407 ± 38) m s^–1 and ν_D(3 nm) = (2496 ± 46) m s^–1, respectively. In the approximation of an isotropic material, the Debye sound velocity is directly related to Young's modulus Y and Poisson's ratio ν by the following relationship [Equation 9]:30

(9)

where h₁(ν) is a function which depends only on ν, with Equation 10:

(10)

and Equation 11:

(11)

From this point, the bulk modulus B can be also expressed as [Equation 12]:

(12)

Then, using the bulk modulus data from high-pressure XRD and the Debye sound-velocity data from NIS, Poisson's ratio for the 115 nm and 3 nm particles can be determined as ν_D(115 nm) = 0.352 ± 0.023 and ν_D(3 nm) = 0.365 ± 0.021, respectively. Using Poisson's ratio, Young's modulus was properly obtained for the two sizes as Y(115 nm) = (21.3 ± 1) GPa and Y(3 nm) = (24.1 ± 1.2) GPa. The shear modulus G can also be calculated, using Equation 13:

(13)

The calculated shear moduli of the 115 nm and 3 nm particles are G (115 nm) = (7.9 ± 0.5) GPa and G (3 nm) = (8.8 ± 0.6) GPa, respectively. Finally, the transversal ν_t and longitudinal ν_l sound velocities can be calculated from Equations 14 and 15:

(14)

(15)

The various elastic moduli for the two particle sizes are summarized in Table 2. Notably, due to the isotropic nature of the lattice, it is not necessary to characterize the elastic moduli in all spatial directions.

Table 2. Elastic properties of PBA nanoparticles

Ni3[Fe(CN)6]2		3 nm	115 nm
Debye temperature [K]26	Θ D	225 ± 15	187 ± 12
Young's modulus [GPa]	Y	24.1 ± 1.2	21.3 ± 1
Bulk modulus [GPa]	B	30.3 ± 3.8	24.5 ± 3.2
Shear modulus [GPa]	G	8.8 ± 0.6	7.9 ± 0.5
Poisson's ratio	ν	0.365 ± 0.021	0.352 ± 0.023
Debye sound velocity [m s^–1]	ν D	2496 ± 46	2407 ± 38
Transversal sound velocity [m s^–1]	ν t	2216 ± 34	2140 ± 27
Longitudinal sound velocity [m s^–1]	ν l	4806 ± 369	4489 ± 326

Overall, the results in Table 2 are consistent with the XRD patterns, which describe a change of the density with the size, but a conservation of the face-centered structure. The different measurements and extracted parameters show, in a coherent way, that the size reduction indeed affects the elastic properties of the nanoparticles. The lattice stiffening in the small particles can be clearly deduced from the concomitant increase of the Debye temperature (Mössbauer spectroscopy),26 the bulk modulus (XRD) and the sound velocity (NIS), despite the differences remain close to the error bars. The calculated Young modulus and shear modulus data confirm also this trend of lattice stiffening. Poisson's ratio does not give information about the stiffness, but rather, the relation between Young's, the bulk, and the shear moduli. The observed values (0.35–36) can be considered as typical for inorganic crystalline materials. The fact that Poisson's ratio does not change with the size is perfectly in line with the preservation of the face-centered cubic crystal structure.

Conclusion

High-pressure XRD and NIS have allowed the size dependence of both the structure and the vibrational density of states to be monitored for Ni/[Fe(CN)₆] particles. These data were used to extract lattice dynamical parameters, such as vibration amplitude, mean force constants, or vibrational entropy and internal energy, and to determine the elastic moduli (bulk, Young's, and shear moduli), as well as Poisson's ratio. We have shown that the chemical reduction at the surface of the smallest nanoparticles leads to a diminution of the lattice parameter, as well as to a shift to higher energies of the metal–ligand stretching mode. This contributes to an increase of the global stiffness of the smallest particles, which is characterized by an increase of all the elastic parameters (Θ_D, ν_D, B, Y, and G). Access to the elastic moduli of PBA nanoparticles, which are largely unknown in molecule-based materials (bulk and nano-object), is of paramount importance for developing applications based on the mechanical properties of molecular materials, such as micro-electromechanical systems (MEMS)36, 37 or mechanically coupled nano-heterostructures.22, 24, 38 It is important to stress also that in this work, we provide an accurate value of Poisson's ratio. The mechanical characterization of coordination networks is challenging, especially at the nanoscale, but it is necessary for the development of new applications and for understanding phase-change phenomena governed by electron–lattice coupling. In this context, high-pressure XRD and NIS appear as suitable techniques to overcome the usual restrictions of other methods.

Experimental Section

The synthesis of the samples is described in ref.26 The synchrotron powder X-ray diffraction experiments were carried out at PSICHÉ-SOLEIL and PETRA III-DESY beamlines. The samples were placed in diamond anvil cells, and the pressure was measured by using the ruby fluorescence technique. At PSICHÉ-SOLEIL, the pressure-transmission medium was water/ethanol/methanol liquid, and the used beam wavelength was λ = 0.477 Å. At PETRA III-DESY, the pressure-transmission medium was neon gas, and the used beam wavelength was λ = 0.29056 Å. NIS experiments were performed at the Nuclear Resonance beamline ID1839 at the European Synchrotron Radiation Facility (ESRF, Grenoble) in the hybrid bunch modes at room temperature, using non-enriched samples with natural ⁵⁷Fe.

Acknowledgements

The authors would like to thank for the financial support from the Agence Nationale de la Recherche (NANOHYBRID project, ANR-13-BS07-0020-01) and from the Federal University of Toulouse (project IDEX Emergence NEMSCOOP). We are indebted to Leonid Dubrovinsky (Bayerisches Geoinstitut) for his kind help.