Introduction

Today, significant progress has been made in nanomaterial research, nanobiotechnology, and nanophotonics, including the work on nanostructures, which is relevant for the interaction of biological samples with nanomaterials. Research on metallic nanostructures is greatly encouraged due to the unique properties of nanosized materials. These properties are highly dependent on the size and shape of the nanoparticle [1,2,3,4,5,6,7,8,9]. It is well known that nanomaterials behave very differently from bulk materials; fine tuning of nanoparticle properties can only be possible when they are monodispersed and structure–property relationships can be understood. Numerous techniques have been designed to generate nanoparticles; chemical synthesis methods include coprecipitation methods, sol–gel methods, and solvothermal processes [10]. Physical approaches, such as pulsed-laser deposition [11], mechanosynthesis [12], and flame spray pyrolysis [13], are also used to synthesize nanoparticles. In comparison with these methods, pulsed laser ablation in liquids possesses fundamental advantages because it combines the physical processes of the laser-matter interaction with chemical mechanisms due to the use of liquid solvents. Among all these synthesis techniques, the one that offers a great challenge, due to its high polydispersity, is the laser ablation technique [14, 15]. This technique provides us with the opportunity to generate colloids that will help us to demonstrate the proposal of this work. Additionally, the synthesis of nanoparticles by laser ablation is used to obtain stable nanoparticles in water, which does not need stabilizing molecules or other chemicals. The obtained nanoparticles are highly available for further functionalization or can be used wherever unprotected metal nanoparticles are found, such as in biosensors or the SERS (surface-enhanced Raman spectroscopy) technique. [16]. That is, properties of the nanomaterials may strongly depend on the nanoparticle size and shape [17]; for this reason, a rapid discrimination and/or separation of the nanoparticles by centrifugation or other techniques, at least in sizes, is necessary and important [18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Differential separation is a centrifugal method based on the different sedimentation rates of the nanoparticles in a homogeneous medium. By repeatedly centrifuging a suspension from a low speed to a high speed, the mixture will be gradually separated into two parts: the large ones in the precipitate and the smaller ones in the supernatant [32]. This method is suitable for the samples which contain a big weight difference between the fragments or the difference between sedimentation coefficients [33, 34].

Centrifugation has proven to be a robust process for enhancing sedimentation by using thousands of times the gravitational acceleration of the Earth. Centrifugation makes use of high-speed rotation to generate centrifugal acceleration acting on phases with different densities. Under centrifugal acceleration, heavier phases migrate to a location at a larger radius (toward the container periphery), while lighter phases are displaced to a location at a smaller radius (toward the axis of the container). A centrifuge uses centrifugal force (g-force) to isolate suspended particles from their surrounding medium on either a batch or a continuous-flow basis [1]. The force on the particles (compared to gravity) is called relative centrifugal force (RCF). For example, a RCF of 500 g indicates that the centrifugal force is 500 times greater than earthly gravitational force. Regarding this, S.O. Majekodunmi presents an extensive review on the basics and principle of centrifugation, classes of centrifuges, types of centrifuge separations, different types of density gradient media, classification of industrial centrifuges [35], types of industrial centrifuges and applications of centrifugation in the pharmaceutical industry, and other references [36]. The motion of a particle in a centrifugal field is described by a vectorial summation of the forces acting on the particle and Newton’s second law [37]. If the size of particles is less than 5 µm, they undergo Brownian motion; in such suspension, a stronger centrifugal force is applied to separate the particles. The g-force acting on particles is exponential to the speed of rotation. Doubling the speed of rotation increases the centrifugal force by a factor of four. The centrifugal force also increases with the distance from the axis of rotation. These two parameters are of considerable significance when selecting the appropriate centrifuge. RCF can be calculated by using the following expression: RCF = 11.18 r (rpm/1000)2, where r is the maximum radius of rotor and rpm represents the rotational speed in revolutions per minute [35].

A large number of works can be found in the literature about the centrifugation technique, and many of them show us its versatility and importance. For example, Paramelle et al. carried out a study in which colloidal gold nanoparticles (Au-NPs) or Ag-NPs were centrifuged for 90 min; 40 nm and 50 nm nanoparticles were centrifuged at 1000 g, whereas 10 nm nanoparticles were centrifuged at 15,000 g [38]. On the other hand, in the work reported by Nabika and Deki, the ruby-red solution, which contains silver nanorods, was obtained by centrifugation at 8000 rpm for 30 min. [39]. S. Agnihotri used the technique of centrifugation to remove the reducers that did not react; in this case, the silver nanoparticle suspensions were centrifuged at 12,000 rpm for 15 min and washed three times, followed by redispersion in deionized water and finally stored at 4 °C [40]. Helmlinger and co-workers used ultracentrifugation systems to collect particles by ultracentrifugation (66,000 g, 30 min), and then wash these particles with acetone and ultrapure water several times [41]. Finally, Park et al. propose that the centrifugation method is more time-efficient for denser or larger nanoparticles because of the centrifugal force (12,225 g for 15–75 min). Once the phase transfer to a certain organic solvent is complete, solvent exchange to another organic solvent is possible with a simple boiling method [42]. For these works, different speeds, with arbitrary centrifugation times, were used to separate their nanomaterials.

As can be seen, there is no research work whose objective has been merely the centrifugation protocol. The existing work in which the aforementioned topic has been addressed has resulted in explanations that may be complex and intricate, and thus, difficult to understand. Here, we intend to provide some guidelines for the use of centrifugation with defined centrifugation speed and time that helps us to homogenize the use of this nanoparticle separation method.

In summary, this work is a step-by-step guide in the use of centrifugation systems, which is very useful for students and young researchers. We hope that this text will fulfill the quest of knowledge rendering centrifuge system to biologists, biotechnologists, chemists, physicists, scientists, researchers, and practicing engineers. It also shows us the existence of a trend and correlation between the sizes of nanomaterials and the centrifugal force. Finally, we use a simple mathematical relationship, which could be used as a “size calibration curve” to extrapolate probable nanoparticle separation results with the knowledge of some technical and initial data.

Experimental section

Materials and methods

Silver colloids were synthesized by the laser ablation technique. The pulsed Nd:YAG laser from Quantel, Q-smart 850 model (by Lumibird, Lannion, France), with 1064 nm of wavelength, 10 Hz of repetition rate and 6 ns of pulse duration, was used for this experiment. Silver material (1.00’’ Dia. X 0.125’’ Thick) from Lesker (PA, USA), 99.99% purity, was used as the ablation target. Briefly, the laser beam was focused on the silver surface, which was immersed in a beaker with deionized water (12 ml). The ablation was carried out for a period of 480 s at a frequency of 10 pulses/s. The target was ablated four times to obtain the silver colloid that was used for centrifugation.

For the centrifugation of colloids, a centrifuge Hermle Labortechnik GmbH (Wehingen, Germany) with a rotor, model 221.23V01, with a capacity for 12 tubes of 1.5 ml and a maximum radius of 6.5 cm, was used.

To determine the particle size, each nanoparticle suspension was analyzed with TEM images obtained by transmission electron microscopy Jeol JEM2200FS (JEOL, Ltd., Tokyo, Japan), with spherical correction in the STEM mode using a high-angle annular dark field detector (HAADF), and the plasmonic resonances were obtained with a Cary 5000 spectrophotometer. The Ag-NPs statistic in each sample was acquired through TEM images with the Comptage de particules software version 2.0. The dispersity index (D), which is a dimensionless number indicating the width of the size distribution (values range between 0 and 1), was also obtained from the particle size measurement.

Separation of Ag-NPs by differential centrifugation, successive and cumulative process

It is generally difficult to control the size of the Ag-NPs; hence, it is necessary to use techniques that help us achieve a certain level of monodispersity after synthesis. In this case, this is possible through a centrifugation process carried out in a sequence of successive steps and additively in terms of rotation speed and duration time.

For the separation process, 18 ml of the colloid were divided into twelve 1.5-ml Eppendorf tubes, which were centrifuged in multiple stages at different speeds and successive and cumulative times. When the centrifugation was completed, the content of each Eppendorf tube was divided into three parts: (1) the bottom, which contained the settled and largest nanoparticles, which were separated and stored for further analysis; (2) The middle part, which was discarded because it was considered the interface between two systems, the sediment, and the supernatant; and (3) the supernatant, which is used for the next centrifugation stage (Fig. 1).

Fig. 1
figure 1

Schematic illustration of the differential centrifugation process

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The complete experimental protocol is described in detail in the diagram shown in Fig. 2. Each of the stages is made up of several rounds of centrifugation. In stage 1, four rounds were carried out because there was a total of 72 ml of the original colloid to ensure that there was enough material to obtain the necessary samples and to be able to carry out the different analyzes without problems due to the insufficient amount of material in the sample. From this first round, the original amount of material was decreased mainly due to the extraction of both the sedimented part and the part named “interface.”

Fig. 2
figure 2

Sequential diagram of the five different stages of the centrifugation process

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Basic principles of sedimentation

The effect of sedimentation due to the influence of the Earth’s gravitational field (g = 9.81 m s−2) versus increasing the sedimentation rate in a centrifugal field (G > 9.81 m s−2) is evident. The relative centrifugal field is usually expressed as a multiple of the acceleration due to gravity. When designing a centrifugation protocol, it is important to keep in mind that (1) the rate of particle sedimentation is proportional to the particle size, (2) the sedimentation rate is proportional to the difference in density between the particle and the medium, (3) the sedimentation rate is zero when the particle density is the same as the medium density, (4) the sedimentation rate decreases as the medium viscosity increases, and (5) the sedimentation rate increases as the gravitational force increases. When the conditions for the centrifugal separation of a nanomaterial are described, a detailed listing of rotor speed and radial dimensions of centrifugation must be provided. Essentially, the rate of sedimentation, v, is dependent upon the applied centrifugal field G (measured in cm s−2). G is determined by the radial distance, r, of the particle from the axis of rotation (in cm) and the square of the angular velocity, ω, of the rotor (in radians per second): \(G=\omega^2x\;r\).

The average angular velocity of a rigid body that rotates around a fixed axis is defined as the ratio of the angular displacement in each time interval. One radian, usually abbreviated as 1 rad, represents the angle subtended at the center of a circle by an arc with length equal to the radius of the circle. Since 360° equals 2π radians, one revolution of the rotor can be expressed as 2π rad. Accordingly, the angular velocity of the rotor, given in rad s−1. Note that rad is treated as a scalar and is related to the rotor speed in revolutions per minute (rpm = 1 min−1) by \(\omega =2\pi v\), and therefore, the centrifugal field can be expressed as \(G= 4{\pi }^{2} {v}^{2} r\), where the variable v is the rotor speed (measured in revolutions per minute) and r is the radial distance from the center of rotation. Note that 60 revolutions per minute is the same speed as one revolution per second, i.e., rpm = 60 min−1 = 1 s−1.

The centrifugal field is generally expressed in multiples of the Earth’s gravitational field, g (9.81 m s−2). The relative centrifugal field, RCF (or g-force), is the ratio of the centrifugal acceleration at a specified radius and the speed to the standard acceleration of gravity. The RCF can be calculated from the following equation:

$$\mathrm{RCF}= \frac{G}{g} =\frac{ 4{\pi }^{2} {v}^{2} r}{g}$$
(1)

RCF units are therefore dimensionless as they denote multiples of the gravitational constant g. Grouping numerical constants together leads to a more convenient form of equation:

$$\mathrm{RCF}= \left(1.12 \times {10}^{-5}\right)\left(\frac{r}{1 \mathrm{cm}}\right){\left(\frac{\mathrm{rpm}}{1 {\mathrm{min}}^{-1}}\right)}^{2}$$
(2)

with r given in cm. The relative centrifugal force can easily be calculated and can often be displayed on modern instruments. In this job, the centrifugation protocol was carried out considering the properties of the rotor and characteristics of the particles to be separated. The type of rotor used here is the one known as “fixed angle rotor”; this means that the fixed angle rotor is held at a constant angle during the whole centrifugation period, so this type of rotors are better suited for the separation of particles with more distinct sedimentation coefficients [43]. It is important to note that each centrifuge, due to its maximum speed and rotor radius characteristics, has a minimum separation size limit (d0). Since RCF values are subject to significant changes depending on the rotor radius (distance between the tube and the rotor axis), it is important to understand that an increase in radius simply caused by changes in rotor angulations and rotor diameter has a dramatic effect on the RCF values [44]. For these reasons, it is important to have a basic understanding of RCF values, including the calculations to obtain RCF-rpm by means of the mathematical expression shown above. To avoid confusions, we have shown the rpm-RCF equivalences used in our experiment. And since these values are generally in the order of 103, we have proposed a scaling factor related to the RCF magnitudes to dampen the high values of the previously mentioned order. We denote this scaling factor as “a” because the centrifugal velocity is proportional to the rotation radius of a particle and the radial coordinate of a particle grows exponentially with time [16], which in our case, is closely related to the relative centrifugal force. In this sense, different classes of centrifuges can be established based on their speed: (a) low speed centrifuge, with maximum speed up to 10,000 rpm (7,000 g); (b) high speed centrifuge, with maximum speed up to 28,000 rpm (100,000 g); and (c) ultra-high-speed centrifuge, with maximum speed up to 150,000 rpm (900,000 g).

On the other hand, we propose an exponential expression based on our experimental design to determine the average diameter, “d”, of Ag-NPs as a function of the RCF that can be obtained from a mixture of Ag-NPs, mostly spherical or semi-spherical and with a size distribution between the 10 and 120 nm:

$$d=A {e}^{-ax}+ {d}_{0}$$
(3)

where a is a scaling factor, x is the relative centrifugal force, d0 is a standard value representing the smallest particle size that the centrifugation equipment can separate, and A is an experimental value associated with the synthesis and established from the colorimetric data observed in the colloid (x, y, Y), mainly considering the relation (100-Y), where Y corresponds to the intensity of the color.

Data treatment and evaluation

The synthesized Ag-NPs were obtained by laser ablation. An advantage of using this method is that naked Ag-NPs are obtained; the disadvantage is that high dispersity of sizes occurs, and nanoparticles with irregular shapes are generated. For this reason, a separation by size of the nanoparticles was carried out, which was controlled by the time and rotation speed of the centrifugation system used.

Figure 3 shows the results obtained in which we can see the size distribution (i), the absorption spectra (ii) and the STEM images (iii and iv), of the sedimented (red) and supernatant (blue) nanoparticles in the centrifugation process.

Fig. 3
figure 3

(i) Size distribution, (ii) absorption spectra, (iii) STEM image of sedimented sample and (iv) STEM image of supernatant sample of the Ag-NPs corresponding to the comparison between the sediments (red) and supernatants (blue) proceeding from five stages of the centrifugation process: a 1820 g/5 min, b 7270 g/10 min, c 16,350 g/15 min, d 29,060 g/20 min, and e 45,410 g/25 min

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The sample with the largest size dispersion was the sample centrifuged at 1820 g (Fig. 3a–i). The absorption spectrum for the sedimented one, it shows an extremely long curve toward higher wavelengths (Fig. 3a–ii), while for the supernatant, it shows a much narrower (almost symmetric curve) related to the SPR; this indicates the bigger, and therefore, heavier NPs were pelleted by the effect of centrifugation. It is observed that for the samples of sedimentation, the peak of the SPR is at 404 nm and that the peak of the quadrupole is at approximately 350 nm. For a better visualization, the sedimented sample was separated in two regions: region I, with the sizes of Ag-NPs placed between 10 and 60 nm, and region II, with Ag-NPs diameters up to approximately 120 nm; there are no Ag-NPs smaller than 8 nm. This is corroborated by STEM images (Fig. 3a–iii); the main geometry is spherical, but there are also other shapes like rounded triangular and icosahedral, as can be seen in the respective STEM image and in Fig. 3a–iv for the supernatant sample, most sizes being between 8 and 26 nm in diameter.

A considerable decrease in the range of size distribution can be observed for the sample sedimented at 7270 g (Fig. 3b–i). As can be seen, there is a notable difference between the SPR curve corresponding to the sedimentation (Fig. 3b–ii) with respect to the sedimentation of the previous stage (Fig. 3a–ii). Now, the curve is significantly narrower than its precedent, which is interpreted as the absence of the largest NPs. In addition, the size distribution of Ag-NPs varies mainly from 10 to 30 nm, with the largest population placed between 15 and 25 nm. In this case, the histogram was adjusted to a chi-square distribution, from which the average was obtained. A STEM image shows that most of the NPs are spherical in shape but still have faceted structures, mainly the largest ones (Fig. 3b–iii). The presence of these faceted shapes could be reflected in the results of the following stages of the process of size selection and separation by centrifugation. In the supernatant sample in Fig. 3b–iv, the main distribution is between 14 and 20 nm.

For the centrifugation at 16,350 g for 15 min, the narrowing in the size distribution (Fig. 3c–i) for the sedimented sample varies from 8 to 30 nm, with a greater presence between 12 and 16 nm, while for the supernatant sample, the distribution range is between 8 and 22 nm in diameter, with a greater presence between 12 and 14 nm. This can also be seen in the STEM images, in which uniformity in the size of Ag-NPs from the sedimented sample is noticeable (Fig. 3c–iii). Whereas for the supernatant sample, a decrease in the size dispersion and concentration of NPs is observed (Fig. 3c–iv). The SPR curve of the sedimented sample has a displacement at its peak toward a position of 406 nm (Fig. 3c–ii).

In samples resulting from the centrifugation process at 29,060 g, the range of size distribution does not exceed 25 nm (Fig. 3d–i), having its greater distribution between 10 and 15 nm for the sedimented sample (Fig. 3d–iii), and for the supernatant sample, the largest size distribution is between 10 and 12 nm (Fig. 3d–iv). The SPR curves show a very small difference in the position of the peak (Fig. 3d–ii), being at 403 nm for the supernatant and at 401 nm for sedimentation showing a great uniformity in the distribution of sizes, in addition to a higher concentration of Ag-NPs compared to the supernatant sample.

Finally, for the centrifugation at 45,410 g, the size distribution (Fig. 3e–i), SPR curves (Fig. 3 e–ii), STEM images for sedimentation (Fig. 3e-iii), and STEM images for the supernatant (Fig. 3e-iv), it is possible to observe a great uniformity in the sizes of the Ag-NPs, ranging from 6 to 18 nm and having the greatest distribution between 9 and 14 nm. For supernatants, the distribution of sizes is similar to the shown for the sedimented samples. This can be explained based on the characteristics of the model of the centrifuge used; that is, the maximum centrifugation speed limits of such a centrifuge are not sufficient to pellet the smaller Ag-NPs, for example, those smaller than 12 nm.

As explanation, although most of the geometries present in the samples are spherical shapes, and despite the reduction in the size distribution, due to the process, the concentration of Ag-NPs has decreased with each centrifugation stage; thus, in the last centrifugation processes (processes d and e in Fig. 3), the presence of other geometries is noticeable, such as those with faceted shapes consistent with the absorption peaks of the bands previously mentioned [45], unlike the first samples in which the concentration was much higher, which meant an opacity of the spherical majority over the minority with faceted form.

Table 1 shows the average diameters (Save and Fave) and the D corresponding to the Ag-NPs samples (DS and FS), both sedimented and supernatant respectively, obtained from the different centrifugation speeds. This index is dimensionless and scaled such that values smaller than 0.05 are mainly seen with highly monodisperse standards. D values bigger than 0.7 indicate that the sample has a very broad particle size distribution and is probably not suitable to be analyzed by the dynamic light scattering (DLS) technique, which offers good statistics with respect to the in situ measurements of the size and D of nanocarriers, and it allows particle sizing down to 1 nm in diameter [46,47,48].

Table 1 Experimental design table for the factors and responses
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As can be seen, even though originally there was a wide distribution of sizes, the largest D values do not reach the value of 0.7, necessary to consider a wide distribution of sizes; and for the 4th and 5th centrifugation stages, the D value is less than 0.05, which indicates a significative monodispersity.

Figure 4 shows the trend corresponding to the particle size obtained as a function of the centrifugation rate for both the sedimented and supernatants samples. For the parameter A, different values are assumed to represent different situations in which the largest size of Ag-NPs that is present in a hypothetical colloidal solution would change. And the d0 value remains fixed at 10, since this value was taken as a base, considering it as the minimum size of Ag-NPs that can be obtained by means of the centrifugation equipment used, and that was established from averaging the sizes of the group of Ag-NPs resulting from the last centrifugation step.

Fig. 4
figure 4

Trend lines given by changing the values of A compared with the size distributions obtained by means of the different centrifugation steps for the a sedimented and b supernatant samples

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Based on our results, and considering the different trends tested, the proposed expression was rethought, with the same structure, but numerical values established for each parameter in the equation are as follows:

$$d={40e}^{-0.0001x}+10$$

where the parameter A could take a value of 40 or 50 between the different tested settings; the scaling factor a = 0.0001 is due to the order of magnitude 103 of the RCF; and the value d0 = 10 is because it is considered as the minimum size that can settle with the maximum speed in the centrifugation equipment used.

For example, considering a centrifuge with the parameters (maximum speed = 12,000 rpm, rotor radius = 6.5 cm, and maximum separation capacity of 15 nm) in an approximate way, we could determine the probable size of the nanoparticles that we could separate between 8000 and 10,000 rpm, which would be between 30 and 40 nm. Another example, using a homemade centrifuge (paperfuge) which can reach a maximum speed = 25,000 rpm, radius = 5 cm, and a maximum separation capacity of 15 nm, we could calculate sizes of separated larger particles. Therefore, we believe that it is a viable and helpful proposal for research groups that do not have centrifugation equipment.

Conclusions

This work demonstrates a viable procedure to selectively separate Ag-NPs with different diameters by means of a successive process, by increasing time and speed of centrifugation, managing to separate Ag-NPs of approximately 11 nm in diameter from a colloidal solution whose size distribution varies between 8 and almost 120 nm. Ag-NPs obtained were synthesized by laser ablation, a technique by which naked Ag-NPs can be obtained since the use of any reducing agent is not necessary, which makes them feasible for catalysis applications, among others. In addition, we proposed a mathematical expression based on the experimental results obtained, which will help orient ourselves regarding what is the RCF necessary to separate Ag-NPs with a desired size distribution.

In summary, we have successfully obtained relatively pure Ag-NPs by an easy centrifugation method. The present separation method by centrifugation is simple, rapid, and effective. It can also be used for separation and purification of other metal nanomaterials with different sizes. Furthermore, the assemblies of monodispersed Ag-NPs obtained in this work have great potential applications including sensors and electronic devices.