Core Particle Size Research Articles

The formation of linear aggregates in magnetic hyperthermia: Implications on specific absorption rate and magnetic anisotropy

The design and application of magnetic nanoparticles for use as magnetic hyperthermia agents has garnered increasing interest over the past several years. When designing these systems, the fundamentals of particle design play a key role in the observed specific absorption rate (SAR). This includes the particle's core size, polymer brush length, and colloidal arrangement. While the role of particle core size on the observed SAR has been significantly reported, the role of the polymer brush length has not attracted as much attention. It has recently been reported that for some suspensions linear aggregates form in the presence of an applied external magnetic field, i.e. chains of magnetic particles. The formation of these chains may have the potential for a dramatic impact on the biomedical application of these materials, specifically the efficiency of the particles to transfer magnetic energy to the surrounding cells. In this study we demonstrate the dependence of SAR on magnetite nanoparticle core size and brush length as well as observe the formation of magnetically induced colloidal arrangements. Colloidally stable magnetic nanoparticles were demonstrated to form linear aggregates in an alternating magnetic field. The length and distribution of the aggregates were dependent upon the stabilizing polymer molecular weight. As the molecular weight of the stabilizing layer increased, the magnetic interparticle interactions decreased therefore limiting chain formation. In addition, theoretical calculations demonstrated that interparticle spacing has a significant impact on the magnetic behavior of these materials. This work has several implications for the design of nanoparticle and magnetic hyperthermia systems, while improving understanding of how colloidal arrangement affects SAR.

Property models and theoretical analysis of novel solid oxide fuel cell with triplet nano-composite electrode

Triplet nano-composite electrodes are actively examined experimentally, but there is a shortage of theoretical study. Theoretical models are helpful for understanding the experiments and provide guidance for design optimization of the novel electrode. Here new models for computing the electrode electronic and ionic conductivities, TPB length and hydraulic radius are presented. The novel properties determined by the models are used in a multi-physics numerical model that couples the intricate interdependency among electric conductions, electrochemical reaction and gas transport in SOFC. The theoretical I–V relations and hydraulic radius are in good agreement with the experiments, validatingtheproposed property models. The property models are then used to examine the influence of microstructure and material composition. The results show that: (i) Larger core-particle size and smaller nano-particle size are helpful for improving electrode properties; (ii) The required nano-particle loading is determined by the desired electronic conductivity instead of the desired TPB length.

Toward Design of Magnetic Nanoparticle Clusters Stabilized by Biocompatible Diblock Copolymers for T2-Weighted MRI Contrast

We report the fabrication of magnetic particles comprised of clusters of iron oxide nanoparticles, 7.4 nm mean diameter, stabilized by a biocompatible, amphiphilic diblock copolymer, poly(ethylene oxide-b-D,L-lactide). Particles with quantitative incorporation of up to 40 wt % iron oxide and hydrodynamic sizes in the range of 80-170 nm were prepared. The particles consist of hydrophobically modified iron oxide nanoparticles within the core-forming polylactide block with the poly(ethylene oxide) forming a corona to afford aqueous dispersibility. The transverse relaxivities (r2) increased with average particle size and exceeded 200 s(-1) mM Fe(-1) at 1.4 T and 37 °C for iron oxide loadings above 30 wt %. These experimental relaxivities typically agreed to within 15% with the values predicted using analytical models of transverse relaxivity and cluster (particle core) size distributions derived from cryo-TEM measurements. Our results show that the theoretical models can be used for the rational design of biocompatible MRI contrast agents with tailored compositions and size distributions.

Meso-scale Simulations of Poly(N-isopropylacrylamide) Grafted Architectures

ABSTRACTPoly(N-isopropylacrylamide) (PNIPAM) is a thermosensitive polymer that is well-known for its behavior at a lower critical solution temperature (LCST) around 305 K. Below the LCST, PNIPAM is soluble in water, and above this temperature, polymer chains collapse and transform into a globule state. The conformational dynamics of single chains of polymer in a solution is known to be different from those of grafted structures that comprise of an ensemble of such single chains. In this study, we have carried out MD simulations of a mesoscopic nanostructure of PNIPAM polymer chains consisting of 60 monomer units grafted onto gold nanoparticles of different diameters, to study the effect of temperature and core particle size on the polymer conformations. Additionally, we have also studied the effect of grafting density on the coil-to-globule transition exhibited by PNIPAM through the LCST. The systems investigated consisted of ∼3 and ∼6 million atoms. Simulations were carried out below and above the LCST of PNIPAM, at 275K and 325K. Simulation trajectories were analyzed for radius of gyration of PNIPAM chains.

Water-Soluble Pd Nanoparticles Synthesized from ω-Carboxyl-S-Alkanethiosulfate Ligand Precursors as Unimolecular Micelle Catalysts

This report describes a two-phase synthesis of water-soluble carboxylate-functionalized alkanethiolate-capped Pd nanoparticles from ω-carboxyl-S-alkanethiosulfate sodium salts. The two-phase methodology using the thiosulfate ligand passivation protocol allowed a highly specific control over the surface ligand coverage of these nanoparticles, which are lost in a one-phase aqueous system because of the base-catalyzed hydrolysis of thiosulfate to thiolate. Systematic synthetic variations investigated in this study included the concentration of ω-carboxyl-S-alkanethiosulfate ligand precursors and reducing agent, NaBH4, and the overall ligand chain length. The resulting water-soluble Pd nanoparticles were isolated and characterized by transmission electron microscopy (TEM), thermogravimetric analysis (TGA), (1)H NMR, UV-vis, and FT-IR spectroscopy. Among different variations, a decrease in the molar equivalent of NaBH4 resulted in a reduction in the surface ligand density while maintaining a similar particle core size. Additionally, reducing the chain length of the thiosulfate ligand precursor also led to the formation of stable nanoparticles with a lower surface coverage. Since the metal core size of these Pd nanoparticle variations remained quite consistent, direct correlation studies between ligand properties and catalytic activities against hydrogenation/isomerization of allyl alcohol could be performed. Briefly, Pd nanoparticles dissolved in water favored the hydrogenation of allyl alcohol to 1-propanol whereas Pd nanoparticles heterogeneously dispersed in chloroform exhibited a rather high selectivity towards the isomerization product (propanal). The results suggested that the surrounding ligand environments, such as the ligand structure, conformation, and surface coverage, were crucial in determining the overall activity and selectivity of the Pd nanoparticle catalysts.

Thermal Properties of Particle Brush Materials: Effect of Polymer Graft Architecture on the Glass Transition Temperature in Polymer‐Grafted Colloidal Systems

SummaryThe governing parameters controlling the glass transition temperature of polymer‐grafted particle systems are analyzed for the particular case of polystyrene (PS)‐grafted silica colloids in the dense grafting limit. At a given degree of polymerization of surface‐grafted chains the glass transition temperature is found to increase as compared to linear chain polymers of equivalent degree of polymerization. The difference in the glass transition temperature between polymer‐grafted particle systems and their respective linear polymer analogs increases with decreasing degree of polymerization of surface‐grafted chains and levels off at similar plateau values for particle brushes of distinct particle core size. The trend toward increased glass transition temperature is interpreted as a consequence of the increased steric hindrance in polymer‐grafted particles that counteracts the relaxation of surface‐grafted polymer chains. The increase in glass transition temperature is shown to be approximately consistent with the chain conformational regimes that are predicted on the basis of a Daoud‐Cotton type scaling model.

Particle size, surface charge and concentration dependent ecotoxicity of three organo-coated silver nanoparticles: Comparison between general linear model-predicted and observed toxicity

Mechanism underlying nanotoxicity has remained elusive. Hence, efforts to understand whether nanoparticle properties might explain its toxicity are ongoing. Considering three different types of organo-coated silver nanoparticles (AgNPs): citrate-coated AgNP, polyvinylpyrrolidone-coated AgNP, and branched polyethyleneimine-coated AgNP, with different surface charge scenarios and core particle sizes, herein we systematically evaluate the potential role of particle size and surface charge on the toxicity of the three types of AgNPs against two model organisms, Escherichia coli and Daphnia magna. We find particle size, surface charge, and concentration dependent toxicity of all the three types of AgNPs against both the test organisms. Notably, Ag+ (as added AgNO3) toxicity is greater than each type of AgNPs tested and the toxicity follows the trend: AgNO3>BPEI-AgNP>Citrate-AgNP>PVP-AgNP. Modeling particle properties using the general linear model (GLM), a significant interaction effect of primary particle size and surface charge emerges that can explain empirically-derived acute toxicity with great precision. The model explains 99.9% variation of toxicity in E. coli and 99.8% variation of toxicity in D. magna, revealing satisfactory predictability of the regression models developed to predict the toxicity of the three organo-coated AgNPs. We anticipate that the use of GLM to satisfactorily predict the toxicity based on nanoparticle physico-chemical characteristics could contribute to our understanding of nanotoxicology and underscores the need to consider potential interactions among nanoparticle properties when explaining nanotoxicity.

Synthesis and Enhanced Colloidal Stability of Cationic Gold Nanoparticles using Polyethyleneimine and Carbon Dioxide

Employing green methods in the design and synthesis of functionalized metallic nanoparticles poses significant challenges in terms of maintaining product integrity (size, shape, dispersity, and colloidal stability). In this study, the direct synthesis of cationic gold nanoparticles (GNPs) capped by low molecular weight (Mw ∼ 600) polyethylenimine was investigated. Specifically, three separate HAuCl4 reduction pathways were used to produce robust GNPs with sizes ranging from 4 to 20 nm in diameter with excellent size control. The inclusion of carbon dioxide as a nontoxic, nonflammable, and inexpensive component led to decreases in particle size and an increase in the colloidal stability of the GNPs. Furthermore, the thermally reversible reaction of CO2 with amines provides means to control the solvent pH through carbamate structures and, hence, the controllable formation of particle aggregates. The effects of pH, PEI concentration, and reduction method on the particle core size and stability were determine...

Gold Nanoparticle Delivery of Modified CpG Stimulates Macrophages and Inhibits Tumor Growth for Enhanced Immunotherapy

Gold nanoparticle accumulation in immune cells has commonly been viewed as a side effect for cancer therapeutic delivery; however, this phenomenon can be utilized for developing gold nanoparticle mediated immunotherapy. Here, we conjugated a modified CpG oligodeoxynucleotide immune stimulant to gold nanoparticles using a simple and scalable self-assembled monolayer scheme that enhanced the functionality of CpG in vitro and in vivo. Nanoparticles can attenuate systemic side effects by enhancing CpG delivery passively to innate effector cells. The use of a triethylene glycol (TEG) spacer on top of the traditional poly-thymidine spacer increased CpG macrophage stimulatory effects without sacrificing DNA content on the nanoparticle, which directly correlates to particle uptake. In addition, the immune effects of modified CpG-AuNPs were altered by the core particle size, with smaller 15 nm AuNPs generating maximum immune response. These TEG modified CpG-AuNP complexes induced macrophage and dendritic cell tumor infiltration, significantly inhibited tumor growth, and promoted survival in mice when compared to treatments with free CpG.

CdSe Quantum Dot Growth on Magnetic Nickel Nanoparticles

This paper reports the formation of Ni–CdSe core–shell structures. The Ni–CdSe core–shell structures were synthesized via a seed-mediated growth method where a CdSe shell was grown on preformed nickel core particles. The types of nickel cores include nickel nanocubes of 13 nm in size and spherical nickel nanoparticles of 11 and 45 nm in size. The coating of 37 nm nickel carbide is also included for comparison. The resultant thickness and morphology of the CdSe shell layer varied depending on the size, shape, and composition of the nickel-based core particles used. High-resolution transmission electron microscopy characterization and analysis of the Ni–CdSe interface led to further understanding of the factors governing the growth between these two phases. Optical and magnetic characterizations were also carried out on selected samples. The use of nickel-based cores with different size, shape, and composition allows the thickness and morphology of the CdSe shell layer to be controlled. This led to the changes in the CdSe optical properties.

알킬 아크릴레이트와 관능성 단량체계 다중 Core-Shell 복합입자의 제조

Core 입자로 methyl methacrylate (MMA), ethyl methacrylate (EMA)를 사용하고 shell 입자로 MMA, EMA, 2-hydroxyl ethyl methacrylate (2-HEMA), glycidyl methacrylate (GMA) 및 methacrylic acid (MAA)를 각각 사용하여 개시제 ammonium persulfate (APS), 유화제로 sodium dodecyl benzene sulfonate (SDBS)의 농도(0.01~0.03 wt%), 단량체의 종류와 조성을 변화시켜 수용성 유화중합으로 다중 core-shell복합입자를 제조하여 전환율, 입자경 및 입도분포, 평균분자량, 분자구조, 유리전이온도, 플라즈마 처리후의 접촉각, 등온열분해속도 및 인장강도를 각각 측정하여 다음과 같은 결론을 얻었다. SDBS 농도 0.02 wt%에서 MMA core-(EMA/GMA) shell 복합입자의 전환율이 98.5%로 우수하였고, 입자직경은 SDBS 농도 0.03 wt%에서 EMA core-(MMA/GMA) shell의 복합입자가 <TEX>$0.48{\mu}m$</TEX>로 높게 나왔다. 유리전이온도 측정으로부터 공중합체는 1~2개의 전이곡선 그리고 다중 core-shell 복합입자는 3개의 전이곡선을 얻었다. 전체적으로 접착박리강도의 크기는 shell 단량체가 MMA core인 경우 EMA/MAA > EMA/2-HEMA > EMA/GMA의 순으로 되었다. Multi core-shell composite particles were prepared by the water-born emulsion polymerization of various core monomers such as methyl methacrylate (MMA), ethyl methacrylate (EMA) and shell monomers such as MMA, EMA, 2-hydroxyl ethyl methacrylate (2-HEMA), glycidyl methacrylate (GMA) and methacrylic acid (MAA) in the presence of different concentrations of sodium dodecyl benzene sulfonate (SDBS). The following conclusions are drawn from the conversion, particle size and distribution, average molecular weight, molecular structure, glass transition temperature with DSC, contact angle after plasma treatment, tensile strength and isothermal decomposition kinetics. In the case of the concentration of 0.02 wt% SDBS, the conversion of MMA core-(EMA/GMA) shell composite particles was excellent as 98.5%. In the case of the concentration of 0.03 wt% SDBS, the particle size of EMA core-(MMA/GMA) shell composite particles was high as <TEX>$0.48{\mu}m$</TEX>. We confirmed that 3 points of glass transition temperatures appear for multi core-shell composite particles compared to 1~2 points of glass transition temperatures appear for general copolymer particles. Overall, the adhesion strength of shell composite particles was in the order of EMA/MAA > EMA/2-HEMA > EMA/GMA.

Silver nanoparticle toxicity in the embryonic zebrafish is governed by particle dispersion and ionic environment

The mechanism of action of silver nanoparticles (AgNPs) is unclear due to the particles’ strong tendency to agglomerate. Preventing agglomeration could offer precise control of the physicochemical properties that drive biological response to AgNPs. In an attempt to control agglomeration, we exposed zebrafish embryos to AgNPs of 20 or 110 nm core size, and polypyrrolidone (PVP) or citrate surface coatings in media of varying ionic strength. AgNPs remained unagglomerated in 62.5 μM CaCl2 (CaCl2) and ultrapure water (UP), but not in standard zebrafish embryo medium (EM). Zebrafish embryos developed normally in the low ionic strength environments of CaCl2 and UP. Exposure of embryos to AgNPs suspended in UP and CaCl2 resulted in higher toxicity than suspensions in EM. 20 nm AgNPs were more toxic than 110 nm AgNPs, and the PVP coating was more toxic than the citrate coating at the same particle core size. The silver tissue burden correlated well with observed toxicity but only for those exposures where the AgNPs remained unagglomerated. Our results demonstrate that size- and surface coating-dependent toxicity is a result of AgNPs remaining unagglomerated, and thus a critical-design consideration for experiments to offer meaningful evaluations of AgNP toxicity.

미니에멀젼 중합에 의한 파라핀 왁스의 나노캡슐화 및 상변환물질로서의 열적 특성

폴리스티렌을 쉘 물질로 하여 상변환물질인 파라핀 왁스 코어 물질을 나노캡슐화하는 방법과 형성된 입자들의 열적 특성을 연구하였다. 나노캡슐화의 방법으로는 미니에멀젼 중합법을 채택하였다. 중합시 사용된 가교제의 성질, 유화제의 양, 상변환 물질과 단량체의 비율, 개시제의 친수성 등이 나노캡슐화된 상변환물질의 열적 특성에 미치는 영향을 분석하였다. 오스왈드 숙성에 의한 입자의 크기 변화, 형태 변화는 캡슐 입자의 열적 특성에 가장 큰 영향을 미치는 것으로 나타났으며, 또한 폴리스티렌 쉘의 친수성과 가교밀도도 캡슐 입자의 형태 변화와 이로 인한 열적 특성에 영향을 주는 것으로 나타났다. Encapsulation of a paraffin wax core as a phase change material with polystyrene shell and the its thermal characteristics caused by the encapsulation were studied. For the nanoencapsulation, the miniemulsion polymerization method was selected. The factors affecting the thermal properties of prepared nanocapsule particles of phase change material were analysed in aspect of the structure of crosslinking agents, amounts of surfactant, ratio of paraffin wax to monomer, and hydrophilicity of initiators. It was assumed that Oswald ripening plays the most important role in the changes of particle size, particle morphology, and thermal capacity of nanocapsule core. It was elucidated that the thermal capacity was also dependent on the hydrophilicity and crosslinking density of polystyrene shell components.

Comparison of cytotoxicity and expression of metal regulatory genes in zebrafish (Danio rerio) liver cells exposed to cadmium sulfate, zinc sulfate and quantum dots

Recent advances in the ability to manufacture and manipulate materials at the nanometer scale have led to increased production and use of many types of nanoparticles. Quantum dots (QDs) are small, fluorescent nanoparticles composed of a core of semiconductor material (e.g. cadmium selenide, zinc sulfide) and shells or dopants of other elements. Particle core composition, size, shell, and surface chemistry have all been found to influence toxicity in cells. The aim of this study was to compare the toxicities of ionic cadmium (Cd) and zinc (Zn) and Cd- and Zn-containing QDs in zebrafish liver cells (ZFL). As expected, Cd(2+) was more toxic than Zn(2+), and the general trend of IC50-24 h values of QDs was determined to be CdTe < CdSe/ZnS or InP/ZnS, suggesting that ZnS-shelled CdSe/ZnS QDs were more cytocompatible than bare core CdTe crystals. Smaller QDs showed greater toxicity than larger QDs. Isolated mRNA from these exposures was used to measure the expression of metal response genes including metallothionein (MT), metal response element-binding transcription factor (MTF-1), divalent metal transporter (DMT-1), zrt and irt like protein (ZIP-1) and the zinc transporter, ZnT-1. CdTe exposure induced expression of these genes in a dose dependent manner similar to that of CdSO4 exposure. However, CdSe/ZnS and InP/ZnS altered gene expression of metal homeostasis genes in a manner different from that of the corresponding Cd or Zn salts. This implies that ZnS shells reduce QD toxicity attributed to the release of Cd(2+), but do not eliminate toxic effects caused by the nanoparticles themselves.

Size Distribution and Magnetization Optimization of Single-Core Iron Oxide Nanoparticles by Exploiting Design of Experiment Methodology

The synthesis of single-core superparamagnetic iron oxide nanoparticles (SPIONs) via high temperature decomposition of the self-synthesized Fe(III)-oleate was studied by exploiting factorial design of experiment methodology to investigate the influence of Fe(III)-oleate concentration, reaction temperature and time, and heating rate on the particle core and hydrodynamic size distributions and magnetization. This approach enabled us to establish a reliable and reproducible protocol for the synthesis of monodisperse SPIONs with high magnetic performance. The structural and magnetic properties of SPIONs were characterized utilizing a variety of methods. By applying a multiple linear regression model, a simple and robust empirical growth model was found for the particle hydrodynamic diameter, presenting its dependencies on reaction temperature and time, and Fe(III)-oleate concentration. Having studied the thermal decomposition behavior of Fe(III)-oleate, the synthesis of highly monodisperse particles with a core size of ~ 12-14 nm and suitable magnetic properties was attributed to burst nucleation which is followed by a rapidly terminating growth. In contrast, the particles with a large primary core size of ~ 22-24 nm, crystallized via a gradual and low temperature nucleation accompanied by a slow growth and Ostwald ripening, show a broader or multi-modal size distribution with relatively poor magnetic performance.

Controllability of the Coulomb charging energy in close-packed nanoparticle arrays

We studied the electronic transport properties of metal nanoparticle arrays, particularly focused on the Coulomb charging energy. By comparison, we confirmed that it is more reasonable to estimate the Coulomb charging energy using the activation energy from the temperature-dependent zero-voltage conductance. Based on this, we systematically and comprehensively investigated the parameters that could be used to tune the Coulomb charging energy in nanoparticle arrays. We found that four parameters, including the particle core size, the inter-particle distance, the nearest neighboring number, and the dielectric constant of ligand molecules, could significantly tune the Coulomb charging energy.

Micro-magnetic simulation study on the magnetic particle imaging performance of anisotropic mono-domain particles

The performance of magnetic mono-domain particles is of crucial importance in magnetic particle imaging (MPI). So far, the behavior of mono-domain particles has been modeled within the framework of Langevin theory. This theory predicts the dependence of the MPI signal on the particle core size, but cannot account for the influence of the shape, i.e. the anisotropy of the particle core. In this study we present the first micro-magnetic ab initio simulation of spectra of anisotropic particles with different core diameters in an oscillating magnetic field at 25 and 100 kHz. We find that the MPI signal strongly depends on the anisotropy of the magnetic core. Thus, a difference of 3 nm between the principal axes of a prolate ellipsoid with the volume of a 30 nm sphere can result in a complete loss of the MPI signal. Smaller anisotropies, however, can increase the MPI performance of the particle. The simulations show that the effect of the anisotropy on the MPI signal depends on the frequency of the oscillating magnetic field. At 100 kHz, the optimal signal is found at smaller anisotropies than at 25 kHz. Furthermore, the simulations show that experimental spectroscopic results for Resovist® can only be explained quantitatively by particles with a magnetic core size of at least 25 nm.

Controlling Surface Ligand Density and Core Size of Alkanethiolate-Capped Pd Nanoparticles and Their Effects on Catalysis

This article presents systematic investigations on the relationship between the catalytic property and the surface ligand density/core size of thiolate ligand-capped Pd nanoparticles (PdNPs). The systematic variations in the two-phase synthesis of PdNPs generated from sodium S-dodecylthiosulfate were performed. The resulting PdNPs were characterized by transmission electron microscopy (TEM), thermogravimetric analysis (TGA), and (1)H NMR and UV-vis spectroscopy. The decrease in the molar equivalent of sodium S-dodecylthiosulfate (Bunte salts) resulted in the formation of nanoparticles with lower surface ligand density and larger particle core size. A decrease in the molar equivalent of tetra-n-octylammonium bromide or an increase in reaction temperature generated nanoparticles with higher surface ligand density and smaller particle core size. As the molar equivalent of NaBH(4) decreased, the particle core size increased. The catalysis studies on various PdNPs with different surface ligand density and average core size showed a strong correlation between the PdNP composition and the turnover frequency (TOF) of the isomerization of allyl alcohol. Optimized "good" PdNPs with lower surface ligand coverage and larger core size catalyzed the isomerization of various allyl alcohols to carbonyl analogues with high activity and selectivity.

Characterization of spherical core–shell particles by static light scattering. Estimation of the core- and particle-size distributions

A numerical method is proposed for the characterization of core–shell spherical particles from static light scattering (SLS) measurements. The method is able to estimate the core size distribution (CSD) and the particle size distribution (PSD), through the following two-step procedure: (i) the estimation of the bivariate core–particle size distribution (C–PSD), by solving a linear ill-conditioned inverse problem through a generalized Tikhonov regularization strategy, and (ii) the calculation of the CSD and the PSD from the estimated C–PSD. First, the method was evaluated on the basis of several simulated examples, with polystyrene–poly(methyl methacrylate) core–shell particles of different CSDs and PSDs. Then, two samples of hematite–Yttrium basic carbonate core–shell particles were successfully characterized. In all analyzed examples, acceptable estimates of the PSD and the average diameter of the CSD were obtained. Based on the single-scattering Mie theory, the proposed method is an effective tool for characterizing core–shell colloidal particles larger than their Rayleigh limits without requiring any a-priori assumption on the shapes of the size distributions. Under such conditions, the PSDs can always be adequately estimated, while acceptable CSD estimates are obtained when the core/shell particles exhibit either a high optical contrast, or a moderate optical contrast but with a high ‘average core diameter’/‘average particle diameter’ ratio.

Rapid growth of gas-giant cores by pebble accretion

The observed lifetimes of gaseous protoplanetary discs place strong constraints on gas and ice giant formation in the core accretion scenario. The approximately 10-Earth-mass solid core responsible for the attraction of the gaseous envelope has to form before gas dissipation in the protoplanetary disc is completed within 1-10 million years. Building up the core by collisions between km-sized planetesimals fails to meet this time-scale constraint, especially at wide stellar separations. Nonetheless, gas-giant planets are detected by direct imaging at wide orbital distances. In this paper, we numerically study the growth of cores by the accretion of cm-sized pebbles loosely coupled to the gas. We measure the accretion rate onto seed masses ranging from a large planetesimal to a fully grown 10-Earth-mass core and test different particle sizes. The numerical results are in good agreement with our analytic expressions, indicating the existence of two accretion regimes, one set by the azimuthal and radial particle drift for the lower seed masses and the other, for higher masses, by the velocity at the edge of the Hill sphere. In the former, the optimally accreted particle size increases with core mass, while in the latter the optimal size is centimeters, independent of core mass. We discuss the implications for rapid core growth of gas-giant and ice-giant cores. We conclude that pebble accretion can resolve the long-standing core accretion time-scale conflict. This requires a near-unity dust-to-gas ratio in the midplane, particle growth to mm and cm and the formation of massive planetesimals or low radial pressure support. The core growth time-scale is shortened by a factor 30-1,000 at 5 AU and by a factor 100-10,000 at 50 AU, compared to the gravitationally focused accretion of, respectively, low-scale-height planetesimal fragments or standard km-sized planetesimals.