# Neutron Study of Multilevel Structures of Diamond Gels

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Samples and Scattering Experiments

^{+}) were obtained using the following method [7]. The industrial DND-powder was annealed in hydrogen flux at 500 °C, mixed with deionized water, and sonicated. The particles of small-sized fraction were separated by centrifugation and their size distribution was controlled by dynamic light scattering (DLS). Hydrosol (concentration C ~ 1 wt %) in the process water evaporation in a vacuum rotary evaporator at temperature 50–60 °C was transformed into the gel with a diamond content of С

_{1}= 5.05 wt % when the critical concentration С ~ C* ~ 4.2 wt % was exceeded. A detail description of the synthesis and characteristics of gels are given in previous articles [14,15]. This gel was used to obtain the systems with the lower concentrations of diamonds, С

_{2}= 2.25 wt %, С

_{3}= 1.13 wt %, by water addition (Table 1).

^{−1}for the samples with diamond concentrations С

_{1}, С

_{2}, and С

_{3}at ambient temperature and atmospheric pressure. A thin layer of gel or diluted systems (1 mm) was used to minimize the multiple scattering effects. The data were normalized to the samples’ thickness and the intensities measured for the Vanadium-standard of incoherent scattering. These data corrected for the solvent contribution gave the desirable coherent cross sections of the samples per unit solid angle and cm

^{3}of the volume.

#### 2.2. Discussion

#### 2.2.1. Dependencies of Cross Sections on Momentum Transfer

^{1}nm. At the same time, the dilution leads to the decrease of cross sections σ(q) according to the decrement of concentration, while at first glance, the behavior of σ(q) does not change. Hence, the gel’s ordering achieved above the critical point (С

_{1}> C* ~ 4.2 wt %) is rather stable and exists after two- and four-fold dilution.

_{min}~ 10

^{2}nm that is defined by the minimal momentum transfer q

_{min}. Such a micro-gel exists in the range of concentrations from C*/4 ≤ С ≤ С*. It is remarkable that all the curves of σ(q) demonstrate a kink at the momentum transfer q ~ q* ~ 1 nm

^{−1}(Figure 1). Obviously, at q ≥ q* the scattering on single particles dominate, whereas at q ≤ q* the interference is observed in scattering at the distances 2π/q ≥ d

_{s}exceeding the particles’ diameter d

_{s}~ 2π/q* ~ 6 nm. In these regions the cross sections show the exponential behaviors, σ(q) ~ 1/q

^{D}, with the parameters D = D

_{1}~ 4 and D = D

_{2}~ 2 where the first one corresponds to Porod’s law observed for the particles with sharp borders and the second one indicates chain-like aggregates of particles (Figure 1).

^{2}, since this scattering law is established from the statistics of flexible chains (polymers) [16]. At the same time, except for linear fragments, the observed structures may include some branched (e.g., dendrimer-like) fragments characterized by the exponent D

_{2}> 2 [16,17].

^{−1}and q < q* for the cross sections σ

_{N}(q) = σ(q)/C normalized onto the diamonds’ concentrations, given in Figure 1 (inserts).

_{int}) between the centers of neighboring particles coupled in network structure of gel. The spacing d

_{int}defines the position of the peak, q

_{int}≈ 1.1 nm

^{−1}= X/d

_{int}where the coefficient X corresponds to the maximum X = q

_{int}d

_{int}≈ 7.72 of Debye scattering functions $\frac{\text{sin}(q{d}_{int})}{q{d}_{int}}$ for a pair of particles at the distance d

_{int}. The value of d

_{int}≈ 7 nm exceeds a characteristic diameter of a diamond particle, d

_{int}− d

_{S}≈ 1 nm, i.e., the particles in the network structure of gel contact via water shells with a thickness of two molecular layers. The first (q

_{int}) and second maxima (~2q

_{int}) on the curves for different concentrations confirm a short range order of particles in gels.

_{w}= X/q

_{w}≈ 40 nm where q

_{w}≈ 0.18 nm

^{−1}is the maximum position for a broad peak at q ~ 0.1–0.3 nm

^{−1}. This spacing should be attributed already to the size of the cells in gels’ multilevel structures, which are discussed below.

_{i}/q

^{Di}, i = 1, 2

_{i}showing the scattering ability of the observed objects and the parameters D

_{i}which characterize their geometry (Figure 2).

_{1}, J

_{2}indicates a change of the amount of particles’ or aggregates by the variation of concentration, when their geometry does not undergo any substantial transformation (Figure 2a). Only a weak linear decrease of the geometric parameters D

_{1}, D

_{2}is observed (Figure 2b). The D

_{2}reflects the geometry of particles’ surface. This parameter declines by dilution, and in the limit, C → 0, the value of D

_{2}= D

_{02}= 3.98 ± 0.04 is very close to the exponent in Porod’s law describing the scattering for the particles with sharp borders [18]. It confirms a perfection of the diamond crystals’ facets that is of principal importance for the configuration of potentials inducing the diamonds’ electrostatic attraction due to different electric charges of the facets [19,20]. Meanwhile, even at a moderate concentration (С

_{3}= 5.05 wt %) the scattering from the particles’ surface is disturbed due to the interference in scattering from neighboring particles that gives D

_{02}> 4 (Figure 2b). This must be distinguished from the results [5,12] for the suspensions of diamonds in graphene shells, which showed a deviation of scattering from Porod’s law.

_{1}~ 2.3–2.4 > 2 (Figure 2b). Such a magnitude in the range 2 < D

_{1}< 3 is inherent in the structures formed via a diffusion limited aggregation (DLA) which creates mass fractals [17]. For pristine gel the fractal dimension D

_{1}~ 2.4 exceeds the parameter for a linear Gaussian polymer chain D

_{G}= 2 [16], that confirms the existence of branched aggregates forming a network of gel. A dilution of gel retains mostly the exponent D

_{1}decreasing linearly with concentration (Figure 2b). In the limit С → 0, the extrapolated D

_{1}= D

_{01}= 2.35 ± 0.01 is substantially greater than the D

_{G}= 2 for linear chains. This shows a good stability of local structure of gel grown at the volume fraction of diamonds φ

_{1}= 1.44% keeping in its integrity at their low content, φ

_{3}= 0.22%. It is noticeable that this ordering is realized in the ensembles of particles with relatively broad size-distributions. For a comprehensive analysis of these structures, it is necessary to rebuild the size-distributions of diamonds.

#### 2.2.2. Size-Distributions of Diamonds

_{1m}~ 2.4–2.6 nm that agrees well with the DLS data for the diamonds prepared by the described method [14].

_{1m}~ 2.6 nm), in the spectrum of the gel (Figure 3) the additional peak is seen at R

_{2m}~ 5.1 nm ~ 2R

_{1m}. Its position lies in the range of radii comparable to the particles’ diameter. This indicates the particles’ association within the first coordination sphere. The dilution of the gel leads to disappearance of the secondary peak which overlaps with the first one that makes it complicated to distinguish single particles and small aggregates. Besides, all the samples show an aggregation at distances of double or triple the diameter of particle, R ~ 10–15 nm (Figure 3). To describe the ordering of the particles in gel, this model may serve as the first approximation which considers only spherical objects which scatter independently. However, various spatial correlations of particles and their aggregates should be taken into account.

#### 2.2.3. Correlation Functions

**R**) = <δρ(

**r**)·δρ(

**r + R**)> is the averaged product of the deviations, δρ(

**r**) = ρ(

**r**) − <ρ>, δρ(

**r + R**) = ρ(

**r + R**) − <ρ>, in scattering length densities from the mean value <ρ> in the sample. A pair correlation is considered for two points at the distance R. The magnitude of <ρ> = Σ(b

_{i}N

_{i}) is defined by the contributions of nuclei with scattering lengths b

_{i}and concentrations N

_{i}. For isotropic samples, the functions γ(R) being the Fourier-transforms of scattering data [18] depend on the modulus of vector

**R**,

^{3}∫σ(q)[sin(qR)/(qR)]4πq

^{2}dq.

_{1}(R) = g

_{1}exp(−R/r

_{1}). It includes the correlation length r

_{1}comparable to the radius of particles and the coefficient g

_{1}proportional to the concentration of diamonds (Table 1). Since the functions γ

_{1}(R) describe the atomic correlations inside the particles, the parameter r

_{1}is the averaged correlation radius of particles r

_{1}~ R

_{1m}which is slightly larger than their geometric radius (Figure 3). This follows from the comparison of γ

_{1}(R) and the correlation function γ

_{S}~ [1 − (3/4)(R/r

_{S}) + (1/16)(R/r

_{S})

^{3}] of a sphere [18] with the radius r

_{S}= R

_{1m}.

_{1}, the γ

_{1}(R) is approximately a linear function, γ

_{1}~ [1 − R/r

_{1}]. At the same time, the linear term γ

_{S}~ [1 − (3/4)(R/r

_{S})] dominates in the correlation function γ

_{S}(R) at R ≤ r

_{S}. The identity γ

_{1}(R) ≡ γ

_{S}(R) is fulfilled if r

_{S}= (3/4)r

_{1}. Thus, in gel the particles with a correlation radius r

_{1}= 3.40 ± 0.01 nm have the geometric radius r

_{S}= (3/4)r

_{1}≈ 2.6 nm ≈ R

_{1m}in accordance with experimental data (Figure 3). In addition, the γ

_{1}(R) yields the information on the contrast factor of diamonds Δρ

_{D}relative to the surrounding medium. The Δρ

_{D}depends on the scattering lengths and the concentrations of chemical elements in the particles and surroundings. This gives the indication of how tightly packed the particles are. In pristine gel with the volume fraction of diamonds φ = 1.44%, the parameter g

_{1}= (Δρ

_{D})

^{2}φ (Table 1) affords the estimate Δρ

_{D}= (10.36 ± 0.01) × 10

^{10}cm

^{−2}. The Δρ

_{D}= ρ

_{D}− <ρ> = (1 − φ)(ρ

_{D}− ρ

_{W}) is the difference between the scattering length densities ρ

_{D}of the carbon material and a similar parameter <ρ> for the whole volume of the sample <ρ> = φρ

_{D}+ (1 − φ)ρ

_{W}, where ρ

_{W}= −0.56 × 10

^{10}cm

^{−1}is the scattering length density for light water.

_{D}= Δρ

_{D}/(1 − φ) + ρ

_{W}= (9.95 ± 0.01) × 10

^{10}cm

^{−2}is 15% lower than a similar parameter, ρ

_{DI}= 11.7 × 10

^{10}cm

^{−2}, for the crystals of diamonds with a characteristic density of 3.5 g/cm

^{3}. The deficit can be explained by the presence of volume (surface) defects (vacancies) in diamonds. On the other hand, we neglected the possible influence of the gel’s structure factor, taken as S(q → 0) = 1. Even at a low volume content of particles, their coordination may cause a deviation of this factor from unity, S(q) < 1.

_{2}(R) = g

_{2}exp(−R/r

_{2}). The coefficient g

_{2}is proportional to the particles’ concentration but the correlation radius r

_{2}~ 7 nm remains almost constant (Table 1). This confirms a stability of the local ordering even by a substantial dilution of gel.

_{1}(R), γ

_{2}(R) describing the correlations inside the particles and in the first coordination sphere, one can compare the correlation volumes, V

_{1}= 4π∫exp(−R/r

_{1})R

^{2}dR = 8πr

_{1}

^{3}≈ 990 nm

^{3}and V

_{2}= 8πr

_{2}

^{3}≈ 9.3 × 10

^{3}nm

^{3}. The volume V

_{2}exceeds V

_{1}by an order of magnitude but the first coordination sphere is not completely filled with particles. This is evident from a comparison of the forward cross sections σ

_{2}and σ

_{1}calculated in the limit q → 0 using the functions γ

_{1}, γ

_{2}.

_{1}= (Δρ

_{D})

^{2}φV

_{1}= g

_{1}V

_{1}is defined by the parameters of single particles while the magnitude of σ

_{2}= (Δρ

_{D})

^{2}φ(mV

_{1}) = g

_{2}V

_{2}is proportional to the number of particles (m) within the correlation volume V

_{2}. The ratio of cross sections gives the value m = (g

_{2}V

_{С})/(g

_{1}V

_{1}) = 3.6 ± 0.1. Hence, a particle in the gel is connected with two to three neighboring particles. This short linear or branched fragment (aggregation number m) is really stable even in the case of a four-fold dilution of gel where only a slight reduction in the aggregation number is observed (~10%).

_{3}(R) = g

_{3}exp[−(R/r

_{3})

^{2}] at R ≥ 20 nm. It corresponds to Guinier’s function exp[−(qR

_{G})

^{2}/3] with the gyration radius R

_{G}= (3/4)

^{1/2}r

_{3}being proportional to the correlation length r

_{3}. In gel and diluted systems, the lengths r

_{3}~ 20 nm are the same (Table 1), as well as the gyration radii R

_{G}= (3/4)

^{1/2}r

_{3}~ 17 nm. One may imagine the structure of these samples being composed of spherical domains having the diameter of ~2R

_{G}~ 40 nm.

^{−1}(Figure 1) while in the extended interval, q ≤ 1 nm

^{−1}, these domains are characterized as chain fractal aggregates (Table 1). The distribution of distances between the particles inside them is described by the function G(R) = γ

_{3}(R)R

^{2}having the maximum at R = r

_{3}as the most probable distance between the units in aggregates. For a Gaussian chain connecting two particles at the distance r

_{3}~ 17 nm, the number of units is equal to n = (r

_{3}

^{2}/d

_{S}

^{2}) + 1 ≈ 16, and the contour length L

_{C}= nd

_{S}≈ 80 nm. The latter should be compared to the spacing between the chains in the gel that is set by their numerical concentration, N

_{L}= φ/(nπd

_{S}

^{3}/6) ≈ 1.2 × 10

^{16}cm

^{−3}for the particles’ volume fraction, φ = 1.44%. For uniform distribution of the chains in the gel, the average distance between them, D

_{w}≈ N

_{L}

^{−1/3}≈ 40 nm ~ 2r

_{3}, is approximately half of the contour length. The chains principally overlap and form the interconnected structure of the gel with cells’ diameter ~D

_{w}~ 40 nm. This is illustrated in Figure 5 where three levels of gel structure are shown. Single particles (size r

_{1}~ 3 nm, the first level) create small linear (branched) fragments (correlation radii r

_{2}~ 7 nm, the second level) associated into chain structures (size r

_{3}~ 20 nm) forming the cells (diameter D

_{w}~ 2r

_{3}~ 40 nm) which are integrated into a gel network.

_{C}= m

_{L}/n can be found using the amount of particles (m

_{L}) per a cell. This value is included in the forward cross section σ

_{3}= (Δρ

_{D})

^{2}φ(πd

_{S}

^{3}/6)m

_{L}S

_{L}= g

_{3}V

_{L}which is defined by the parameters of the correlation function (g

_{3}, r

_{3}) where V

_{L}= 4π∫γ

_{3}(R)R

^{2}dR = π

^{3/2}r

_{3}

^{3}is the correlation volume and S

_{L}(q → 0) is the structure factor of the systems of the cells. The model of random contacts of spherical objects in the sample volume [18] was used to evaluate the S

_{L}= 1 − 8V

_{L}/V

_{1L}, where V

_{1L}= f

_{C}/N

_{L}is the volume per cell that is determined by the numerical concentration of the cells (N

_{L}). The parameter V

_{L}= f

_{C}V

_{n}/β includes the volume of the chain’s region, V

_{n}= πr

_{3}

^{3}/6, and the coefficient β characterizing the density of the chains’ package in a cell. If the regions of chains fill the total volume, β = 1. If between the regions any cavities exist, β < 1, e.g., in the case of spheres, the most dense packing β = β

_{S}≈ 0.74. Further we used the average β

_{a}= (1 + β

_{S})/2 ≈ 0.87 to estimate the S

_{L}= 1 − 8(φ/nβ

_{a})(r

_{3}/d

_{S})

^{3}≈ 0.54. Under these assumptions the amount of particles in a cell and the functionality of the network junctions is determined, m

_{L}= (g

_{3}V

_{L})/[(Δρ

_{D})

^{2}φ(πd

_{S}

^{3}/6)S

_{L}] ≈ 70 and f

_{C}= m

_{L}/n ≈ 4.These parameters are attributed to the original and diluted systems, saving almost their initial structural features at different scales (from one to tens of diameter of particle) characterizing the multilevel order detected in gels.

## 3. Conclusions

^{2}phase on the surface of the diamond particles. In the case of hydrosol produced by another method [14], we have observed a giant increase of viscosity at a concentration of diamond particles more than 4–5 wt % [21].

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Cross sections σ(q) for the samples with diamond contents of 5.05, 2.25, and 1.13 wt % (1–3) vs. momentum transfer. Straight lines show characteristic slopes for cross sections’ q-dependencies. The fitting curves for the model of spherical particles are plotted also. Inserts show the Kratky- and Porod-plots (left, right) for the cross sections σ

_{N}(q) normalized to the diamonds’ concentrations.

**Figure 2.**Parameters of function (1): (

**a**) J

_{1}, J

_{2}—the characteristics of the scattering ability of fractal structures and single particles; (

**b**) D

_{1}—the fractal dimension of aggregates, D

_{2}—the exponent describing the geometry of particles’ surface.

**Figure 3.**Volume fractions Φ(R) of scattering objects vs. their radii: diamonds’ content of 5.05, 2.25, and 1.13 wt % (1–3). Arrows indicate the main (R

_{1m}) and secondary maxima (R

_{2m}).

**Figure 4.**Functions γ(R) for the samples with 5.05, 2.25 and 1.13 wt % of diamonds (1–3). The lines show the fitting functions γ

_{1}, γ

_{2}, γ

_{3}(1–3) for the lowest concentration (see text).

**Figure 5.**Structural levels in gel: the single particles (correlation radius r

_{1}), the first coordination sphere (radius r

_{2}), and the chains (size r

_{3}) associated into the cells (diameter D

_{w}).

**Table 1.**Parameters of partial correlation functions γ

_{i}(R) of the samples with different contents of diamond (C): coefficients g

_{i}and correlation radii r

_{i}(i = 1, 2, 3).

С, wt % | 1.13 | 2.25 | 5.05 |
---|---|---|---|

g_{1}·10^{3}, cm^{−1}·nm^{−3} | 36.4 ± 0.1 | 74.5 ± 0.2 | 154.8 ± 0.3 |

g_{2}·10^{3}, cm^{−1}·nm^{−3} | 14.3 ± 0.1 | 27.9 ± 0.2 | 59.0 ± 0.5 |

g_{3}·10^{4}, cm^{−1}·nm^{−3} | 22.8 ± 0.4 | 45.1 ± 0.8 | 94.6 ± 1.8 |

r_{1}, nm | 3.51 ± 0.01 | 3.48 ± 0.01 | 3.40 ± 0.01 |

r_{2}, nm | 7.12 ± 0.03 | 7.31 ± 0.03 | 7.19 ± 0.03 |

r_{3}, nm | 19.7 ± 0.1 | 20.2 ± 0.1 | 19.9 ± 0.1 |

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**MDPI and ACS Style**

Lebedev, V.; Kulvelis, Y.; Kuklin, A.; Vul, A.
Neutron Study of Multilevel Structures of Diamond Gels. *Condens. Matter* **2016**, *1*, 10.
https://doi.org/10.3390/condmat1010010

**AMA Style**

Lebedev V, Kulvelis Y, Kuklin A, Vul A.
Neutron Study of Multilevel Structures of Diamond Gels. *Condensed Matter*. 2016; 1(1):10.
https://doi.org/10.3390/condmat1010010

**Chicago/Turabian Style**

Lebedev, Vasily, Yury Kulvelis, Alexander Kuklin, and Alexander Vul.
2016. "Neutron Study of Multilevel Structures of Diamond Gels" *Condensed Matter* 1, no. 1: 10.
https://doi.org/10.3390/condmat1010010