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1.
J Phys Chem C Nanomater Interfaces ; 126(14): 6308-6317, 2022 Apr 14.
Artigo em Inglês | MEDLINE | ID: mdl-35449522

RESUMO

Plasmonic nanoparticles are renowned as efficient heaters due to their capability to resonantly absorb and concentrate electromagnetic radiation, trigger excitation of highly energetic (hot) carriers, and locally convert their excess energy into heat via ultrafast nonradiative relaxation processes. Furthermore, in assembly configurations (i.e., suprastructures), collective effects can even enhance the heating performance. Here, we report on the dynamics of photothermal conversion and the related nonlinear optical response from water-soluble nanoeggs consisting of a Au nanocrystal assembly trapped in a water-soluble shell of ferrite nanocrystals (also called colloidosome) of ∼250-300 nm in size. This nanoegg configuration of the plasmonic assembly enables control of the size of the gold suprastructure core by changing the Au concentration in the chemical synthesis. Different metal concentrations are analyzed by means of ultrafast pump-probe spectroscopy and semiclassical modeling of photothermal dynamics from the onset of hot-carrier photogeneration (few picosecond time scale) to the heating of the matrix ligands in the suprastructure core (hundreds of nanoseconds). Results show the possibility to design and tailor the photothermal properties of the nanoeggs by acting on the core size and indicate superior performances (both in terms of peak temperatures and thermalization speed) compared to conventional (unstructured) nanoheaters of comparable size and chemical composition.

2.
Small ; 16(48): e2005701, 2020 12.
Artigo em Inglês | MEDLINE | ID: mdl-33169513

RESUMO

Directed assembly of nanocrystals from conventional templates suffers from poor control over the periodicity of the nanocrystal assembly, which is largely due to the fact that the template exists prior to the assembly and is not generally adaptive. Herein, small organic molecules (tris-amide triarylamines, TATA) are demonstrated as conceptual templates from self-assembly through noncovalent interactions. The as-formed supramolecular structures with terminated alkyl chains, resembling the structure of as-synthesized nanocrystals capped with alkyl chains, are able to interact with nanocrystals through van der Waals attractive forces, thereby enabling directed assembly of nanocrystals into ordered superlattices. Specifically, it is found that, as determined by the substituted alkyl chains of TATA, either H or J-aggregates of TATA can be achieved, which eventually produce several distinct supramolecular structures, from rods to spindles, to rings, and to spheres, serving as on-pathway intermediate that directs the assembly of nanocrystals into diverse nanocrystal superlattices. The approach described can be applicable to produce ordered nanocrystal assemblies of a wide range of nanocrystals, independent of size and shape and without ligand exchange process. Strikingly, a helical TATA stacking can direct assembly of binary nanocrystal mixtures into NaZn13 binary superhelix.

3.
ACS Nano ; 14(5): 5517-5528, 2020 May 26.
Artigo em Inglês | MEDLINE | ID: mdl-32374985

RESUMO

Although self-assembly across multiple length scales has been well recognized and intensively investigated in natural biological system, the design of artificial heterostructures enabled by integrative self-assembly is still in its infancy. Here we report a strategy toward the growth of discrete supracrystalline heterostructures from inorganic nanocrystals and porous organic cages (CC3-R), which in principle relies on the host-guest interactions between alkyl chains coated on nanocrystals and the cavity of cage molecules. Density functional theory calculation indicates that an attractive energy of ∼-2 kBT is present between an alkyl chain and the cavity of a CC3-R molecule, which is responsible for the assembly of nanocrystal superlattices on the CC3-R octahedral crystals. Of particular interest is that, determined by the shape of the nanocrystals, two distinct assembly modes can be controlled at the mesoscale level, which eventually produce either a core/shell or heterodimer supracrystalline structure. Our results highlight opportunities for the development of such a noncovalent integrative self-assembly not limited to a particular length scale and that could be generally applicable for flexible integration of supramolecular systems.

4.
Proc Natl Acad Sci U S A ; 116(17): 8161-8166, 2019 04 23.
Artigo em Inglês | MEDLINE | ID: mdl-30952788

RESUMO

We investigate, with a combination of ultrafast optical spectroscopy and semiclassical modeling, the photothermal properties of various water-soluble nanocrystal assemblies. Broadband pump-probe experiments with ∼100-fs time resolution in the visible and near infrared reveal a complex scenario for their transient optical response that is dictated by their hybrid composition at the nanoscale, comprising metallic (Au) or semiconducting ([Formula: see text]) nanostructures and a matrix of organic ligands. We track the whole chain of energy flow that starts from light absorption by the individual nanocrystals and subsequent excitation of out-of-equilibrium carriers followed by the electron-phonon equilibration, occurring in a few picoseconds, and then by the heat release to the matrix on the 100-ps timescale. Two-dimensional finite-element method electromagnetic simulations of the composite nanostructure and multitemperature modeling of the energy flow dynamics enable us to identify the key mechanism presiding over the light-heat conversion in these kinds of nanomaterials. We demonstrate that hybrid (organic-inorganic) nanocrystal assemblies can operate as efficient nanoheaters by exploiting the high absorption from the individual nanocrystals, enabled by the dilution of the inorganic phase that is followed by a relatively fast heating of the embedding organic matrix, occurring on the 100-ps timescale.

5.
ACS Nano ; 13(1): 573-581, 2019 Jan 22.
Artigo em Inglês | MEDLINE | ID: mdl-30557505

RESUMO

Physical properties of nanocrystals self-assembled into 3D superlattices called supracrystals are highly specific with unexpected behavior. The best example to support such a claim was given through STM/STS experiments at low temperature of very thick supracrystals (around 1000 layers) where it was possible to image the surpracrystal surface and study their electronic properties. From previous studies, we know the optical properties of Ag nanocrystals self-assembled in a hexagonal network two-dimensional (2D) or by forming small 3D superlattices (from around 2 to 7 layers) are governed by dipolar interactions. Here, we challenge to study the optical properties of Ag supracrystals film characterized by large thicknesses (from around 27 to 180 Ag nanocrystals layers). In such experimental conditions, according to the classical Beer-Lambert law, the absorption of Ag films is expected to be very large, and the film transmission is close to zero. Very surprisingly, we observe reduced transmission intensity with an increase of the notch line width, in the 300-800 nm wavelength range, as the supracrystal film thickness increased. By calculating the transmission through the supracrystal films, we deduced that the films were dominated by the presence of cracks with wetting layers existing at their bottoms. This result was also confirmed by optical micrographs. The cracks widths increased with increasing the film thickness leading to more complex wetting layers. We also demonstrated the formation of small Ag clusters at the nanocrystal surface. These results provide some implications toward the design of plasmonic materials.

6.
Nanoscale ; 10(32): 15371-15378, 2018 Aug 16.
Artigo em Inglês | MEDLINE | ID: mdl-30083696

RESUMO

Vicinal Au supracrystal surfaces were prepared from Ausingle single domain nanocrystals (NCs), whereas by replacing Ausingle with their polycrystalline counterparts common low-energy supracrystal surfaces were produced. By analogy to atomic crystalline surfaces, we propose a mechanism to explain the formation of such unexpected supracrystal vicinal surfaces, composed of only Ausingle NCs which are non-compact (bct structure) with a coherent alignment of the atomic planes oriented along the [111] superlattice axis and a slight tilt configuration (8.1°) of NCs. In the presence of Co(ε) NCs, these Ausingle supracrystals lose both the slightly tilted configuration of NCs and their orientational order leading to a superlattice transition from bct to fcc. In contrast, supracrystals of Aupoly NCs are insensitive to the presence of Co(ε) NCs. These intriguing structural changes obtained with Ausingle compared to Aupoly supracrystals in the absence and presence of Co(ε) NCs could explain the formation of vicinal surfaces. Note that the solvent used to disperse the nanocrystals plays a key role in the formation of supracrystal vicinal surfaces. Here, a new analogy between supracrystals and atomic crystals is presented.

7.
Phys Chem Chem Phys ; 19(35): 23887-23897, 2017 Sep 13.
Artigo em Inglês | MEDLINE | ID: mdl-28829072

RESUMO

The Young's modulus of three-dimensional self-assembled Ag nanocrystals, as so-called supracrystals, is correlated with the type of coating agent as well as the nanocrystal morphology. The Young's moduli of supracrystals of icosahedral Ag nanocrystals are measured in the range of tens to hundreds of megapascals revealing an extremely soft mechanical behavior. The alkylamine molecules used as coating agents weakly interact with the Ag nanocrystal surface favoring translational and orientational ordering of atomic lattice planes of nanocrystals. Under such experimental conditions, both the average distance between nanocrystals and the increase of the nanocrystal diameter control the measured Young's modulus: it increases with decreasing interparticle distance and increasing nanocrystal diameter. When dodecylamine (C12NH2) molecules are replaced by dodecanethiol (C12SH), the orientational ordering between nanocrystals, produced from the same batch as C12NH2, disappears by inducing a drop in the Young's modulus. This is attributed to the formation of a "skin" at the nanocrystal surface causing a transition from shaped to spherical nanocrystals. Finally, by comparing with various studies performed in our group with Co and Au nanocrystals, we explain the formation of such extremely soft materials with Ag nanocrystals by both the strength of the binding between nanocrystals and coating agents and the ligand-ligand interactions.

8.
Acc Chem Res ; 50(8): 1946-1955, 2017 08 15.
Artigo em Inglês | MEDLINE | ID: mdl-28726381

RESUMO

The spontaneous assembly of uniform-sized globular entities into ordered arrays is a universal phenomenon observed for objects with diameters spanning a broad range of length scales. These extend from the atomic scale (10-8 cm), through molecular and macromolecular scales with proteins, synthetic low polymers, and colloidal crystals (∼10-6 cm), to the wavelength of visible light (∼10-5 cm). The associated concepts of sphere packing have had an influence in diverse fields ranging from pure geometrical analysis to architectural models or ideals. Self-assembly of atoms, supramolecules, or nanocrystals into ordered functional superstructures is a universal process and prevalent topic in science. About five billion years ago in the early solar system, highly uniform magnetite particles of a few hundred nanometers in size were assembled in 3D arrays.1 Thirty million years ago, silicate particles with submicrometer size were self-organized in the form of opal.2 Opal is colorless when composed of disordered silicate microparticles whereas it shows specific reflectivity when particles order in arrays. Nowadays, nanocrystals, characterized by a narrow size distribution and coated with alkyl chains to maintain their integrity, self-assemble to form crystallographic orders called supracrystals. Nanocrystals and supracrystals are arrangements of highly ordered atoms and nanocrystals, respectively. The morphologies of nanocrystals, supracrystals, and minerals are similar at various scales from nanometer to millimeter scale.3,4 Such suprastructures, which enable the design of novel materials, are expected to become one of the main driving forces in material research for the 21st century.5,6 Nanocrystals vibrate coherently in a supracrystal as atoms in a nanocrystal. Longitudinal acoustic phonons are detected in supracrystals as with atomic crystals, where longitudinal acoustic phonons propagate through coherent movements of atoms of the lattice out of their equilibrium positions. These vibrational properties show a full analogy with atomic crystals: In supracrystals, atoms are replaced by (uncompressible) nanocrystals and atomic bonds by coating agents (carbon chains), which act like mechanical springs holding together the nanocrystals. Electronic properties of very thick (more than a few micrometers) supracrystals reveal homogeneous conductance with the fingerprint of the isolated nanocrystal. Triangular single crystals formed by heat-induced (50 °C) coalescence of thin supracrystals deposited on a substrate as epitaxial growth of metal particles on a substrate with specific orientation produced by ultrahigh vacuum (UHV). Here we demonstrate that marked changes can occur in the chemical and physical properties of nanocrystals differing by their nanocrystallinity, that is, their crystalline structure. Furthermore, the properties (mechanical, growth processes) of supracrystals also change with the nanocrystallinity of the nanoparticles used as building blocks.

9.
ACS Nano ; 11(8): 7797-7806, 2017 08 22.
Artigo em Inglês | MEDLINE | ID: mdl-28745866

RESUMO

Hydrophobic Au nanocrystal assemblies (both ordered and amorphous) were dispersed in aqueous solution via the assistance of lipid vesicles. The intertwine between vesicles and Au assemblies was made possible through a careful selection of the length of alkyl chains on Au nanocrystals. Extinction spectra of Au assemblies showed two peaks that were assigned to a scattering mode that red-shifted with increasing the assembly size and an absorption mode associated with localized surface plasmon that was independent of their size. This plasmon fingerprint could be used as a probe for investigating the optical properties of such assemblies. Our water-soluble assemblies enable exploring a variety of potential applications including solar energy and biomedicine.

10.
Langmuir ; 32(47): 12231-12241, 2016 11 29.
Artigo em Inglês | MEDLINE | ID: mdl-27934509
11.
Langmuir ; 32(31): 7683-96, 2016 08 09.
Artigo em Inglês | MEDLINE | ID: mdl-27412075

RESUMO

N-Heterocyclic carbenes (NHCs) have emerged as a new class of ligands for materials chemistry that appears particularly relevant for the stabilization and functionalization of metal nanoparticles (NPs). The particular properties and high synthetic flexibility of NHCs make them highly attractive tools for the development of new (nano)materials and the fundamental study of their properties. The relationships between the NHC structure and NP structure/properties, including physical, biological, and self-assembly properties, remain largely unknown. In the past decade, many efforts have been made to gain more fundamental understanding in this area. In this feature article, we present our contribution in this field focusing on the formation of NHC-coated Au nanocrystals (NCs), their stability, and their ability to self-assemble into 3D crystalline structures called supracrystals. First, the formation of NHC-stabilized Au NCs is discussed by comparing different NHC structures, NHC-based Au precursors, and synthesis methods. This study shows the major role of the NHC structure in obtaining both stable NHC-coated Au NCs and narrow size distributions. In a second part, a comparative study of the oxygen resistance of NHC- and thiol-coated NCs is presented, demonstrating the enhanced stability of NHC-coated Au NCs to oxygen-based treatments. Finally, the self-assembly of NHC-coated Au NCs into 3D Au superlattices is presented. The formation of large organized domains of several micrometers is described from the design of NHCs tailored with long alkyl chains. In these different contexts, efforts have been made to gain a more in-depth understanding of the behavior of NHC ligands at the surface of NCs. These results show that the NHC-based approach to nanomaterials has many assets for opening a new research area in the supracrystal world.

12.
Langmuir ; 32(47): 12370-12377, 2016 Nov 29.
Artigo em Inglês | MEDLINE | ID: mdl-27243268

RESUMO

Since both liposomes and nanoparticles have shown great potential in application for clinical diagnostics and therapeutics, the perfect combination of the two materials is appealing for further improving the theranostic effect. Therefore, fabrication of liposomes loaded with nanoparticles in a controllable manner is desirable. Detection of various factors affecting encapsulation needs to be assigned. Here, we use zwitterionic Au nanoparticles (Au±NPs) coated with a mixture of 11-mercaptoundecanoic acid and N,N,N-trimethyl(11-mercaptoundecyl) ammonium chloride to study their encapsulation behavior by reversed phase evaporation (REV) method. To produce a reverse emulsion, an organic solution of dipalmitoylphosphatidylcholine (DPPC) and 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-[methoxy(poly(ethylene glycol))-2000] (ammonium salt) (PEG2000-DOPE) is mixed with an aqueous Au±NP colloidal solution under sonication. The pH of the colloidal solution controls the surface charge of the Au±NPs and then tunes the interactions between Au±NPs and phospholipids. At lower pH, the positive surface charges favor Au±NP transfer into the organic phase and consequently prevent their encapsulation into liposomes. The efficiency in encapsulation is markedly improved by increasing the pH of the Au±NP colloidal solution. The highest efficiency is obtained at a pH value slightly larger than the isoelectric point. Further pH increase induces a decrease in encapsulation efficiency. This is due to increase of the repulsive forces between Au±NPs and phospholipids indicating that both the nature (positive or negative) and the amount of surface charge are key parameters in the encapsulation efficiency. We also find that the increase in Au±NP concentration favors the encapsulation process.


Assuntos
Ouro , Lipossomos/química , Nanopartículas Metálicas , Fosfolipídeos/química , Tamanho da Partícula , Polietilenoglicóis
13.
J Am Chem Soc ; 138(10): 3493-500, 2016 Mar 16.
Artigo em Inglês | MEDLINE | ID: mdl-26908091

RESUMO

The concept of template-confined chemical reactions allows the synthesis of complex molecules that would hardly be producible through conventional method. This idea was developed to produce high quality nanocrystals more than 20 years ago. However, template-mediated assembly of colloidal nanocrystals is still at an elementary level, not only because of the limited templates suitable for colloidal assemblies, but also because of the poor control over the assembly of nanocrystals within a confined space. Here, we report the design of a new system called "supracrystalline colloidal eggs" formed by controlled assembly of nanocrystals into complex colloidal supracrystals through superlattice-matched epitaxial overgrowth along the existing colloidosomes. Then, with this concept, we extend the supracrystalline growth to lattice-mismatched binary nanocrystal superlattices, in order to reach anisotropic superlattice growths, yielding freestanding binary nanocrystal supracrystals that could not be produced previously.

14.
ACS Nano ; 10(2): 2277-86, 2016 Feb 23.
Artigo em Inglês | MEDLINE | ID: mdl-26812588

RESUMO

Assembly of nanoparticles into supracrystals provides a class of materials with interesting optical and magnetic properties. However, supracrystals are mostly obtained from hydrophobic particles and therefore cannot be manipulated in aqueous systems, limiting their range of applications. Here, we show that hydrophobic-shaped supracrystals self-assembled from 8.2 nm cobalt nanoparticles can be dispersed in water by coating the supracrystals with lipid vesicles. A careful characterization of these composite objects provides insights into their structure at different length scales. This composite, suspended in water, retains the crystalline structure and paramagnetic properties of the starting material, which can be moved with an applied magnetic field.


Assuntos
Interações Hidrofóbicas e Hidrofílicas , Lipossomos/química , Nanopartículas Metálicas/química , Cobalto/química , Ácido Oleico/química , Fosfolipídeos/química
15.
J Am Chem Soc ; 137(46): 14773-84, 2015 Nov 25.
Artigo em Inglês | MEDLINE | ID: mdl-26549642

RESUMO

The surface chemistry in colloidal nanocrystals on the final crystalline structure of binary superlattices produced by self-assembly of two sets of nanocrystals is hereby demonstrated. By mixing nanocrystals having two different sizes and the same coating agent, oleylamine (OAM), the binary nanocrystal superlattices that are produced, such as NaCl, AlB2, NaZn13, and MgZn2, are well in agreement with the crystalline structures predicted by the hard-sphere model, their formation being purely driven by entropic forces. By opposition, when large and small nanocrystals are coated with two different ligands [OAM and dodecanethiol (DDT), respectively] while keeping all other experimental conditions unchanged, the final binary structures markedly change and various structures with lower packing densities, such as Cu3Au, CaB6, and quasicrystals, are observed. This effect of the nanocrystals' coating agents could also be extended to other binary systems, such as Ag-Au and CoFe2O4-Ag supracrystalline binary lattices. In order to understand this effect, a mechanism based on ligand exchange process is proposed. Ligand exchange mechanism is believed to affect the thermodynamics in the formation of binary systems composed of two sets of nanocrystals with different sizes and bearing two different coating agents. Hence, the formation of binary superlattices with lower packing densities may be favored kinetically because the required energetic penalty is smaller than that of a denser structure.

16.
Langmuir ; 31(47): 12873-82, 2015 Dec 01.
Artigo em Inglês | MEDLINE | ID: mdl-26550843

RESUMO

The stability of Au nanocrystals (NCs) coated with different N-heterocyclic carbenes (NHCs) or dodecanethiol (DDT) to oxygen-based treatments was investigated. A dominant effect of the ligand type was observed with a significantly greater oxygen resistance of NHC-coated Au NCs compared to that of the thiol-based analogues. NHC-coated Au NCs are stable to 10 W oxygen plasma etching for up to 180 s whereas the integrity of DDT-coated Au NCs is strongly affected by the same treatment from 60-80 s. In the latter case, the average size of the NCs (from 2.6 to 6.3 nm) and the method of synthesis have no effect on the stability. NHC-coated Au NCs were found to generate of a smaller quantity of ligand-derived species under molecular oxygen treatment, which could account for the increased stability.


Assuntos
Ouro/química , Metano/análogos & derivados , Nanopartículas/química , Oxigênio/química , Metano/química , Estereoisomerismo
17.
Beilstein J Nanotechnol ; 6: 1229-36, 2015.
Artigo em Inglês | MEDLINE | ID: mdl-26114081

RESUMO

BACKGROUND: Highly ordered three-dimensional colloidal crystals (supracrystals) comprised of 7.4 nm diameter Au nanocrystals (with a 5% size dispersion) have been imaged and analysed using a combination of scanning tunnelling microscopy and dynamic force microscopy. RESULTS: By exploring the evolution of both the force and tunnel current with respect to tip-sample separation, we arrive at the surprising finding that single nanocrystal resolution is readily obtained in tunnelling microscopy images acquired more than 1 nm into the repulsive (i.e., positive force) regime of the probe-nanocrystal interaction potential. Constant height force microscopy has been used to map tip-sample interactions in this regime, revealing inhomogeneities which arise from the convolution of the tip structure with the ligand distribution at the nanocrystal surface. CONCLUSION: Our combined STM-AFM measurements show that the contrast mechanism underpinning high resolution imaging of nanoparticle supracrystals involves a form of nanoscale contact imaging, rather than the through-vacuum tunnelling which underpins traditional tunnelling microscopy and spectroscopy.

18.
J Am Chem Soc ; 137(13): 4487-93, 2015 Apr 08.
Artigo em Inglês | MEDLINE | ID: mdl-25785302

RESUMO

Here, it is shown that binary superlattices of Co/Ag nanocrystals with the same size, surface coating, differing by their type of crystallinity are governed by Co-Co magnetic interactions. By using 9 nm amorphous-phase Co nanocrystals and 4 nm polycrystalline Ag nanocrystals at 25 °C, triangle-shaped NaCl-type binary nanocrystal superlattices are produced driven by the entropic force, maximizing the packing density. By contrast, using ferromagnetic 9 nm single domain (hcp) Co nanocrystals instead of amorphous-phase Co, dodecagonal quasicrystalline order is obtained, together with less-packed phases such as the CoAg13 (NaZn13-type), CoAg (AuCu-type), and CoAg3 (AuCu3-type) structures. On increasing temperature to 65 °C, 9 nm hcp Co nanocrystals become superparamagnetic, and the system yields the CoAg3 (AuCu3-type) and CoAg2 (AlB2-type) structures, as observed with 9 nm amorphous Co nanocrystals. Furthermore, by decreasing the Co nanocrystal size from 9 to 7 nm, stable AlB2-type binary nanocrystal superlattices are produced, which remain independent of the crystallinity of Co nanocrystals with the superparamagnetic state.

19.
Nano Lett ; 14(11): 6632-8, 2014 Nov 12.
Artigo em Inglês | MEDLINE | ID: mdl-25337812

RESUMO

Crystals of nanocrystals, also called supracrystals and nanocrystal superlattices, are expected to exhibit specific properties that differ from both the corresponding bulk material and nanosized elementary units. In particular, their surfaces have a great potential as nanoscale interaction plateforms. However, control of the symmetry, compacity, and roughness of their surfaces remains an open question. Here, we describe the spontaneous formation of upper vicinal surfaces for supracrystals of Au nanocrystals grown on a sublayer of ordered Co nanocrystals. Stepped or kinked surfaces vicinal to the {100}, {110}, and {111} planes are observed to be extended on the micrometer range. The formation of such high-index planes is explained by a heteroepitaxial relationship between both Co and Au nanocrystal superlattice.

20.
J Phys Chem B ; 118(49): 14070-5, 2014 Dec 11.
Artigo em Inglês | MEDLINE | ID: mdl-25198062

RESUMO

Silver nanocrystals (NCs) stabilized using amine-terminated coating agents (oleylamine or dodecylamine), their size ranging between 2 and 12 nm in diameter, are synthesized by hot injection methods. Their dispersion in size is relatively low (typically below 10%) without the need for a postsynthesis size segregation process. The amine-terminated coating agents are replaced by thiol-terminated molecules (dodecanethiol or hexadecanethiol) by ligand exchange, allowing the formation of alkanethiol coated Ag colloids. All NCs with various surface coatings are dispersed in toluene. Regardless of the nature of the coating agent, the surface plasmon resonance (SPR) is red-shifted with decreasing the NC size. For a given size, the SPR peak of thiol-stabilized NCs is shifted to lower energies compared to that of amine-stabilized NCs. Furthermore, with thiol-stabilized Ag NCs, the position of the SPR peak was found to be sensitive to the length of the alkyl chains of the coating agent, whereas minor differences are detected for Ag NCs coated with amines terminated with differing alkyl chain lengths.

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