2.1.

Chemicals and Materials

Europium(III) nitrate $\mathrm{Eu}{\left(\mathrm{N}{\mathrm{O}}_3\right)}_3\bullet 5{\mathrm{H}}_2\mathrm{O}$ and gadolinium(III) nitrate $\mathrm{Gd}{\left(\mathrm{N}{\mathrm{O}}_3\right)}_3\bullet 6{\mathrm{H}}_2\mathrm{O}$ were purchased from Sigma for the synthesis of luminescent $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ . Polystyrene microspheres doped with Eu chelates were purchased from Seradyn Inc. Avidin and bovine serum albumin (BSA) were obtained from Sigma. Immunopure biotinylated BSA (BSA-b, $8\text{moles}$ biotin∕mol protein) was purchased from Pierce. The 5-carboxyrhodamine 6G, succinimidyl ester was obtained from Sigma. Avidin-rhodamine 6G was prepared by a conjugation reaction between the amine groups of the protein and the succinimidyl ester of the 5-carboxyrhodamine 6G following a standard procedure recommended by Molecular Probes.¹⁰ The degree of labeling obtained was $2\text{moles}$ rhodamine∕mol avidin. Deionized water $\left(18\mathrm{M}\Omega \mathrm{cm}\right)$ was obtained using a Millipore purification system. Phosphate buffered saline (PBS) (pH 7.5) was $10\text{‐}\mathrm{m}M$ phosphate buffer, 0.8% saline. Black 96-well plates from Nunc were used for fluorescence measurements. The polydimethylsiloxane (PDMS) stamp used in microcontact printing had a linewidth of $5\mu \mathrm{m}$ . An $n$ -doped Si wafer was used as a substrate for microcontact printing and atomic force microscopy (AFM) characterization.

2.2.

Instrumentation

The size and the shape of the $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles were determined using a Philips CM-12 transmission electron microscope (TEM). An Opolette™ tunable pulsed optical parametric oscillator (OPO) laser (Opotek, California) was used for fluorescence excitation of the nanoparticles and their luminescence spectra were recorded using a Spectra Pro 300i gated intensified spectrometer (Princeton Instruments Inc). A Spectramax M2 microplate reader (Molecular Devices, Sunnyvale, California) was used for the detection of rhodamine fluorescence. An ultrasonic bath 75D (VWR) was used for treating the nanoparticle suspensions and for cleaning procedures. The centrifuges used for nanoparticle sizing were a Sorvall® RC5B Plus Centrifuge (Kendro Laboratory Products) and a Centrifuge 5415D (Eppendorf).

Fluorescence images were obtained by an epifluorescent microscope (Nicon Microphot-SA) equipped with a $400\text{‐}\mathrm{nm}$ dichroic mirror cube (DM 400) and a $100\text{‐}\mathrm{W}$ mercury lamp. A computer-controlled CCD camera RT Color, model 2.2.1 from Diagnostic Instruments Inc. was coupled to the fluorescent microscope and was used for digital visualization of the fluorescent images. A dichroic cube with a threshold wavelength of $400\mathrm{nm}$ was used to separate the UV excitation $({\lambda }_{\mathrm{ex}}<400\mathrm{nm})$ from the visible luminescent emission $({\lambda }_{\mathrm{em}}>400\mathrm{nm})$ . The CCD camera, image capture, and analysis were controlled by “Spot” software.

The AFM used for this work is a MFP-3D (Asylum Research, Santa Barbara, California). The ultrasharpened ${\mathrm{Si}}_3{\mathrm{N}}_4$ AFM tip was obtained from Veeco (Santa Barbara, California) with a spring constant of $0.1\mathrm{nN}{\mathrm{m}}^{-1}$ . The AFM images were obtained using the contact mode at a scan speed of $1\text{to}2\mathrm{Hz}$ . The image force was under $5\mathrm{nN}$ . The topography and lateral force images were taken in air at room temperature.

2.3.

Synthesis of $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ Nanoparticles

Luminescent $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles were synthesized by a spray pyrolysis method that is described in detail elsewhere.¹¹ Briefly, an ethanol solution containing $20\text{‐}\mathrm{m}M$ $\mathrm{Eu}{\left(\mathrm{N}{\mathrm{O}}_3\right)}_3$ and $80\text{‐}\mathrm{m}M$ $\mathrm{Gd}{\left(\mathrm{N}{\mathrm{O}}_3\right)}_3$ was sprayed into a hydrogen diffusion flame through a nebuilizer. The flame was formed by an ${\mathrm{H}}_2$ flow at $2\mathrm{L}{\min }^{-1}$ and an air coflow at $10\mathrm{L}{\min }^{-1}$ , surrounding the outlet of the nebulizer. A flame temperature above $2100°\mathrm{C}$ was reached. Oxidation reactions took place within the flame to form $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles. A cold finger was used to collect the $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ particles thermophoretically. The production rate of this synthesis procedure was about $400\text{to}500\mathrm{mg}{\mathrm{h}}^{-1}$ . The as-synthesized particles were suspended in methanol in an ultrasonic bath for $30\min$ to break any weak agglomerates formed during the collection process. Particles larger than $200\mathrm{nm}$ in diameter were settled from the primary suspension by means of selective centrifugation at $1000\times g$ for $2\min$ . The nanoparticles from the supernatant (diameter below $200\mathrm{nm}$ ) were extracted and dried for subsequent use.

2.4.

Coating of $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ Nanoparticles with Avidin

$\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles $\left(1\mathrm{mg}\right)$ were suspended using an ultrasonic bath in $1\mathrm{ml}$ of $25\text{‐}\mathrm{m}M$ carbonate-bicarbonate buffer, pH 8.6, in a polypropylene tube, previously coated with 0.5% BSA to avoid loss of avidin to the tube walls. A solution of $2\mathrm{mg}{\mathrm{ml}}^{-1}$ avidin $\left(100\mu \mathrm{l}\right)$ was added to the particle suspension and incubated in a rotating mill overnight at room temperature. The suspension was then centrifuged at $\mathrm{15,000}\times g$ for $3\min$ . The supernatant was discarded and the nanoparticle pellet was resuspended in the same buffer to wash off the excess protein. This washing procedure was repeated 3 times. To ensure that no bare particle surface remained, the avidin- $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles were incubated in $1\mathrm{ml}$ of $0.5\mathrm{mg}{\mathrm{ml}}^{-1}$ BSA solution in $25\text{‐}\mathrm{m}M$ phosphate buffer for $1\mathrm{h}$ at room temperature in the rotating mill. After three consecutive washings by centrifugation and resuspension, the avidin- $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles were used for the micropattern detection assays.

The surface saturation capacity of the nanoparticles with respect to avidin was evaluated following the same coating procedure, using avidin-rhodamine complex instead of avidin. After efficient washing of the excess avidin-rhodamine from the coating solution, the nanoparticle pellet was resuspended in $100\mu \mathrm{l}$ of carbonate-bicarbonate buffer. The fluorescence of rhodamine (excitation, $520\mathrm{nm}$ ; emission, $550\mathrm{nm}$ ) adsorbed on the particle surface was measured on the microplate reader and was used as an indication for the amount of adsorbed avidin molecules.

An identical coating with neutravidin rather than with avidin produced a smaller amount of protein adsorbed to the nanoparticle’s surface. This result strongly suggests that the higher positive charge of hen egg avidin (isoelectric point $\mathrm{PI}=10.0$ ), compared to that of neutravidin $(\mathrm{PI}=6.3)$ , contributes to the stronger adsorption to the negatively charged particle surface. This observation is consistent with other studies on protein adsorption onto quantum dots based on electrostatic interactions.¹²

2.5.

Microcontact Printing of Biotinylated BSA: Substrate, Stamp, and Sample Preparation

A silicon (100) wafer was used as a solid substrate for microcontact printing of proteins and the consecutive specific interactions. Prior to use, the silicon wafer was thoroughly washed in an ultrasound bath in acetone, ethanol, and deionized water for $10\min$ each. Finally, it was dried under a nitrogen stream. The PDMS stamp was washed by sonication in ethanol $(3\times 10\min )$ , dried under nitrogen and exposed to the solution of the inking protein ( $50\mu \mathrm{g}{\mathrm{ml}}^{-1}$ BSA-b in PBS) for $40\min$ . Excess solution was removed and the stamp was dried under a stream of nitrogen gas. After inking, the stamp was brought into contact with the silicon wafer substrate; a very small amount of force was applied to make a good contact between both surfaces. The stamp was removed after $2\min$ , and the wafer was rinsed with PBS and deionized water and dried under nitrogen. The stamp could be used about 50 times without degradation of the printing capability when rinsed with water, water:ethanol mixture (80:20) and cleaned by sonication in ethanol $(3\times 10\min )$ after each inking and printing cycle.

The BSA-b micropattern produced by microcontact printing is presented schematically in Fig. 1(a) . After microcontact printing of the BSA-b, the uncovered areas on the silicon substrate were blocked by immersing the wafer into a $2\mathrm{mg}{\mathrm{ml}}^{-1}$ BSA solution in PBS for $1\mathrm{h}$ [Fig. 1(b)] to avoid further nonspecific binding and∕or sticking. After washing and drying, the wafer was incubated with a suspension of the avidin-coated $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles in carbonate-bicarbonate buffer for $1\mathrm{h}$ in a shaker to enable specific interaction between the avidin and biotin [Fig. 1(c)]. After the interaction took place, the substrate was rinsed with buffer and water, and dried under nitrogen before fluorescence and AFM studies.

3.1.

Properties of the $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ Nanoparticles

The particle size distribution was determined by TEM analysis.¹¹ Most of the as-synthesized particles were nearly spherical with diameters between 5 and $500\mathrm{nm}$ . A narrower size distribution between 5 and $200\mathrm{nm}$ was obtained by means of selective centrifugation. Figure 2 represents a typical bright-field TEM micrograph of the sized nanoparticles. Three particles with sizes about $100\mathrm{nm}$ can be observed in the figure and one with a diameter of $15\mathrm{nm}$ . They have a dense morphology and approximately spherical shape.

Fig. 2

Bright-field electron micrograph of $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles obtained by spray pyrolysis.

The optical properties of the $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles were studied by using laser-induced luminescence. A colloidal solution of $1\mathrm{mg}{\mathrm{ml}}^{-1}$ in methanol was excited with a pulsed laser beam at a $260\text{‐}\mathrm{nm}$ wavelength. The slit width of the spectrophotometer was $0.8\mathrm{mm}$ . The luminescence emission spectrum in Fig. 3(a) shows an emission peak at $612\mathrm{nm}$ , which corresponds to the ${}^5\mathrm{D}_0\to {}^7\mathrm{F}_2$ electronic transition of the ${\mathrm{Eu}}^{3+}$ ion in the ${\mathrm{Gd}}_2{\mathrm{O}}_3$ crystal lattice.³ The half-width of the peak is $5\mathrm{nm}$ , which is very narrow in comparison with other fluorophores. The excitation spectrum from Fig. 3(b) reveals multiple excitation peaks far from the emission at $612\mathrm{nm}$ . The most efficient excitation is in the UV range between 250 and $280\mathrm{nm}$ . Some additional excitation peaks are distributed in the range between 350 to 400 and $450\text{to}550\mathrm{nm}$ . This offers the possibility to choose between different excitation sources according to the experimental setup and the application.

Fig. 3

(a) Luminescent emission spectrum of $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles (inset shows the luminescent lifetime) and (b) excitation spectrum of the $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles.

In addition to the Eu doping into the ${\mathrm{Gd}}_2{\mathrm{O}}_3$ host, we performed doping with other lanthanide ions such as Tb, Dy, Sm, and Tm (data not shown). This is achieved by using the same synthesis process and changing the doping precursor [e.g., $\mathrm{Tb}{\left(\mathrm{N}{\mathrm{O}}_3\right)}_3$ instead of $\mathrm{Eu}{\left(\mathrm{N}{\mathrm{O}}_3\right)}_3$ ]. This enables us to synthesize a variety of gadolinium oxide particles with different luminescent spectra and identical morphology and surface properties.

The photostability of $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles was evaluated by subjecting a $1\mathrm{mg}{\mathrm{ml}}^{-1}$ colloidal solution to a $10\text{‐}\mathrm{ns}$ pulsed UV laser beam $(\lambda =260\mathrm{nm})$ with an energy of $30\mu \mathrm{J}$ and frequency of $20\mathrm{Hz}$ . The luminescence intensity of the nanoparticles was measured after different irradiation times and was compared to the initial intensity. For comparison, the same experiment was performed with a colloidal solution of commercially available polystyrene nanoparticles (diameter $92\mathrm{nm}$ ) doped with Eu chelates (Seradyn™), which have very good brightness and spectral properties that are nearly identical to the $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles. We measured the luminescence intensity of the polystyrene nanoparticles to be about 10 times higher than that of $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles. Figure 4 shows a comparison of the relative luminescence intensity changes of the two samples after UV laser irradiation for up to $10\min$ . The intensity of $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles was not affected by the laser irradiation during the experiment. On the other hand, the polystyrene nanoparticles showed progressive photobleaching as the radiation was extended. Although the luminescence intensity of the chelate particles was not significantly affected after irradiation up to $1\min$ , it was reduced twofold after $6\min$ of irradiation and decreased by 40% of the initial value after $10\min$ of irradiation. Similar results regarding the bleaching of the same polystyrene particles were reported previously.¹³ Consecutive TEM studies showed that the size and shape of the polystyrene particles were not changed after the laser exposure. Therefore, we conclude that the Eu chelate complex was affected by the prolonged laser exposure.

Fig. 4

Luminescence intensity change due to irradiation with UV laser $(\lambda =260\mathrm{nm})$ for $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles and polystyrene spheres impregnated with Eu chelates.

The difference in the photostability between the polystyrene-chelate and the oxide particles was also visually observed on a fluorescent microscope, under excitation by a $100\text{‐}\mathrm{W}$ Hg lamp. The fluorescence of the polystyrene particles quickly decayed and the fluorescent image disappeared completely in less than $2\min$ . On the other hand, although oxide particles provided less bright images, the emission was stable for an effectively unlimited time of observation.

These results demonstrate one of the big advantages offered by the oxide particles—the high photostability, which makes them suitable for labels in a large variety of applications. In the case of microscopy applications, good photostability will provide enough time for image optimization and acquisition, while for quantitative measurements it will reduce the errors due to bleaching.

3.2.

Coating of the Nanoparticles with Avidin

Because the avidin-biotin interaction has a high binding constant, this assay is widely used in molecular biology, immunoassay, diagnostics, and biosensor research. A demonstration using the avidin and biotin reaction system is the logical first step in an evaluation of a new format for biosensors, protein chips, and other miniaturized analytical devices. Thus, we chose the avidin-biotin system as a model system in our studies to demonstrate the effectiveness of $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles as luminescent labels for micropattern imaging.

Avidin has been found to adsorb tightly to a variety of surfaces, such as dye-doped silica nanoparticles,¹⁴ carbon nanotubes,¹⁵ and quantum dots,^{12, 16} due to electrostatic and∕or hydrophobic interactions. Here, we take advantage of the negatively charged surface of the $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles to coat them with the avidin molecules using electrostatic interactions. The coating was carried out according to the experimental procedure already described. There are several advantages offered by this coating method: it is a one-step procedure (thereby avoiding chemical functionalization and conjugation steps); proteins retain their activity; conjugates are stable in a variety of buffers; the number of binding sites on the surface can be controlled by varying the coating concentration and they can be quantified easily; the nanoparticle’s luminescence is not affected by the protein layer; and the nanoparticle surface can be efficiently blocked to avoid nonspecific binding in immunoassays.

For evaluation of the avidin coating, $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ particles were coated with rhodamine-labeled avidin, following the procedure described in the experimental section. Using the measured fluorescence intensity of rhodamine, the amount of avidin that was adsorbed on the particle’s surface was evaluated for different coating concentrations of avidin-rhodamine per $1\mathrm{mg}$ nanoparticles (Fig. 5 ). For concentrations up to $200\mu \mathrm{g}{\mathrm{ml}}^{-1}$ , the amount of adsorbed avidin increased proportionally to the coating concentration, showing that the amounts of avidin were not enough to form dense monolayers on the particles' surfaces. For concentrations higher than $200\mu \mathrm{g}{\mathrm{ml}}^{-1}$ , the adsorption of avidin did not depend on the coating concentration, indicating saturation of the particles’ surfaces with avidin, suggesting the formation of a monolayer.

Fig. 5

Luminescent intensity of avidin-rhodamine adsorbed on $1\text{‐}\mathrm{mg}$ $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles, measured for different avidin-rhodamine coating concentrations.

We estimated the formation of the adsorbed layer thickness on spherical nanoparticles by assuming a $100\text{‐}\mathrm{nm}$ average particle diameter and a $6\times 6\text{‐}\mathrm{nm}$ footprint of the avidin molecule.¹⁷ For $1\mathrm{mg}$ of particles with density $7.4g{\mathrm{cm}}^{-3}$ (for ${\mathrm{Gd}}_2{\mathrm{O}}_3$ ), the total surface area is $4\times {10}^{-3}{\mathrm{m}}^2$ . This surface can be covered by $3.2\times {10}^{10}$ molecules $\left(185\mathrm{pmol}\right)$ of avidin with a footprint area of $36\mathrm{nm}$ .² In other words, $185\mathrm{pmol}$ of avidin are necessary to form a densely packed monolayer on the surface of $1\text{‐}\mathrm{mg}$ nanoparticles in our case. This number is in excellent agreement with the experimentally obtained saturation level in Fig. 5, therefore supporting the assertion that a monolayer formed. Based on these results, we chose $200\mu \mathrm{g}$ of avidin per $1\mathrm{mg}$ of nanoparticles as coating concentrations for further experiments.

3.3.

Fluorescence Imaging of BSA-Biotin Micropatterns with Avidin-Coated $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ Nanoparticles

The preparation of the BSA-biotin micropatterns and their interaction with the avidin coated $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles is schematically presented in Fig. 1 (see Sec. 2). The corresponding luminescent image is shown in Fig. 6 . A series of alternating bright strips and dark strips can be observed. The actual width of the strips is $5\mu \mathrm{m}$ , which corresponds to the features of the PDMS stamp used for BSA-b printing. The bright strips correspond to the luminescent $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ -avidin complexes that were specifically bound to the printed BSA-b. The dark strips, on the other hand, correspond to the blocked space between the printed strips where no avidin-coated particles were present. Although not all the bright strips had perfectly uniform fluorescent intensity, they were covered with specifically bound luminescent particles without large gaps. Densely packed bound particles showed up as higher intensity spots, while areas with lower surface density of bound particles were dimmer. On the other hand, very few particles could be observed in the BSA-passivated areas, demonstrating very low nonspecific binding in this case. The presence of clearly distinguishable fluorescent strips showed that the specific binding of avidin to biotin was not disturbed by the particles. In addition, the absence of particles between the strips confirmed that the blocking of the silicon substrate and the nanoparticles with BSA was sufficient to prevent non-specific binding of nanoparticles onto the substrate.

Fig. 6

Luminescence image of specifically immobilized avidin- $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles on printed BSA-biotin.

Fig. 1

Schematic description of microcontact printing of BSA-biotin (a), blocking with BSA (b), and specific interaction biotin with avidin- $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ (c).

Another important conclusion is that the nonuniform size distribution of the nanoparticles may not disturb the specific binding and does not lead to nonspecific binding. This makes possible the use of polydispersed particles instead of expensive monodispersed particles. The lack of photobleaching of the $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles enabled the fluorescent image to be observed for an unlimited period of time facilitating image optimization.

3.4.

AFM Characterization of BSA-Biotin Micropatterns with Avidin-Coated $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ Nanoparticles

We employed an AFM to image the immobilized avidin- $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ on a single particle level. The presence of solid nanoparticle labels with sizes larger than those of proteins on the substrate surface made it possible to evaluate the density of specifically and nonspecifically bound particles. An AFM topographic image of a randomly chosen region with a scan size of $40\times 40\mu {\mathrm{m}}^2$ of the substrate is shown in top part of Fig. 7 . Four vertical bright strips were visualized within the scan area as shown in Fig. 7. The surface density of specifically bound particles on the strips was much higher than the density of nonspecifically bound particles shown as scattered bright spots between the line patterns. The selectivity achieved here is the basis for applying this approach as a detection technique.

Fig. 7

AFM characterization of BSA-biotin micropatterns with avidin- $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ nanoparticles. Top: AFM topographic image of a micropatterned BSA-b after incubation with avidin- $\mathrm{Eu}:{\mathrm{Gd}}_2{\mathrm{O}}_3$ particle for $1\mathrm{h}$ (image size is $40\times 40\mu {\mathrm{m}}^2$ ) and bottom: the cursor profile along the line in the top image shows that the height of the particles is in the range of $10\text{to}250\mathrm{nm}$ .

The nonspecific binding of particles could be minimized possibly by further optimization of the coating and washing processes. We believe that the nonuniform density of the specifically bound particles could be partially due to reduced diffusion near the substrate surface and partially due to formation of small aggregates of particles during the incubation. This effect could be avoided by optimization of the incubation protocols. A cursor profile, which was taken along the white line in the AFM image, is shown in the bottom part of Fig. 7. According to the cursor profile, the particles on the strips have heights between about 10 and $250\mathrm{nm}$ . The majority of the particles exhibit heights between 50 and $100\mathrm{nm}$ . This height measurement is consistent with the particle size known from TEM studies. It is clear that these particles are the immobilized avidin-coated nanoparticles. The spatial resolution and the uniformity of protein micropattern visualization could be improved by using smaller, nearly monodispersed particles.