Simultaneous Block Copolymer and Magnetic Nanoparticle Assembly in Nanocomposite Films

Icosahedral supracrystal assembly from polymer-grafted nanoparticles via interplay of interfacial energy and confinement effect

Self-assembly of nanoparticles (NPs) in drying emulsion droplets paves the way for intricate three-dimensional (3D) superstructures, given the myriad of control parameters for fine-tuning assembly conditions. With their substantial energetic dynamics that are acutely responsive to emulsion confinements, polymeric ligands incorporated into a system can enrich its structural diversity. Here, we demonstrate the assembly of soft polymer-grafted NPs into Mackay icosahedrons beyond spherical body-centered cubic (BCC) packing structures commonly observed for these soft spheres. This behavior is governed by the free energy minimization within emulsions through the interplay of the oil-water interfacial energy and confinement effect as demonstrated by the experimental observations of structural transitions between icosahedrons and BCC crystals and by corresponding free energy calculations. The anisotropic surface of the icosahedral supracrystals provides the capability of guiding the position of a secondary constituent, creating unique hybrid patchy icosahedrons with the potential to develop into multifunctional 3D clusters that combine the benefits of both polymers and conventional colloids.

Effect of shape anisotropy on the precipitation of dimeric nanoparticles

We use grand canonical transition matrix Monte Carlo simulations to study the precipitation of dimeric nanoparticles. The dimers are composed of two particles having different chemical features and separated by a fixed distance. The non-attractive and attractive parts of the dimer are modeled using hard-sphere and square-well potentials, respectively. The shape anisotropy is altered by changing the relative sizes of the two particles. We observe that the stability of the nanosuspension increases with the increase in the size of the non-attractive part of the dimer. The precipitates of dimers having larger non-attractive parts have lower packing densities, contain large cavities, and show evidence of self-assembly in the bulk and on the surface. We also use the results from our simulations and the classical nucleation theory to study the kinetics of precipitation. At a given temperature and relative supersaturation, the rate of homogeneous nucleation increases with the increase in the size of the non-attractive parts. Finally, we use an example to show how our results can guide the design of nanosuspensions containing chemically anisotropic dimers that are stable under particular conditions.

Softness- and Size-Dependent Packing Symmetries of Polymer-Grafted Nanoparticles

Achieving ordered arrays of nanoparticles (NPs) with controlled packing symmetry and interparticle spacing is of great importance to design complex metamaterials. Herein, we report softness- and size-dependent self-assembly behavior of polystyrene-grafted Au NPs (Au@PS NPs). We varied the core size of Au NPs from 1.9 to 9.6 nm and the number-average molecular weight (M_n) of thiol-terminated polystyrene from 1.8 to 7.9 kg mol^-1. The optimal packing model based on an "effective softness" parameter λ_eff that accounts for close-packed and semidilute brush regimes could predict the effective radius of Au@PS NPs (within ±9%) for a wide range of PS M_n, grafting density, and Au core size. With increasing λ_eff, the self-assembled Au@PS NP superlattices undergo a symmetry transition from hexagonal close packed (hcp) to body-centered tetragonal (bct) to body-centered cubic (bcc). This work demonstrates the effective softness model as a simple but robust tool for the design of NP superlattices with precisely controlled interparticle distance and packing symmetry, both of which are critical for the development of sophisticated materials through control of nanoscale structure.

The interfacial structure and dynamics in a polymer nanocomposite containing small attractive nanoparticles: a full atomistic molecular dynamics simulation study

We study the interfacial structure and dynamics of a polymer nanocomposite (PNC) composed of octaaminophenyl polyhedral oligomeric silsesquioxane (OAPS) and poly(2-vinylpyridine) (P2VP) by performing full atomistic molecular dynamics simulations. There are eight aminophenyl groups grafted on the surface of the OAPS particle and the particle has a size comparable to the Kuhn segment of P2VP. These aminophenyl groups can form hydrogen bonds (HBs) with pyridine rings from surrounding P2VP chains. We found that OAPS can form ∼2 HBs on average with surrounding polymer chains. The effect of the HBs is investigated in detail by either switching on or off these HBs in our simulation. By analyzing the interfacial static packing structure and dynamic properties, we demonstrate that the system has an ∼1 nm interface width, similar to the OAPS particle size. We also found that HBs can prevent the further penetration of polymers into the inner zone (grafting layer) of the OAPS, and therefore keep the P2VP chains in the outer layer (>1 nm), remaining bulk-like, which is well consistent with experimental results. In addition, we found that NP diffusion is coupled to the absorbed polymer chains, which also dramatically slows down the diffusion of polymer segments in return. The core-shell model in which the NP and absorbed polymers diffuse as a single object is validated here at the full atomistic level. These results provide atomistic insights into the unique structure and dynamics in the small attractive NP-polymer interfacial region. We hope these results will be helpful for the understanding of peculiar phenomena in attractive polymer nanocomposites containing small NPs.

Segregation of Maghemite Nanoparticles within Symmetric Diblock Copolymer and Triblock Terpolymer Patterns under Solvent Vapor Annealing

Block copolymers (BCPs), through their self-assembly, provide an excellent guiding platform for precise controlled localization of maghemite nanoparticles (MNPs). Diblock copolymers (di/BCP) represent the most applied matrix to host filler components due to their morphological simplicity. A series of nanocomposites based on diblock copolymer or triblock terpolymer matrices and magnetic nanoparticles were prepared to study and compare the influence of an additional block into the BCP matrix. MNPs were grafted with low molecular weight polystyrene (PS) chains in order to be segregated in a specific phase of the matrix to induce selective localization. After the mixing of the BCPs with 10% w/v PS-g-MNPs, nanocomposite thin films were formed by spin coating. Solvent vapor annealing (SVA) enabled the PS-g-MNPs selective placement within the PS domains of the BCPs, as revealed by atomic force microscopy (AFM). The recorded images have proven that high amounts of functionalized MNPs can be controllably localized within the same block (PS), despite the architecture of the BCPs (AB vs. ABC). The adopted lamellar structure of the “neat” BCP thin films was maintained for MNPs loading approximately up to 10% w/v, while, for higher content, the BCP adopted lamellar morphology is partially disrupted, or even disappears for both AB and ABC architectures.

Perspective: Outstanding theoretical questions in polymer-nanoparticle hybrids

This topical review discusses the theoretical progress made in the field of polymer nanocomposites, i.e., hybrid materials created by mixing (typically inorganic) nanoparticles (NPs) with organic polymers. It primarily focuses on the outstanding issues in this field and is structured around five separate topics: (i) the synthesis of functionalized nanoparticles; (ii) their phase behavior when mixed with a homopolymer matrix and their assembly into well-defined superstructures; (iii) the role of processing on the structures realized by these hybrid materials and the role of the mobilities of the different constituents; (iv) the role of external fields (electric, magnetic) in the active assembly of the NPs; and (v) the engineering properties that result and the factors that control them. While the most is known about topic (ii), we believe that significant progress needs to be made in the other four topics before the practical promise offered by these materials can be realized. This review delineates the most pressing issues on these topics and poses specific questions that we believe need to be addressed in the immediate future.

Dispersion of PMMA-grafted, mesoscopic iron-oxide rods in polymer films

This study investigates the parameters that affect the dispersion of polymer grafted mesoscopic iron-oxide rods (FeMRs) in polymer matrices. FeMRs (212 nm long by 36 nm in diameter) are grafted with poly(methyl methacrylate) (PMMA) at three different brush molecular weights: 3.7 kg mol(-1), 32 kg mol(-1), and 160 kg mol(-1). Each FeMR sample was cast in a polymer thin film consisting of either PMMA or poly(ethylene oxide) (PEO) each at a molecular weight much higher or much lower than the brush molecular weight. We find that the FeMRs with 160 kg mol(-1) brush disperse in all matrices while the FeMRs with 32 kg mol(-1) and 3.7 kg mol(-1) brushes aggregate in all matrices. We perform simple free energy calculations, taking into account steric repulsion from the brush and van der Waals attraction between FeMRs. We find that there is a barrier for aggregation for the FeMRs with the largest brush, while there is no barrier for the other FeMRs. Therefore, for these mesoscopic particles, the brush size is the main factor that determines the dispersion state of FeMRs in polymer matrices with athermal or weakly attractive brush-matrix interactions. These studies provide new insight into the mechanisms that affect dispersion in polymer matrices of mesoscopic particles and therefore guide the design of composite films with well-dispersed mesoscopic particles.

Stress–strain behavior of block-copolymers and their nanocomposites filled with uniform or Janus nanoparticles under shear: a molecular dynamics simulation

We adopted molecular dynamics simulation to study the relation between the ordered structures and the resulting mechanical properties of block copolymers filled with uniform or Janus nanoparticles.

Self-Assembled Nanoparticle Arrays on Chemical Nanopatterns Prepared Using Block Copolymer Lithography

We present a high-throughput and inexpensive fabrication approach that uses self-assembled block copolymer (BCP) films as templates to generate dense nanoscale chemical patterns of polymer brushes for the selective immobilization of Au nanoparticles (NPs). A cross-linked random copolymer mat that contains styrene and methyl methacrylate units serves both as a base layer for perpendicular assembly of nanoscale domains of poly(styrene-block-methyl methacrylate) (PS-b-PMMA) films and as a nonadsorbing background layer that surrounds the chemical patterns. The selective removal of the PMMA block and the underlying mat via oxygen plasma etching generates binding sites which are then functionalized with poly(2-vinylpyridine) (P2VP) brushes. Au NPs with a diameter of 13 nm selectively immobilize on the patterned P2VP brushes. An essential aspect in fabricating high quality chemical patterns is the superior behavior of methyl methacrylate containing cross-linked mats in retaining their chemistry during the grafting of P2VP brushes. The use of BCPs with different molecular weights and volume fractions allows for preparation of chemical patterns with different geometries, sizes, and pitches for generating arrays of single particles that hold great promise for applications that range from molecular sensing to optical devices.

Synthesis and characterization of nanostructured PS-b-P4VP/Fe2O3 thin films with magnetic properties prepared by solvent vapor annealing

In this work hybrid organic/inorganic nanocomposites with magnetic properties, based on a PS-b-P4VP block copolymer and Fe₂O₃ maghemite nanoparticles have been prepared.

Influence of magnetic nanoparticle size on the particle dispersion and phase separation in an ABA triblock copolymer

Oleic acid modified iron oxide nanoparticles (IONs) with different sizes were synthesized and mixed with styrene-butadiene-styrene block copolymer (SBS) with a lamellar structure. The octadecene segments on the oleic acid molecules have chemical affinity with the polybutadiene (PB) blocks, which makes IONs tend to be selectively confined in the microphase-separated PB domains. However, the dispersion state strongly depends on the ratio of the particle diameter (d) to the lamellar thickness (l) of the PB domains, which further changes the phase separation of SBS. When d/l ∼0.5, most of IONs are concentrated in the middle of the PB layers at low particle loading. Upon increasing the particle loading, part of IONs contact each other to form long strings due to their strong magnetic interactions. Away from the strings, IONs are either selectively dispersed in the middle and at the interfaces of the PB domains, or randomly distributed at some regions in which the phase separation of SBS is suppressed. The phase separation of SBS transforms from the lamellar structure to a cylinder structure when the IONs loading is higher than 16.7 wt %. As d is comparable to l, IONs aggregate to form clusters of 100 to 300 nm in size, but within the clusters IONs are still selectively dispersed in the PB domains instead of forming macroscopic phase separation. It is interpreted in terms of the relatively small conformational entropy of the middle blocks of SBS; thus, incorporation of nanoparticles does not lead to much loss of conformational entropy. Although incorporation of IONs with d/l ∼1 significantly increases the interfacial curvature and roughness, it has less influence on the phase separation structure of SBS due to the inhomogeneous dispersion. When d is larger than l, IONs are macroscopically separated from the SBS matrix to form clusters of hundreds of nanometers to several micrometers. More interestingly, the phase separation of SBS transforms from the lamellar structure to a two-phase co-continuous structure, probably due to the rearrangement of SBS molecules to cover the clusters with PB segments and the strong magnetic interaction exerting additional force on the SBS matrix during the evaporation of the solvent and the subsequent thermal annealing process.

Thermal annealing as an easy tool for the controlled arrangement of gold nanoparticles in block-copolymer thin films

Thermal annealing was used for the bottom-up fabrication of morphologically controlled gold-block-copolymer (Au-BC) nanocomposites. Three different blends formed by polystyrene (PS) homopolymer and PS-coated gold nanoparticles (PSSH@Au) were used as modifiers of asymmetric polystyrene-b-polymethylmethacrylate (PS-b-PMMA): PS26/PS26SH@Au, PS75/PS75SH@Au and PS167/PS167SH@Au (where the subscripts refer to the number of styrene monomeric units).The results indicated that all three blends used as modifiers (PSn/PSnSH@Au) were successfully located in the PS phase during thermally induced BC self-assembly for a composition range from 5 to 43 wt% without macro-phase separation. The PSnSH@Au moiety experienced molecular desorption, nanocrystal core coalescence and partial molecular re-encapsulation processes during thermal annealing, leading to sphere-like gold NPs with a larger average size (without exceeding an interdomain distance). Ligand chain length regulated the degree of coalescence and re-encapsulation, defining ultimate core size. Furthermore, proper combination of chain length and composition enabled tuning of NP partitioning and arrangement on different length scales through thermally activated cooperative assembly processes. These results have not only significant impact for establishing thermal processing as a useful tool for the precise control of NP size and distribution, but also much broader implications for many nanoparticle-based technologies.

Three-Dimensional Periodically Ordered Nanohetero Metallic Materials from Self-Assembled Block Copolymer Composites

We report a synthetic route to inorganic nanoheterostructures with tunable size and morphology via self-assembled block copolymer mesophase templates. Two Fe precursors, tricarbonyl(cyclooctatetraene) iron and acetylacetonate iron(III), and one Pt precursor, platinum dimethylcyclooctadiene, were selectively introduced into separate polymer blocks of a block copolymer, polystyrene-b-poly-4-vinylpyridine, and then the block copolymer templates were removed by pyrolysis. Self-assembled inorganic nanoheterostructures (spheres, hexagonal cylinders, and layers of hard magnetic phase FePt in a matrix of soft magnetic phase α-Fe) were produced. The morphology and domain size of the nanoheterostructures could be tailored by controlling the molecular weight and relative block lengths of the block copolymers. The controlled size and morphology of the inorganic nanoheterostructures demonstrate the method's utility for producing highly functional materials.

Directed polystyrene/poly(methyl methacrylate) phase separation and nanoparticle ordering on transparent chemically patterned substrates

We investigate the surface-directed phase separation of spin-coated polystyrene/poly(methyl methacrylate) (PS/PMMA) blends on prepatterned octadecyltrichlorosilane (OTS)-glass substrates under various experimental conditions. As a result of tandem processes of spinodal decomposition and selective wetting of polymer components during spin-coating, low-energy OTS stripes and high-energy glass surfaces laterally arrange the phase-separated polymers according to the chemical pattern on the substrate. Optimal pattern replication was achieved when the length scale of phase separation, controlled via the polymer concentration of the spin-coating solution, matched the smallest feature dimension in a striped chemical pattern possessing two alternating distances between stripes. It was also shown that polymer blend patterns were most closely registered with the underlying substrate when the PS/PMMA composition ratio (30/70, w/w) matched the areal fraction of OTS on the glass surface (∼30%). The influence of solvents demonstrated that a solvent with a relatively low volatility, such toluene, was required for patterning so that domain feature sizes were able to coarsen to the size of the patterned features before film vitrification. As well, we showed that the technique and optimized conditions developed in this study could be applied to pattern photoluminescent CdS quantum dots into microscale arrays of parallel lines via spin-coating onto transparent OTS-glass substrates.

Structure of nanoparticles embedded in micellar polycrystals

We investigate by scattering techniques the structure of water-based soft composite materials comprising a crystal made of Pluronic block-copolymer micelles arranged in a face-centered cubic lattice and a small amount (at most 2% by volume) of silica nanoparticles, of size comparable to that of the micelles. The copolymer is thermosensitive: it is hydrophilic and fully dissolved in water at low temperature (T ~ 0 °C), and self-assembles into micelles at room temperature, where the block-copolymer is amphiphilic. We use contrast matching small-angle neuron scattering experiments to independently probe the structure of the nanoparticles and that of the polymer. We find that the nanoparticles do not perturb the crystalline order. In addition, a structure peak is measured for the silica nanoparticles dispersed in the polycrystalline samples. This implies that the samples are spatially heterogeneous and comprise, without macroscopic phase separation, silica-poor and silica-rich regions. We show that the nanoparticle concentration in the silica-rich regions is about 10-fold the average concentration. These regions are grain boundaries between crystallites, where nanoparticles concentrate, as shown by static light scattering and by light microscopy imaging of the samples. We show that the temperature rate at which the sample is prepared strongly influence the segregation of the nanoparticles in the grain-boundaries.

Hybridization of inorganic nanoparticles and polymers to create regular and reversible self-assembly architectures

Inorganic nanoparticles (NPs) with diversified functionalities are promising candidates in future optoelectronic and biomedical applications, which greatly depend on the capability to arrange NPs into higher-order architectures in a controllable way. This issue is considered to be solved by means of self-assembly. NPs can participate in self-assembly in different manners, such as smart self-organization with blended molecules, as the carriers of host molecules for assembly and disassembly with guest molecules, as netpoints to endow the architectures specific functionalities, and so forth. To enhance the structural stability of the as-prepared assembly architectures, polymers have been utilized to create NP-polymer composites. Meanwhile, such a strategy also demonstrates the possibility of integrating the functionalities of NPs and/or polymers by forming regular architectures. The emerging interest in the current optoelectronic and biological areas strongly demands intelligent nanocomposites, which are produced by combination of the excellent functionalities of NPs and the responsiveness of polymers. On the basis of the recent progress in fabricating NP-polymer composites, this critical review summarizes the development of new methods for fabricating regular self-assembly architectures, highlights the reversible assembly and disassembly behavior, and indicates the potential applications.

Gold nanorods dispersed in homopolymer films: optical properties controlled by self-assembly and percolation of nanorods

In this paper, polymer nanocomposite films containing gold nanorods (AuNRs) and poly(2-vinyl pyridine) (P2VP) have been investigated for their structure-optical property relationship. Using transmission electron microscopy (TEM), the assembly of AuNRs (7.9 nm × 28.4 nm) grafted with a P2VP brush in P2VP films is examined as a function of the AuNR volume fraction Ø(AuNRs) and film thickness h. For h ∼ 40 nm, AuNRs are confined to align parallel to the film and uniformly dispersed at low Ø(AuNRs). Upon increasing Ø(AuNRs), nanorods form discrete aggregates containing mainly side-by-side arrays due to depletion-attraction forces. For Ø(AuNRs) = 2.7%, AuNRs assemble into a 2D network where the discrete aggregates are connected by end-to-end linked nanorods. As Ø(AuNRs) further increases, the polymer-rich regions of the network fill in with nanorods and rod overlap is observed. Monte Carlo simulations capture the experimentally observed morphologies. The effect of film thickness is investigated at Ø(AuNRs) = 2.7%, where thicker films (40 and 70 nm) show a dense array of percolated nanorods and thinner films (20 nm) exhibit mainly isolated nanorods. Using Rutherford backscattering spectrometry (RBS), the AuNRs are observed to segregate near the substrate during spin-casting. Optically, the longitudinal surface plasmon resonance (LSPR) peaks are correlated with the local orientation of the AuNRs, where side-by-side and end-to-end alignments induce blue and red shifts, respectively. The LSPR undergoes a red shift up to 51 nm as Ø(AuNRs) increases from 1.6 to 2.7%. These studies indicate that the optical properties of polymer nanocomposite films containing gold nanorods can be fine-tuned by changing Ø(AuNRs) and h. These results are broadly applicable and provide guidelines for dispersing other functional nanoparticles, such as quantum dots and carbon nanotubes.

Hexagonal-to-cubic phase transformation in composite thin films induced by FePt nanoparticles located at PS/PEO interfaces

The organization process of asymmetric poly(styrene-block-ethylene oxide) (PS-b-PEO) copolymer thin films blended with FePt nanoparticles is studied. In a first step, it is shown that FePt nanoparticles stabilized by oleic acid ligands are distributed within the PS matrix phase, whereas the same particles partially covered with short dopamine-terminated-methoxy poly(ethylene oxide) (mPEO-Dopa) are located at PS/PEO interfaces. The swelling of PS domains, induced by FePt_oleic acid nanoparticles during the solvent annealing process, results in formation of a disordered microstructure in comparison to the well-organized hexagonally close-packed (HCP) cylinder phase formed in the neat PS-b-PEO copolymer. The evolution of the microstructure of PS-b-PEO/FePt_mPEO-Dopa composite has been investigated for different solvent annealing treatments. Under high-humidity conditions during the vapor annealing process, the addition of FePt nanoparticles results in formation of spheres in the film split into terraces. The upper and lower terraces are occupied by spheres organized in an unusual square and HCP phases, respectively. Under low-humidity conditions, undulated PEO cylinders oriented parallel to substrate are formed in the presence of FePt nanoparticles. In this case, we observe that most of the nanoparticles accumulate within the core of topological defects, which induces a low nanoparticle concentration at the PS/PEO interfaces and so stabilizes an intermediate undulated cylinder phase.

Morphology of thin nanocomposite films of asymmetric diblock copolymer and magnetite nanoparticles

Thin self-assembled nanocomposite films of an asymmetric diblock copolymer and nanoparticles are fabricated. The morphologies of the films of the diblock copolymer poly(styrene-block-n-butyl methacrylate), P(Sd-b-BMA), with different volume fractions of large magnetite Fe(3)O(4) nanoparticles are studied before and after annealing. Neutron reflectometry reveals remarkable evidence that confining asymmetric copolymer to a limit of two layers forces the film, after the annealing, to form a mixed cylindrical-lamellar two-layer structure. This complex morphology is very stable and is preserved after the incorporation of nanoparticles up to 10% volume fraction. The other striking result is that the monodispersed nanoparticles with affinity to the polystyrene (PS) domain and with a size of 10 nm, which is close to the size of the PS chains, are assembled by the diblock copolymer matrix, so the distribution of the nanoparticles is reduced solely to the PS domain of the film. Our studies demonstrate that for asymmetric block copolymers in thin film geometry the self-assembly is strongly influenced by the interfacial and surface energies of the blocks and substrate.

Patterning block copolymer aggregates via Langmuir-Blodgett transfer to microcontact-printed substrates

We demonstrate a new strategy for producing hierarchical polymer nanostructures, which combines nanoscale self-assembly of amphiphilic block copolymers at the air-water interface with microscale templated assembly of the resulting aggregates on chemically patterned substrates. Aggregates are formed via interfacial self-assembly of 141k polystyrene-b-poly(ethylene oxide) (PS-b-PEO, MW = 141k, 11.4 wt % PEO) or a blend of 185k PS-b-PEO (MW = 185k, 18.9 wt % PEO) and PS-coated CdS nanoparticles to form strandlike copolymer or copolymer-nanoparticle aggregates. Using Langmuir-Blodgett (LB) techniques, the aggregates are then transferred to patterned substrates possessing alternating hydrophilic/hydrophobic stripes, obtained by microcontact printing octadecyltrichlorosilane (OTS) on glass. The aggregates are transferred under various conditions of surface pressure, orientation of the patterned substrate, and withdrawal speed. Templated assembly of aggregates into the hydrophilic substrate domains is achieved when the hydrophilic/hydrophobic stripes are oriented perpendicular to the water surface during LB transfer; this is explained by surface energy heterogeneities along the subphase-substrate contact line, which induce selective dewetting and concomitant monolayer rearrangement at the drying front. In contrast, parallel orientation of stripes results in nonselective transfer of the monolayer without registration to the underlying surface pattern. By studying the effect of surface pressure, we show that packing constraints imposed by compression of aggregates to high surface densities prevent the formation of patterned LB films that match the established periodicity of the OTS-patterned glass. As well, it is shown that efficient transfer of aggregates to the patterned glass requires slower substrate withdrawal speeds compared to transfer to unpatterned hydrophilic glass.

End-Grafted Polymer Chains onto Inorganic Nano-Objects

Organic/inorganic nanohybrid materials have attracted particular scientific and technological interest because they combine the properties of the organic and the inorganic component. Inorganic nanoparticles exhibit interesting electrical, optical, magnetic and/or catalytic properties, which are related with their nano-scale dimensions. However, their high surface-to-volume ratio often induces agglomeration and leads to the loss of their attractive properties. Surface modification of the inorganic nano-objects with physically or chemically end-tethered polymer chains has been employed to overcome this problem. Covalent tethered polymer chains are realized by three different approaches: the “grafting to”, the “grafting from” and the “grafting through” method. This article reviews the synthesis of end-grafted polymer chains onto inorganic nanoparticles using “controlled/living” polymerization techniques, which allow control over the polymer characteristics and the grafting density of the end-tethered polymer chains.