Spontaneous Phase Separation in Polymer Solution by Janus Nanoparticle Compatibilizers: An Atomistic Molecular Dynamics Study

Abstract

We perform atomistic molecular dynamics simulations in order to investigate the structural and dynamic properties of Janus gold nanoparticles within diverse solvent environments, specifically pure polyethylene, pure water, and mixed polyethylene/water matrices. The nanoparticles, approximately 7 nm in diameter and almost totally surrounded by (211) surfaces at the thermodynamic limit, are functionalized with hydrophobic polyethylene chains on one hemisphere and hydrophilic poly­(ethylene oxide) chains on the other. In mixed systems, the Janus nanoparticle acts as an effective stabilizer, directing and stabilizing phase separation in the solvent, where water and polyethylene concentrate around their respective preferred functionalized hemispheres. The resulting interface between the two matrix components was measured at approximately 0.82 nm in thickness. The study characterizes these systems through density profiles, radii of gyration, and second-rank bond order parameters. Regarding the local dynamics of the polymer matrix, the simulations reveal that while polyethylene chains maintain a random orientation in the bulk, they adopt a predominantly parallel orientation relative to the nanoparticle surface when within a distance of 3.2–3.9 nm. Additionally, while the radius of gyration of matrix chains remains stable across systems, the relative shape anisotropy increases significantly in the mixed environments. These molecular-level insights into structural and dynamic organization underscore the potential of gold Janus nanoparticles to control interfacial behavior and stabilize immiscible blends in nanotechnology and biomedicine.

Introduction

Nanoparticles (NPs) are material particles ranging from few to few tens of nanometers in size. Due to their small size, they exhibit unique physical and chemical properties compared to their bulk counterparts, and are therefore highly versatile and useful in various fields. In medicine, nanoparticles are used for targeted drug delivery and imaging. In engineering, they enhance materials’ strength and durability. They also play a crucial role in catalysis, environmental remediation, and electronics. Their ability to interact at the molecular level opens up numerous innovative applications, making them a key component in advancing technology and science.

Nanoparticles can be classified in many ways based on their shape, size, and chemical characteristics: they can be metallic, ceramic, lipid-based, semiconductor, or polymeric. Metal type NPs can be made of several precursors such as Cu, Ag, Zn, or Au, and their facet-, size-, and shape-controlled syntheses are essential in modern materials. , Especially, gold NPs (AuNPs) have nowadays been the subject of extended research due to their optical, electrical, and molecular recognition capabilities with a lot of innovative uses in fields including electron microscopy, electronics, nanotechnology, biomedicine, and materials science because of their unique physical and chemical properties as well as their ability to absorb and scatter light in the visible and near-infrared spectrum range. Moreover, AuNPs are considered ideal for nanomedicine applications due to their unique biocompatibility and stability. Regarding their formation, nanoparticles are classified into two major categories: (a) NPs that are formed themselves by immiscible materials and (b) coated NPs with grafted chains of various kinds on their surface, with various synthesis methods.

The name Janus nanoparticles (Janus NPs) comes from the ancient Roman god Janus, who is usually depicted having two faces, because these nanoparticles consist of two or more different regions having various chemical and physical properties, for example, hydrophilic and hydrophobic regions. Janus particles integrate the colloidal characteristics of conventional particle stabilizerssuch as high desorption energy at fluid interfaceswith the molecular attributes of surfactants, namely amphiphilicity and the ability to lower interfacial tension. For instance, Glaser et al. demonstrated that uniform spherical nanoparticles (7 nm iron or 10 nm gold) decrease the hexane/water interfacial tension from 48 mN/m to approximately 33 mN/m. In contrast, comparably sized gold–iron Janus particles reduce the interfacial tension more significantly, to about 22.5 mN/m. This amphiphilic nature enables Janus NPs to engage with materials possessing dissimilar physicochemical properties; thus, they are attractive materials for a lot of applications: − biological sensors, drug delivery, − optical sensing devices, − nano micromotors, − electronic displays, − two-phase stabilizers ,, and controllable stabilized oil/water emulsion separation. Various methods have been developed for the synthesis of Janus nanoparticles as well as for conventional ones. −

Especially, Au Janus NPs (AuJNPs), which possess one hydrophilic and one hydrophobic side, have been studied with great attention due to their potential advantages in biomedical applications. On the other hand, integrating both morphological and surface anisotropy can further promote the interfacial adsorption of Janus particles, thereby amplifying their surface activities. For example, Ruhland et al. fabricated Janus particles with varied geometries using block terpolymers and found that spherical Janus particles (50 nm diameter) decreased the toluene/water interfacial tension from 36 mN/m to approximately 18 mN/m. In comparison, cylindrical Janus particles (23 nm in diameter and 2300 nm in length) achieved an even greater reduction, lowering the interfacial tension to around 14 mN/m.

Various theoretical studies examine the behavior of Janus NPs as compatibilizers for polymer blends. For example, Estridge and Jayaraman, using coarse-grained molecular simulations, showed that homopolymer-grafted Janus NPs have the largest desorption energy if they are used as compatibilizers inside polymer blends compared to diblock copolymer-grafted NPs or diblock copolymers, due to their deeper penetration of the grafted beads into the two domains of the blend. Guo et al. studied the effect of Janus nanospheres on the phase separation dynamics of binary immiscible polymer blends and took homogeneous nanospheres as a reference for comparison, by using dissipative particle dynamics (DPD) simulations. They concluded that Janus nanospheres can be used as a more effective emulsifying or stabilizing agent than homogeneous nanospheres for immiscible polymer blends.

A lot of experimental studies have also been performed on the use of Janus NPs as compatibilizers as well as for efficient demulsification of oil/water emulsions: Virgilio and Favis investigated the effect of Janus nanoparticle affinity and found that particles composed of polystyrene and poly­(methyl methacrylate) spherical caps are interfacially active, localizing at the high-density polyethylene/polypropylene interface. On the other hand, demulsification of the crude oil/water emulsions is a problem for the petroleum industry, and Janus NPs formed by two faces with different wettabilities have been excellent candidates to solve the issue. Hong et al. primarily prepared the amphiphilic SiO₂-based Janus NPs (Janus-C18), and Jia et al. used them to enhance oil recovery from oil-in-water emulsions. They found that the hydrophilic–lipophilic balance of the modified Janus-SiO₂ should be the dominant factor to achieve the highest demulsification efficiency. Furthermore, the demulsification mechanism of Janus-C18 was attributed to both the hydrophile–lipophile transition and electrostatic interaction. The amphiphilic Janus-C18 could spontaneously enter the oil/water interface to disturb the initial emulsifier arrangement, and its high positive charges further neutralize the charges of the original emulsifier to enhance the demulsification efficiency.

A main challenge in the field of Janus-NP-based polymer nanocomposites lies in providing detailed information on the behavior of such systems at the atomic/molecular scale, focusing on the (distinct) polymer/nanoparticle interfaces. In this work, we will examine a Janus AuNP (AuJNP) half-coated by hydrophobic polyethylene (PE) and hydrophilic poly­(ethylene oxide) (PEO) chains. We provide a detailed investigation of the structural and dynamical behavior of polymer/water solutions with Janus Au nanoparticles as compared to reference polymer and water systems. Our simulation domain contains a single Janus Au nanoparticle (AuJNP) in a mixed matrix of polyethylene chains (PEs) and water molecules (H₂O) as a model of well-dispersed AuJNPs. For comparison reasons, simulations were also performed on bulk systems that do not include any AuJNPs but PE chains only.

Models and Simulation Methodology

Construction of Model Systems

The Au Nanoparticle (AuNP)

The construction of the Au nanoparticle (AuNP) was based on Wulff’s theorem, which connects the shape of a particle with its Gibbs free energy G, given by

$$G=G_{\mathit{bulk}}+\sum _{\mathit{hkl}}{\gamma }_{\mathit{hkl}}{A}_{\mathit{hkl}}$$ 1

where G _bulk is the free energy of the bulk material, γ _hkl is the energy required for the creation of a unit area parallel to the (hkl) plane of the crystal (surface tension), and A _hkl is the total area parallel to the respective plane. Specifically, since G _bulk is constant, minimizing the second term of eq under thermodynamic equilibrium will give a certain shape for a certain quantity of matter. G. Wulff suggested that the shape that minimizes eq is such that the distance d _hkl of each face from the center of the nanoparticle is proportional to the surface tension γ _hkl of the respective (hkl) surface according to

$$d_{\mathit{hkl}}\sim {\gamma }_{\mathit{hkl}}$$ 2

therefore the lower the total surface tension, the shorter the distance between the center of mass of the nanoparticle and its various surfaces.

The Au nanoparticle (AuNP) (Figure (a)) has been modeled by a code developed in our team, which is based on the atomistic Wulff construction and has been used in previous works to model Au nanoparticles with adsorbed ligands. , This code uses the equations of planes that enclose the nanoparticle and takes advantage of the cubic symmetry of the fcc crystal structure of Au, working only with atoms of the first octant in order to reduce the execution time. It consists of 6860 Au atoms, and its diameter is ∼7 nm. Its shape is very close to a deltoidal icositetrahedron and has been chosen because it is almost totally surrounded by (211) surfaces at the thermodynamic limit. This is the equilibrium shape of AuNP when thiols are attached at their surfaces, as discussed in refs , . ,

1.

Janus Au nanoparticle (AuJNP) (a) is modeled by attaching hydrophilic PEO (b) and hydrophobic PE (c) molecules on the two opposite sides of a Au nanoparticle.

The Janus Au Nanoparticle (AuJNP)

We employ poly­(ethylene oxide) (PEO) as a model hydrophilic polymer (Figure (b)) and polyethylene (PE) as a model hydrophobic polymer (Figure (c)). In the present study, grafted PE is modeled by −S­(CH₂)₁₉CH₃ (eicosanethiolate), i.e., a chain of 20 monomers terminated by a S atom on one side. The S atom is then connected to the Au atoms of the nanoparticle. Grafted PEO is modeled by −S­(CH₂CH₂O)₆H, which has similar length and molecular weight to the molecule that is used to model PE. Using a plane of symmetry of the nanoparticle, we split its surface into two equal parts. In each part, 108 PE or PEO model molecules are attached, creating a hydrophobic and hydrophilic side, respectively. The above configuration gives a grafting density of ∼1.4 chains/nm². In this study, we consider the x-y plane as the symmetry plane with hydrophilic molecules on the negative x-axis and hydrophobic molecules on the positive x-axis (Figure ).

The initial state of our AuJNP system is demonstrated in Figure . The chains were grafted at bridge site positions between two Au atoms over the steps of (211) surfaces of the nanoparticle, according to Barmparis et al., as shown in Figure (right). Performing DFT calculations, they showed that for Au surfaces, grafting methanethiols on bridge sites is more stable than on top or hollow sites for Au(211) surfaces. The same behavior is also observed in the other crystallographic directions. The ligands between the AuNP surface and chains were chosen to be S atoms, as it has been proven that they create a very strong Au–S bond, which is useful to our simulation. Several typical PEO chains are demonstrated in Figure (right).

2.

(a) Janus Au nanoparticle in the initial state formed by grafting PE and PEO chains. The poly­(ethylene oxides) are grafted at bridge positions on the left side (x < 0) of the nanoparticle, while the polyethylenes are grafted on the right side (x > 0) at similar positions. (b) PEO chains grafted at bridge site positions over a step of a (211) surface of the nanoparticle.

The Matrices

In this work, three types of matrixes are considered for our simulations: One matrix that contains PE molecules modeled in the present study by CH₃(CH₂)₂₀CH₃, a second matrix that consists of water molecules (H₂O), and a third one that contains both PE and water molecules. Their details are demonstrated in Table .

1. Systems Configuration.

		AuJNP			matrix
system ID	description	Au atoms	–S(CH2)19CH3	–S[(CH2)2O]6H	CH3(CH2)20CH3	H2O
PE	Bulk PE matrix				5040
PE_W	Bulk PE and water matrix				3730	64,239
AuJNP_PE	Janus AuNP in PE matrix	6860	108	108	5040
AuJNP_W	Janus AuNP in H2O matrix	6860	108	108		127,526
AuJNP_PE_W	Janus AuNP in PE and H2O matrix	6860	108	108	3730	64,239

The initial simulation box sizes are selected to be 16 × 16 × 16 nm³ for all examined systems. The boxes were chosen to be a little more than twice the nanoparticle diameter (∼7 nm), ensuring a minimum nanoparticle surface-to-periodic-image distance of ∼8.5–10 nm that is several times larger than the size of the polymer chains and the radius of the AuJNPs. At the same time, the density profiles of polymer and water molecules, shown in Figures , , and , reach the corresponding bulk density values well before the box boundaries in all systems. Based on that, we do not expect system size effects in the simulated systems. Moreover, the corresponding NP volume fraction is approximately less than 5%, which falls within the range of experimentally studied polymer nanocomposites. The final sizes of the boxes are obtained after the NPT simulation processes.

5.

Average density distribution of the matrix (PE), PE- and PEO-grafted chains for the AuJNP_PE system, at various distances from the center of mass of the nanoparticle.

7.

Average density distribution of the matrix (H₂O), the PE- and PEO-grafted chains for the AuJNP_W system, at various distances from the center of mass of the nanoparticle.

9.

Average density distribution of the matrix that contains both PEs and (H₂O), the PE- and PEO-grafted chains for the AuJNP_PE_W system, at various distances from the center of mass of the nanoparticle.

Interface layer between PE and water in the final configuration of the AuJNP system treated as Gibbs surface. Notice that the interface layer is located at x = −1.75 nm and its thickness is 0.82 nm. The PE density starts at 0.00 and ends at 0.72 gr/cm³.

The Complete Systems

We consider five different systems: two bulk polymers and three AuJNP/polymer nanocomposites. The key parameters of these systems are given in Table . Bulk systems include pure PE and a PE/water mixture. Nanocomposites include a AuJNP fixed at the center of the simulation box, surrounded by PE or H₂O or a PE/H₂O mixture. In the case of the PE/H₂O mixture, these two matrix components were initialized in a fully mixed configuration with no predefined interface. The observed phase separation and the resulting interface were formed spontaneously during the simulation. The last five columns of Table contain the number of atoms and chains for each component of the specified system.

Atomistic Model and Simulations

Potential Parameters

All gold atoms comprising the NP were maintained at their initial positions throughout the simulation by applying the corresponding positional constraint parameter in the simulation software. The potential parameters for the intramolecular interactions were chosen to be those of Ghorai et al. and TRaPPE (trappe.oit.umn.edu). The ligand sulfur atoms were not fixed to the gold surface during the simulation; they were also free to move. For the Au–S interactions, we used the Morse formula

$$V_{\mathit{Morse}}(r)=D{(1-e^{-b(r-\beta )})}^2$$ 3

as this type of potential describes better the interaction between the two atoms.

The nonbonding interactions of S–CH _x (x = 2,3), CH _x –CH _y (x, y = 2,3), and Au–CH _x (x = 2,3) were described by the typical equation of 12–6 Lennard-Jones potential

$$V_{\mathit{LJ}}(r)=4{ϵ}_{\mathit{ij}}({\left(\frac{{\sigma }_{\mathit{ij}}}{r_{\mathit{ij}}}\right)}^{12}-{\left(\frac{{\sigma }_{\mathit{ij}}}{r_{\mathit{ij}}}\right)}^6)$$ 4

where the estimation of the proper values for ϵ _ij and σ _ij was based on Lorentz–Berthelot rules ( ${\sigma }_{\mathit{ij}}=\frac{1}{2}({\sigma }_{\mathit{ii}}+{\sigma }_{\mathit{jj}})$ and ${ϵ}_{\mathit{ij}}=\sqrt{{ϵ}_{\mathit{ii}}{ϵ}_{\mathit{jj}}}$ ).

Furthermore, we used the potential of the Au–S–CH₂–CH₂ dihedral angle over a (211) Au surface that was calculated in a previous work, as coefficients of a 11th-grade polynomial. For the H₂O molecules in the matrix, we used the well-known SPC/E model.

The molecular dynamics simulations (MD) were performed using the GROMACS software (version 2023.1). We used a cubic box with periodic boundary conditions in all three directions. We also used: grid as the neighbor searching type and the Verlet cutoff scheme, the Nosé-Hoover thermostat , at 350 K temperature, the Parinello-Rahman barostat , with isotropic behavior, at a reference pressure level of 1 Atm (1.0.1325 bar), and the compressibility at a value of 2.5 × 10^–5 bar^–1. For the electrostatic interactions, we used the Particle-mesh Ewald (PME) method. All the potential parameters are demonstrated in Table . The time step was 1 fs (10^–6 ns) for 2 × 10⁸ steps, while the coordinates and velocities of particles were written in the trajectory file with a 10000 calculation steps interval. For each system, after the initial energy minimization, we performed NVT and then NPT simulation processes until full relaxation. The initial velocities were stochastically produced.

2. Interaction Parameters of the Molecular Models Used in Simulations.

type of interaction and potential function	interacting parameters
	Bond type: Stretching (Morse potential)			V stretch (r) = D(1–e –b(r–β))2
Bonds	β (nm)	$D\left(\frac{\mathrm{kJ}}{\mathrm{mol}}\right)$	b (nm–1)
Au–S	0.252	36.664392	0.147
	Bond type: Stretching (Harmonic Oscillator potential)			$V_{\mathrm{stretch}}(r)=\frac{1}{2}{k}_s{(r-r_0)}^2$
	r 0 (nm)	$k_s(\frac{\mathrm{kJ}}{\mathrm{mol}}\times {\mathrm{nm}}^{-2})$
S-CH2	0.181	92884.80
CH2–CH x (x = 2,3)	0.154	108784.00
CH2–O	0.141	267900.00
O–H	0.0945	267900.00
Au–Au (NP)	frozen
	Bond type: Bending			$V_{\mathrm{bend}}(\theta )=\frac{1}{2}{k}_b{(\theta -{\theta }_0)}^2$
Bonds	θ0 (deg)	$k_b(\frac{\mathrm{kJ}}{\mathrm{mol}}\times {\mathrm{rad}}^{-2})$
Au–S–CH2	110.1	261.5
S–CH2–CH2	114.4	263.6
CH2–CH2–CH x (x = 2,3)	114.0	519.611
CH2–CH2–O	112.0	418.218
CH2–O–CH2	112.0	502.194
O–CH2–CH x (x = 2,3)	112.0	418.218
CH2–O–H	112.0	418.218
Au–Au–Au (NP)	frozen
	Bond type: Dihedrals (Ryckaert-Bellemans function)			$V_{\mathrm{tor}}(\varphi )=\sum _{i=0}^5{\alpha }_i\cos (\varphi )$ ,	${\alpha }_i\left(\frac{\mathrm{kJ}}{\mathrm{mol}}\right)$
Bonds	α0	α1	α2	α3	α4	α5
S–CH2–CH2– CH2	9.2759	12.1545	–13.1168	–3.0585	26.2378	–31.4929
CH2–CH2–CH2 −CH x (x = 2,3)	9.2759	12.1545	–13.1168	–3.0585	26.2378	–31.4929
S–CH2–CH2–O	9.2759	12.1545	–13.1168	–3.0585	26.2378	–31.4929
CH2–CH2–O–CH2	1.60941	19.79231	–7.82474	–15.72474	6.43215	–4.5435
CH2 – O – CH2 – CH2	1.60941	19.79231	–7.82474	–15.72474	6.43215	–4.5435
O–CH2–CH2–O	2.22267	17.03651	8.29835	–31.2451	5.13025	–1.91522
CH2–CH2–O–H	2.822	2.943	0.485	–6.2501	0.0	0.0
	Tabular potential for Au – S – CH2 – CH2 dihedral angle over the (211) surface				$V_{\mathrm{tor}}(\varphi )=\sum _{i=0}^{11}{\alpha }_i{\varphi }^i$
Polynomial coefficients
a 0	1.938784 × 101
a 1	–1.316188 × 10–1
a 2	–3.480334 × 10–3
a 3	–4.817540 × 10–6
a 4	5.119324 × 10–7
a 5	1.715997 × 10–9
a 6	–3.462513 × 10–11
a 7	–7.556656 × 10–14
a 8	1.006355 × 10–15
a 9	9.766564 × 10–19
a 10	–1.045875 × 10–20
a 11	–5.492738 × 10–25
	Nonbonded interactions			$V_{\mathrm{LJ}}(r)=4{ϵ}_{\mathit{ij}}({\left(\frac{{\sigma }_{\mathit{ij}}}{r_{\mathit{ij}}}\right)}^{12}-{\left(\frac{{\sigma }_{\mathit{ij}}}{r_{\mathit{ij}}}\right)}^6)$
	S	O	CH2	CH3	Au
${ϵ}_{\mathrm{ij}}\left(\frac{\mathrm{kJ}}{\mathrm{mol}}\right)$	1.6628	0.636	0.4937	0.7326	0.1632
σ ij (nm)	0.4250	0.3154	0.3905	0.3905	0.2935
	Charges for electrostatic forces on PEO-grafted chains		$V_{\mathrm{coul}}=\frac{1}{4\pi {ϵ}_0}\frac{q_i{q}_j}{r_{\mathit{ij}}^2}$	q i , q j (e), e = 1.602176565 × 10–19 C
	Bonds
Elements	S–CH2	CH2–O–CH2	CH2–O–H
S	–0.265
CH2	0.265	0.250	0.265
O		–0.500	–0.700
H			0.435
Au	0.000 (In the NP)

Results and Discussion

For all of the systems in Table , similar simulation processes were performed using the model described in Table . A typical snapshot of a simulation at the final equilibrium state is shown in Figure (a). The AuJNP is located at the center of the simulation box, and its final state is shown in Figure (b). We can see that the two components of the matrix have been fully separated: H₂O is located at the hydrophilic side of PEO-grafted chains of the AuJNP, while the PEs are located at the hydrophobic side.

3.

(a) AuJNP_PE_W system final state. Matrix PEs are displayed as blue chains, while for the H₂O molecules, the H atoms are displayed as white balls and the O atoms as red balls. The AuJNP is located at the center of the simulation box. (b) Janus Au nanoparticle in the final state. The poly­(ethylene oxides) are grafted at bridge positions on the left side (x < 0) of the nanoparticle, while the polyethylenes are grafted on the right side (x > 0) at similar positions.

Systems Structural Properties

Density Profiles of Polymer Chains

In order to represent the average density of chains of any kind inside the simulation box, we demonstrate the density grid plots of the PE and water matrices, as well as the PE and PEO chains grafted on the AuJNP as the average density distribution for the systems AuJNP_PE, AuJNP_W, and AuJNP_PE_W, as described in Table . We also plot the average density distribution for the components of these systems at various distances along the x-axis, as it is demonstrated in Figures , , and , using 50 spherical shells from the center of mass of the nanoparticle. Moreover, the statistical uncertainties for every component in each system and for every single shell radius were estimated by using block averaging. Specifically, the equilibrated production trajectory was divided into four consecutive time blocks of equal length. For each block, the density profiles were recalculated independently for every component and spherical shell. The reported density profiles correspond to the average over the four blocks, while the uncertainty was estimated from the standard deviation among the block-averaged values. The maximum uncertainty for each component and shell radius was found to be below 1.0% with a mean value below 0.2%.

First of all, we examined the average density of the PEs of the matrix and chain polymers grafted on AuJNP in the AuJNP_PE system. In the leftmost column of Figure , symmetrical cross sections of the simulation box with respect to the center of mass of the nanoparticle at several distances on both sides of AuJNP on the x-axis (vision axis) are demonstrated. The ”PEO side” (hydrophilic) of the AuJNP is located on the negative values of the vision axis, while the ”PE side” (hydrophobic) is located on the positive values. Given that the dimensions of the cubic simulation box are 15.5 × 15.5 × 15.5 nm³ and each dimension is divided into 100 slices, each one of them has a width of ∼0.155 nm. In the second and third columns, we demonstrate the average density of the matrix on both sides of the AuJNP for the mentioned sections, respectively: the second column refers to the PEO side of the NP, while the third column refers to the PE side. Here, one can see the decrement of the matrix average density while approaching the AuJNP from both sides and the penetration of the chains grafted on it into the matrix. The density decrement is indeed caused by the penetration of PEO- and PE-grafted chains into the matrix. At distances lower than the AuJNP radius, the matrix density is of course zero (the matrix is not extended inside the nanoparticle). In the fourth and fifth columns, the average density of the PEO- and PE-grafted chains is demonstrated on both sides of the AuJNP for the same sections as well: the fourth column refers to the PEO chains of the NP, while the fifth column refers to the PE chains. This picture is complementary to the previous one: the average density of PEs and PEOs increases as we approach the nanoparticle. The first row of the same figure demonstrates the average density at the point where the PE-grafted chains begin to appear, the second one when the PEO-grafted chains begin to appear, the third row near the AuJNP surface, and the fourth row at a position less than the AuJNP radius. The fourth and fifth columns that refer to PEO- and PE-grafted chains, respectively, demonstrate a complementary picture of the previous one, while blue color indicates the absence of them inside or away from the AuJNP.

4.

Left column: Cross sections (green columns) of the simulation box on both sides of AuJNP on x-axis (vision axis) for the AuJNP_PE system. Second and third columns: Density of the matrix (CH₃(CH₂)₂₀CH₃) on both sides of the AuJNP for the sections demonstrated on the leftmost column: the second column refers to the PEO side of the NP, while the third column refers to the PE side. Fourth and fifth columns: Density of the PEO- and PE-grafted chains on both sides of the AuJNP for the sections demonstrated on the leftmost column: the fourth column refers to the PEO chains of NP, while the fifth column refers to the PE chains. Yellow and red colors indicate higher and lower density, respectively, while blue color indicates the absence of matrix chains. The dimensions of each slice are 16 nm × 16 nm, while the origin (0,0) is at the center of the slice. Density is expressed in gr/cm³ and density scales range between 0.0 and 0.8 gr/cm³.

To better elucidate the density of the above components at various distances from the center of mass of the AuJNP, we plot their average density distribution along the x-axis, as it is demonstrated in Figure , using 50 spherical shells from the center of mass of the nanoparticle.

It seems that all the results are in a good agreement, as the average density of the matrix tends to ∼0.7 gr/cm³ in agreement with values reported in the literature. The PE-grafted chains seem to have a density peak (∼0.46 gr/cm³) between 3.3 and 3.5 nm from the center of mass of the AuJNP, while for the PEO-grafted chains, the average density takes its maximum value ∼0.6 gr/cm³ at ∼3.53 nm.

The above analysis was followed for the system of one AuJNP inside a H₂O matrix. Here, the dimensions of the cubic simulation box are 16.2 × 16.2 × 16.2 nm³ and each dimension is divided into 100 slices; each one of them has a width of ∼0.162 nm. The grid density plots are demonstrated in Figure .

6.

Left column: Cross sections (green columns) of the simulation box on both sides of AuJNP on x-axis (vision axis) for the AuJNP_W system. Second and third columns: Density of the matrix (H₂O) on both sides of the AuJNP for the sections demonstrated on the leftmost column: the second column refers to the PEO side of the NP, while the third column refers to the PE side. Fourth and fifth columns: Density of the PEO- and the PE-grafted chains on both sides of the AuJNP for the sections demonstrated on the leftmost column: the fourth column refers to the PEO chains of the NP, while the fifth column refers to the PE chains. Yellow and red colors indicate higher and lower density, respectively, while blue color indicates the absence of matrix chains. The dimensions of each slice are 16 nm × 16 nm, while the origin (0,0) is at the center of the slice. Density is expressed in gr/cm³ and density scales range between 0.0 and 1.0 gr/cm³.

The corresponding average density distribution plot along the x-axis for the current system is demonstrated in Figure .

In this plot, one can see that the peak of the average density of the PE-grafted chains is equal to 0.74 gr/cm³ and is located at ∼3.6 nm, which is very close to the radius of the AuJNP (3.5 nm). There is also a small value of density of PE-grafted chains on the other side of the nanoparticle at ∼3.6 nm, but it is due to statistical reasons and for a very small number of CH _x monomers of the PE chains. The peak of the PEO-grafted chain’s average density is located at ∼3.5 nm with a value of 0.82 gr/cm³, very close to the surface of the nanoparticle. As for the PE-grafted chains, there is a small peak in the density on the other side at ∼3.8 nm. The average density of the H₂O matrix is ∼0.96 gr/cm³ very close to 1.0.

Identical analysis was applied to the system that consists of the AuJNP and a matrix of both PEs and H₂O molecules. During the simulation, the system from the initial (biased) state ended up almost instantly to the final position (Figure (B)). The final dimensions of the simulation box are 16.99 × 16.99 × 16.99 nm³ and, dividing each dimension into 100 slices, gives a slice width of ∼0.17 nm. In the plots we can see the complete separation of the two components of the matrix: the PE chains were collected on the right side (x > 0) where the PE chains grafted on the nanoparticle are similar to them (hydrophobic) while, on the left side (x < 0), the H₂O were collected due to their intermolecular hydrogen bonds between the O and H atoms of the H₂O and the PEO-grafted chains (hydrophilic). Some of the corresponding grid plots of the various system components’ density are demonstrated in Figure . The sequence of the plots has been selected to elucidate the following interesting observations.

8.

Grid density plots of the AuJNP_PE_W system: upper left: H₂O matrix, upper right: PE matrix, lower left: PEO-grafted chains, lower right: PE-grafted chains. The dimensions of each slice are 16 nm × 16 nm, while the origin (0,0) is at the center of the slice. Density is expressed in gr/cm³, and density scales range between 0.0 and 0.8 gr/cm³.

Although the concentration of the matrix components was 50% w/w for each one and given that the nanoparticle is firmly positioned in the center of the simulation box, the interface between the PE chains and the H₂O molecules is not located exactly in the middle of the nanoparticle (x = 0), but it is shifted to negative x-axis values. Furthermore, observing the upper images in Figure , we see that the interface between the two components is actually a layer where the transition from one component to the other is taking place. The interface, treated as a Gibbs dividing surface − (GDS), is identified from the gradual increase in the PE matrix density when moving from the water phase (left) to the PE matrix phase (right) in the final configuration of the AuJNP system. The GDS is calculated using the formula

$$\rho (x)=\frac{1}{2}({\rho }_1+{\rho }_2)-\frac{1}{2}({\rho }_1-{\rho }_2)\tanh \left(\frac{x-x_0}{d}\right)$$ 5

In eq , the interfacial density, ρ, is represented as a function of the coordinate x that is the axis where the phase transition takes place, and ρ₁ = 0 and ρ₂ = 0.72 gr/cm³ denote the density of the PE matrix located on either side of the Gibbs dividing surface, x ₀. d characterizes the steepness of transition and is related to the bounds of GDS with the formula t _90–10 = 2.1972d, where t _90–10 is the 10–90 interfacial width obtained from the fitted density profile that extends from 10 to 90% of the maximum of ρ₁ and ρ₂ densities. According to this estimation, the Gibbs surface is centered at x = – 1.75 ± 0.01 nm with a characteristic steepness at d = 0.38 ± 0.01 nm, having a thickness of t = 0.82 ± 0.02 nm, as shown in Figure . Given the above, the left bound of the GDS is located at x ₁ = −2.16 ± 0.03 nm, while the right bound is at x ₂ = −1.34 ± 0.03 nm. It is worth noting that, due to the periodic boundary conditions of the simulation, a small fraction of the H₂O molecules of the system appears on the very right edge of the simulation box.

The corresponding average density distribution plot along the x-axis for the current system is demonstrated in Figure . Here, we observe that the peak average density of PEO hair is located at 3.5 nm on the left of the center of mass of the nanoparticle (very close to its surface) with a value of 0.72 gr/cm³, as well as the density of the PE-grafted chains that are located at the symmetrical position 3.5 nm on the right to its center of mass (0.46 gr/cm³).

10.

Average density distribution of the matrix that contains both PEs and (H₂O), the PE- and PEO-grafted chains for the AuJNP_PE_W system, at various distances from the center of mass of the nanoparticle.

The average density of water H₂O reaches the value of 0.85 gr/cm³ at 7.8 nm left to the center of mass of the AuJNP. As for the density of the PE matrix, there is a plateau that is extended from 5.5 to 8.0 nm right to the center of the nanoparticle with a value range around 0.72 g/cm³. There are also a few PE matrix chains (6/3730 = 0.16%) on the hydrophilic side due to the excess of PE molecules that penetrate on the hydrophilic side (Figure ).

Gyration Tensor and Radius of Gyration

In order to determine the average shape and size of the molecules (chains) participating in our systems, we calculate the Gyration Tensor S, whose elements are determined by the formula

$$S_{\mathit{mn}}(r)\stackrel{\mathit{def}}{=}\frac{1}{N}\sum _{i=1}^N(r_m^{(i)}-r_{\mathit{CM}}^{(i)})(r_n^{(i)}-r_{\mathit{CM}}^{(i)})$$ 6

where N is the number of particles (here: united atom, that is, a C atom with two or three bonded H atoms) of the chain, r _m and r _n are the m ^th and n ^th Cartesian coordinates, respectively, of the average position vector r ⁽ⁱ⁾ of the i ^th particle and r _CM is the average position vector of the center of mass of the specific chain the particle belongs to. Because of its symmetry (it is a 3 × 3 matrix), its diagonalization gives a principal axis system where we choose that

$$S=\mathrm{diag}({\lambda }_x^2,{\lambda }_y^2,{\lambda }_z^2)$$ 7

where λ _x , λ _y , and λ _z are the eigenvalues of S and λ _x ≥ λ _y ≥ λ _z .

The size of a linear polymer chain can be characterized by the radius of gyration (R _g ) that is defined as the average square distance between monomers in a given chain and its center of mass. Taking into account eqs and , R _g is given by

$$R_g\stackrel{\mathit{def}}{=}\sqrt{{\lambda }_x^2+{\lambda }_y^2+{\lambda }_z^2}$$ 8

Using S, one can calculate the relative shape anisotropy κ² of a polymer, where 0 ≤ κ² ≤ 1, that reflects the spherical symmetry of a polymer (κ² = 0: spherical symmetry, κ² = 1: linear symmetry). κ² is calculated by

$${\kappa }^2\stackrel{\mathit{def}}{=}\frac{3}{2}\frac{{\lambda }_x^4+{\lambda }_y^4+{\lambda }_z^4}{{({\lambda }_x^2+{\lambda }_y^2+{\lambda }_z^2)}^2}-\frac{1}{2}$$ 9

We also calculate the average end-end distance (R _end–end ) of the participating chains in our systems. The calculations of the above properties for all the considered systems are demonstrated in Table .

3. Eigenvalues (λ _x , λ _y , λ _z ) of the Diagonalized Matrix of the Gyration Tensor for the Various Oligomers of the Considered Systems, Radius of Gyration (R _g ), Spherical Anisotropy (κ ²), and End-to-End Distance of the Various Oligomers (R _end–end ).

	Properties
chain/system	λ x ,λ y ,λ z (nm2)	R g (nm)	κ2	R end–end (nm)
PE matrix (CH3(CH2)20CH3)
Bulk	0.138, 0.137, 0.137	0.642	1.78 × 10–6	1.82 ± 0.40
AuJNP_PE	0.140, 0.140, 0.140	0.648	3.20 × 10–6	1.84 ± 0.40
AuJNP_PE_W	0.142, 0.142, 0.140	0.643	8.86 × 10––4	1.82 ± 0.40
PE-grafted chains (−(CH2)19CH3)
AuJNP_PE	0.125, 0.120, 0.116	0.601	4.16 × 10–4	1.75 ± 0.33
AuJNP_W	0.120, 0.108, 0.103	0.575	2.00 × 10–3	1.57 ± 0.36
AuJNP_PE_W	0.121, 0.121, 0.118	0.601	1.02 × 10–4	1.78 ± 0.33
PEO-grafted chains (−[(CH2)2O]6H)
AuJNP_PE	0.074, 0.067, 0.059	0.447	4.50 × 10–3	1.26 ± 0.24
AuJNP_W	0.080, 0.064, 0.055	0.447	1.22 × 10–2	1.26 ± 0.24
AuJNP_PE_W	0.080, 0.072, 0.044	0.443	2.69 × 10–2	1.30 ± 0.21

According to Table , the difference between the R _g of PE matrix chains of AuJNP_PE, AuJNP_PE_W systems, and the bulk system is under 1%, which means that the radii of gyration of the three systems are almost identical. This is not the same for the relative shape anisotropy, where κ² of the AuJNP_PE_W system is ∼277 times greater than the bulk’s one. Even though both values are very close to zero (that means almost no anisotropy), the AuJNP_PE_W system seems to be slightly more anisotropic than the bulk’s one. AuJNP_PE system κ² value is 80% greater than the bulk’s, but both of them are very close to perfect sphericity.

In the hydrophobic environment of AuJNP_PE (pure PE matrix) as well as on the hydrophobic side of AuJPE_PE_W systems, R _g values of PE-grafted chains are identical at 0.601 nm, while the same value in a pure water matrix (AuJNP_W system) is 0.575 nm. κ² values of the three systems (although they are also near to zero) indicate that PE-grafted chains on AuJNP of the AuJNP_W system are slightly more anisotropic than the other two, while the anisotropy for the same chains decreases when moving to AuJNP_PE_W and to the pure PE matrix (AuJNP_PE) systems. R _g values for PEO-grafted chains in all the considered systems are almost identical, with a slight difference in AuJNP_PE_W. The stability of R _g in both cases indicates that the effect is dominated by conformational entropy, as it is mainly interfacial.

Comparing R _g values for PE- and PEO-grafted chains for the various systems, one can see that generally PE-grafted chains seem to extend more into the corresponding system’s matrix, as it is demonstrated in Figures , , and .

Second-Rank Bond Order Parameter (SRBO)

Second-rank bond order parameter , (SRBO) determines the average orientation of polymer PE chains (in the case of the PE matrix) by examining the orientation of the vector that connects two nonconsecutive C atoms in the polymer chain or with respect to the nanoparticle surface at various distances. SRBO is given by the formula

$$S_{1-3}(r)=\frac{3}{2}⟨{\cos }^2\theta ⟩-\frac{1}{2}$$ 10

where θ is the angle between the radial vector r⃗ from the center of mass of the nanoparticle and the center of the vector v⃗ ^1–3 that connects the two nonconsecutive C or the H–H atoms and r is the distance from the center of mass of the nanoparticle, as shown in Figure . Brackets (⟨ ⟩) denote statistical average on semispherical shells at a distance of r → r + dr from the center of mass of the AuJNP, where dr is 1/50th of the simulation box width. Values of S _1–3: −0.5, 0.0, and 1.0 correspond to perfectly parallel, random, and perpendicular vector orientations relative to the AuNP surface, respectively. According to the previous analysis, x = 0 represents the position of the center of mass on the AuJNP on x-axis, while, as we mentioned before, on the left side (x < 0) PEOs are grafted on the nanoparticle and on the right side there are PEs grafted on it.

11.

Second-rank bond order parameter (SRBO) definition. θ is the angle between the radial vector r⃗ from the center of mass of the nanoparticle and the center of the vector v⃗ ^1–3 that connects the two nonconsecutive C atoms.

In the plot of Figure (a), S _1–3, which refers to the PE chains of the matrix of the AuJNP_PE system, is plotted. As we see, on the PE side of the AuJNP, S _1–3 keeps its value around 0 at a distance of ∼4.50 nm from the center of the nanoparticle to the very right edge of the simulation box (7.5 nm), which means that the orientation of the PE chains in the matrix is randomly distributed with respect to the nanoparticle surface. There is a small peak of the S _1–3 value between 3.9 and 4.5 nm, while for shorter distances (3.15–3.9 nm), the orientation tends to be almost parallel to the surface. On the PEO side of the AuJNP, S _1–3 tends to show random orientation of PE matrix chains from 4.95 to 7.5 nm with a small variance from 3.9 to 4.9 nm. In the area of 3.15 up to 3.9 nm, the chains tend to be parallel to the surface of the AuJNP.

12.

Second-rank bond order parameter (SRBO) for (a) PE matrix on AuJNP_PE and (b) AuJN_PE_W systems.

S _1–3 of the system of AuJNP in the PE/water matrix is plotted in the plot of Figure (b). The plots are referred to the PE molecules of the matrix. As we see, on the PE side of the AuJNP, S _1–3 keeps its value around 0 (random orientation) at a distance of ∼3.94 nm from the center of the nanoparticle to the very right edge of the simulation box (8.5 nm). For shorter distances (3.5–3.94 nm), the same orientation tends to be almost parallel to the surface. As we saw previously, some of the PE chains (0.16%) remained on the hydrophilic side of the AuJNP. These few chains preserve their almost random orientation after ∼3.41 nm from the center of the nanoparticle, while there is a small peak in the range from 3.95 to 4.49 nm. Recent polarized resonant soft X-ray scattering experiments directly demonstrated orientational ordering in polymer-grafted gold nanoparticles, consistent with the interfacial chain alignment observed here. −

Flory’s Ratio (C)

In this section, we demonstrate Flory’s characteristic ratio C that, generally, quantifies the stiffness of a polymer chain and compares the mean-square end-to-end distance of a real polymer chain to an idealized freely jointed chain. C = 1 corresponds to an ideal free-jointed chain with no restrictions on bond angles and rotations, while C > 1 is the case for most real polymer chains: reasons that cause this are fixed bond angles (polymer bonds are approximately 109.5° for a carbon backbone) or steric hindrance that is caused by the repulsion between nonbonded atoms or groups along the chain. Flory’s ratio depends on the chemical distance m between two UAs in the same chain. For m → ∞, it is saturated at a finite value C _∞ that is characteristic for a polymer.

Here, we calculate this ratio C(m) for all the backbone united atoms (UA) for the PE matrix, PE- and PEO-grafted chains on the nanoparticle along the chains for all our consider systems that are distant from each other by m chain positions (chemical distance). C(m) is given by the following formula

$$C(m)\equiv \frac{R^2(m)}{m{l}_b^2},R^2(m)\equiv ⟨\frac{1}{n_c(N-m)}\sum _{i=1}^{n_c}\sum _{j=1}^{N-m}{[{\mathrm{r⃗}}_{i,j}-{\mathrm{r⃗}}_{i,j+m}]}^2⟩,0<m<N$$ 11

R ²(m) in eq is the mean-square of the internal distances between two atoms in the same chain for every chemical distance separately (the angular brackets ⟨ ⟩ denote an average over the whole system configuration). Furthermore, n _c is the number of chains, N is the number of repeat units (here, the UA backbone atoms belonging to the same chain), r⃗ _i,j is the position vector of the j-th repeat unit on the i-th chain, and m is the chemical distance of two atoms in the same chain, i.e., it expresses the number of atoms comprising the chain contour between the two considered atoms. C(m) is defined as the reduced value of R ²(m) with respect to the chemical distance m, as well as the squared average value of the bond length between two UA backbone atoms (l _b ).

In the upper plot of Figure , we demonstrate the results for the PE polymers comprising the matrices of the systems AuJNP_PE and AuJNP_PE_W and the bulk system of PE polymers for comparison purposes. The values of C(m) are almost identical for PEs of AuJNP_PE and AuJNP_PE_W systems compared to the bulk one. This indicates that the chain stiffness of the matrix chains in the three systems is the same as the bulk one’s. Moreover, C(m) tends to a value of 7 that is very close to C _inf = 7.4, which is the saturation value of polyethylene according to the literature. This value is not reached because of the small number of monomers in the chain (m = 22).

13.

C(m) for PE matrix chains (upper), grafted PE chains (middle), and grafted PEO chains (lower) in Bulk, AuJNP_PE, and AuJNP_PE_W systems.

The middle plot of Figure demonstrates the C(m) values of the PE chains grafted on the AuJNP for the AuJNP_PE, AuJNP_W, and AuJNP_PE_W systems. In AuJNP_PE and AuJNP_PE_W systems, C(m) values are the same as the ones of the bulk system. In the AuJNP_W system where the nanoparticle is located in a water matrix, C(m) has the same behavior as the previous system, but it tends to a value of 5.8; this indicates that the stiffness of the chains is lower than the bulk’s one. Again, the saturated values are not reached because of the small number of monomers in each chain. The above analysis is repeated for PEO chains grafted onto the AuJNP, and the results are demonstrated in the lower plot of Figure . Here, the C(m) values are almost identical in AuJNP_PE and AuJNP_W systems, while the same value in AuJNP_PE_W is a little higher.

Overall, we observe that C(m) values of PE chains behave almost identically even if the chains constitute the matrix or are grafted on the nanoparticle in a hydrophobic environment. This means that the nanoparticle modifies how chains are arranged and move near its surface, but it does not change the fundamental conformational properties of the polymer backbone. In the hydrophilic environment of AuJNP_W, these C(m) values are lower for m > 4. On the other hand, C(m) of PEO-grafted chains behave almost identically in pure hydrophobic or hydrophilic environments, while on the hydrophilic side of AuJNP_PE_W they give slightly higher values for m > 7.

Dynamic Properties

Time Autocorrelation Function (ACF)

Time autocorrelation function (ACF) determines statistically the rotating ability of polymer chains with respect to time. This is described by the formula

$$C_{\mathit{end}‐\mathit{end}}(t)=\frac{⟨R(t)·R(0)⟩}{⟨\parallel R{\parallel }^2⟩}$$ 12

where R(0) is the end-to-end vector from first to last monomer of a certain chain at t = 0, R(t) is the same vector after time t, ⟨R(t)·R(0)⟩ is the average of their internal product, and ⟨∥R∥²⟩ is the average square of magnitude for the same time. An ACF value of 0 denotes a relaxed system, as the angles between R(0) and R(t) of the various PE chains have a random distribution, while a value near 1 denotes that chains cannot rotate freely.

In this work, we considered spherical shells around the AuJNP with various radii, and the averages were calculated for the time that the center of mass of a specific chain remains inside the same spherical shell. Because of their mobility, chains are permitted to leave their initial shell up to 0.1 nm from its boundaries for a short amount of time (200 ps) before they return to it. If the chain keeps remaining outside the initial boundaries, then it is considered to be located in a new shell and the time starts counting from 0.

The ACF calculation was performed for the PE matrix of the AuJNP_PE_W system. Furthermore, we calculated the ACF of a bulk system that consists only of a matrix of PE chains (without any nanoparticle in it) for comparison reasons. The whole plot of ACF for the various spherical shells as well as for the bulk system is shown in Figure . The boundaries of the first area (spherical shell) extend from 0.0 up to 1.0 nm from the AuJNP’s surface. The shell beyond 4.5 nm from the surface overrides the simulation box edges, so it does not have any statistical meaning, so we cannot perform such calculations in it. To check the accuracy of our results, we calculated the statistical uncertainty for every single time period for all of the spherical shells as well as for the PE bulk system. The statistical uncertainties were estimated by using block averaging. The equilibrated production trajectory was divided into four consecutive time blocks with equal lengths. For each block, the ACF of the PE matrix was recalculated independently for every spherical shell as well as for the PE bulk system for comparison reasons. The reported results correspond to the average ACF over the four blocks, while the uncertainty for each time period ranges between 0.10 and 1.56%, 0.14 and 1.17%, and 0.03 and 0.77% for the shells with radii between 0.0 and 1.0, 1.0 and 4.5, and >4.5 nm, respectively. The average uncertainties of the above are 0.74%, 0.74%, and 0.35%, respectively. The uncertainty for the PE bulk system fluctuates between 0.09 and 1.42% with an average of 0.72%.

14.

Time autocorrelation function for PE matrix for AuJNP_PE_W system and a bulk PE matrix. Bulk PE matrix relaxes after a time of about 1 ns. PE matrix chains that are positioned in a spherical shell up to 1 nm away from the AuJNP’s surface relax more slowly than the same chains in the bulk system. PE chains positioned after this spherical shell are likely to follow the bulk system for relaxation.

Observing the plot in Figure , we see that the bulk PE matrix relaxes after a time of about 1 ns, which has also been shown in other works. PE matrix chains that are positioned in a spherical shell of 1 nm away from the AuJNP’s surface relax more slowly than the same chains in the bulk system. Chains positioned after this spherical shell are likely to follow the bulk system for relaxation; thus, the corresponding plots of the various shells are very similar to the bulk system plot.

Elucidating the previous results, we calculate the corresponding mean relaxation times by fitting the above curves with the Kohlraush-Williams–Watts (KWW) stretch exponential function

$$C_{\mathit{end}‐\mathit{end}}(t)=\exp [-{\left(\frac{t}{{\tau }_{\mathit{KWW}}}\right)}^{\beta }]$$ 13

where τ _KWW is the KWW relaxation time and β the stretch exponent. The mean relaxation time τ _end–end is given by

$${\tau }_{\mathit{end}‐\mathit{end}}=\frac{{\tau }_{\mathit{KWW}}}{\beta }\mathrm{\Gamma }\left(\frac{1}{\beta }\right)$$ 14

where Γ( ) is the γ function. The results of the above calculations are presented in Table .

4. KWW Times, Stretch Exponents, and Mean Relaxation Times of PE Matrix in Bulk and AuJNP_PE_W Systems for the ACF of Rotation.

system ID	t KWW (ps)	β	τ end‑end (ps)
Bulk only (PE matrix)	204.3 ± 0.4	0.938 ± 0.021	210.3
	Area 0.0–1.0 nm from the AuJNP’s surface
AuJNP_PE_W	268.2 ± 1.5	0.786 ± 0.005	307.9
	Area 1.0–4.5 nm from the AuJNP’s surface
	196.4 ± 0.4	0.968 ± 0.003	199.2

According to Table , we observe that the mean relaxation time of PE matrix chains is significantly (46.4%) bigger than the bulk system’s one at distances between 0 and 1 nm from the surface of AuJNP, which indicates the interaction with the PE-grafted chains of the nanoparticle. On the other hand, for longer distances, the relaxation time is slightly (5.3%) smaller, indicating the absence of interactions. This dynamic heterogeneity indicates two regions with respect to τ _end–end : a slow interfacial region with a locally slower relaxation dynamics where chains are more constrained and dissipate stress more slowly and a faster one where the mobility is similar to bulk systems. The slower relaxation dynamics predicted near the AuJNP surface are qualitatively consistent with NMR-observed immobilized polymer interphases in nanoparticle-modified systems.

Mean-Square Displacement (MSD)

Mean-square displacement (MSD) determines the moving ability of the chain monomers statistically; here, it is calculated for the monomers of matrix PE chains on the hydrophobic side of the AuJNP. The formula we used is

$$\mathrm{\Delta }{R}_j(\tau )=⟨{[r(t+\tau )-r(t)]}^2⟩$$ 15

where j is the specific spherical shell, and r(t) and r(t + τ) are the vectors of a chain monomer of the matrix PE in region j having a time distance of τ. Distances are referred to the surface of the nanoparticle, while the statistical average refers to all the monomers within the same region. In the calculation, we considered the same constraints as those used to calculate the Autocorrelation Function in terms of crossing the boundaries of each region by the monomers. The above calculation has also been performed for the same bulk PE matrix system as we did for the autocorrelation function for comparison reasons.

The plot of MSD for the various spherical shells as well as for the bulk system is shown in Figure . According to this plot, the MSD of the PE chain monomers of the PE/water matrix in a spherical shell from 0.0 to 1.0 nm from the surface of the nanoparticle, during the time interval 10–2 × 10³ ps, gradually increases from 0.6 to ∼4 nm². The MSD values for the spherical shell from 1.0 to 4.5 nm from the nanoparticle’s surface and for a time interval from 10 to 10⁴ ps gradually increase from 0.8 to 35 nm². In both cases, the MSD values become chaotic for longer times. The corresponding values of MSD for the bulk PE chain matrix system and for time values from 10 to 3 × 10⁴ ps are much higher, as they increase from 5 to 10² nm². The above results indicate that PE monomers in a system with a AuJNP are generally less mobile than the PE monomers in a bulk system. Moreover, PE monomers located in spherical shells with radii in the range of 1.0–4.5 nm from the AuJNP’s surface are more mobile than the monomers positioned in the range 0.0–1.0 nm from the nanoparticle’s surface. Concerning the diffusion coefficient, we should note that the presented dynamics refer to rather short time scales where a linear (Fickian) diffusion is not expected and therefore a diffusion coefficient cannot be defined; this has been validated from our data. Therefore, the dynamical slowdown near the NP is instead quantified through the KWW relaxation times from the ACF analysis (Table ), which show a 46.4% increase in end-end for chains within 0–1 nm of the NP surface relative to the bulk, and bulk-like relaxation beyond 4.5 nm. The slower relaxation dynamics predicted near the AuJNP surface are qualitatively consistent with NMR-observed immobilized polymer interphases in nanoparticle-modified systems. ,

15.

Mean-square displacement for PE matrix for AuJNP_PE_W system and a bulk PE matrix (distances are referred to the surface of the nanoparticle).

Desorption of Polymer Chains

We also compared the flux of the matrix PE monomers through spherical shells of about the same volume for both the AuJNP_PE_W and PE (bulk) systems via the desorption evolution ratio DE = N(t)/N(0) of the above chains monomers that is the mean relative density, where N(0) and N(t) are the initial number of PE monomers and the remaining of them, after time t, respectively. The selected spherical shells are listed in Table .

5. Description of Spherical Shells for N(t)/N(0) Calculation .

shell #	R int (nm)	R ext (nm)	ΔR (nm)	volume (nm3)
1	0.000	1.500	1.500	344.00
2	1.500	2.417	0.917	344.15
3	2.417	3.113	0.697	343.64
4	3.113	3.688	0.575	344.26

^aDistances R _int and R _ext are measured from the AuJNP’s surface and represent the internal and external radius of a spherical shell, respectively; AuJNP’s radius is 3.5 nm.

In the calculation of the DE evolution ratio, we only took into account the PE monomers that were initially present in a specific spherical shell and that subsequently left it, from either the inner or outer surface of it. For the AuJNP_PE_W system specifically, where the AuJNP is centrally positioned, there is no flow of monomers from the inner surface because of the NP’s existence. As in the case of the time autocorrelation function, monomers are permitted to leave their initial shell up to 0.1 nm from the shell boundaries for a short amount of time (200 ps) before they are considered that they have definitely left it. If a monomer returns to the initial shell within the above time, it is considered that it has never left it. The results are plotted in Figure .

16.

Mean relative density evolution of PE monomers in consecutive spherical ranges from the AuJNP’s surface for AuJNP_PE_W and bulk PE systems.

As we expected, there is an obvious difference in density evolution between the AuJNP_PE_W and the PE bulk system in the innermost shell, where the DE of the first one relaxes slower than the bulk one, due to the presence of the AuJNP (Figure (upper left)). For the rest of the shells located at greater distances, no similar behavior is observed. This is another form of dynamic heterogeneity but now in translational motion rather than rotational relaxation (ACF).

For further quantification of the DE evolution, we calculated the mean relaxation times by fitting a KWW stretch exponential function to our results, as we did before in the ACF case. The various values are listed in Table .

6. KWW Times, Stretch Exponents (β) and Mean Relaxation Times (τ _mean ) of PE Matrix Density in Bulk and AuJNP_PE_W Systems for DE .

	AuJNPPE_W			PE bulk
shell range	τ KWW (ps)	β	τ mean	τ KWW (ps)	β	τ mean
0.000–1.500	185.98 ± 0.47	0.68 ± 0.00	240.54	147.90 ± 0.40	0.85 ± 0.00	161.06
1.500–2.417	72.52 ± 0.14	0.95 ± 0.00	74.39	79.45 ± 0.19	0.93 ± 0.00	82.09
2.417–3.113	52.63 ± 0.20	0.98 ± 0.00	53.06	52.42 ± 0.18	0.94 ± 0.00	53.97
3.113–3.688	44.28 ± 0.11	0.95 ± 0.00	45.28	43.62 ± 0.16	0.97 ± 0.01	44.15

^aRanges are measured from the AuJNP’s surface; AuJNP’s radius is 3.5 nm.

In Figure , the calculated τ _mean values with respect to the spherical shell ranges for both the AuJNP_PE_W and bulk PE systems are demonstrated. As we observe, the difference between the mean times between the considered systems is large at ranges in the first system where the AuJNP is present: more specifically, the relaxation time is larger than the second one, while for larger distances, the mean relaxation times are almost equal.

17.

Mean times of PE density evolution for AuJNP_PE_W and bulk PE systems.

Conclusions

In this work, atomistic molecular dynamics simulations were employed to investigate the structural and dynamical effects induced by a single amphiphilic Janus gold nanoparticle (AuJNP) in polymeric, aqueous, and mixed polymer–water environments. By explicitly resolving the gold core, the chemically distinct grafted ligands, and the surrounding matrix components, the study provides a detailed molecular-level description of how nanoscale chemical anisotropy governs interfacial organization and dynamics in complex soft-matter systems.

The simulations demonstrate that the Janus character of the nanoparticle directs and stabilizes spatial segregation in the mixed PE/water matrix by acting as a structural template that promotes highly organized, nonisotropic polymer conformations. Experimental observations suggest that this stabilization arises from the ability of Janus NP surfaces to dictate polymer-chain orientation, promote the formation of adsorbed polymer layers and associated condensed polymer regions, and impose geometric constraints that drive structural reorganization of the surrounding polymer matrix. These effects collectively reinforce the phase segregation.

Water molecules preferentially occupy the vicinity of the hydrophilic PEO-grafted hemisphere, while polyethylene chains are localized near the hydrophobic PE-grafted side. This leads to phase separation, but instead of forming a sharp boundary, there is a thin transition layer (∼0.82 nm thick) where the densities gradually change. This behavior confirms that amphiphilic Janus nanoparticles act as strong interfacially active inclusions capable of locally stabilizing phase separation through directional interactions. The resulting density profiles reveal the formation of structured interfacial regions whose extent is controlled by the surface chemistry of the grafted ligands rather than by the inorganic core alone. In this way, the Janus nanoparticle acts as a compatibilizer and an emulsion stabilizer simultaneously. It keeps the two immiscible phases separated but also stabilizes the interface between them by anchoring each phase to the side it prefers. The term compatibilizer is used here to describe the local molecular mechanism by which the Janus nanoparticle stabilizes the PE/water interface, rather than the macroscopic behavior of a many-particle compatibilized blend. As a result, the system remains structured and stable, preventing the phases from mixing randomly or separating uncontrollably.

Moreover, the simulations show that polyethylene (PE) matrix chains near the nanoparticle surface do not remain randomly oriented, as they do in the bulk. Within a few nanometers of the surface (especially in the 3.1–3.9 nm region from the center of mass), the chains adopt a preferential orientation that is nearly parallel to the nanoparticle surface. This local alignment reflects an interfacial response typical of polymer nanocomposites, where confinement and attractive interactions with grafted ligands modify chain conformation and mobility. As a result, an interphase region forms around the nanoparticle, characterized by altered orientation and slower relaxation dynamics compared to the bulk. Such surface-induced ordering is a key mechanism by which nanoparticles influence stress transfer, mechanical reinforcement, and overall macroscopic performance in nanocomposite materials. The almost constant Flory ratio indicates that the intrinsic chain stiffness of the polymer does not affect the creation of the interphase around the nanoparticle; it rather arises due to the interactions between chains and the nanoparticle’s surface. This distinction is important in polymer nanocomposites: the interphase is governed by interfacial physics, not by a change in polymer molecular structure. The observed preferential parallel orientation of PE matrix chains near the AuJNP surface is consistent with experimental evidence from related systems: FTIR studies of alkanethiol-coated gold nanoparticles report predominantly all-trans chain conformations for long alkyl chains, , while parallel chain alignment near solid surfaces is qualitatively consistent with experimental X-ray scattering studies reporting ordered interphase regions in polymer nanocomposites containing nanoparticles. ,−

In addition to static structural effects, the presence of the AuJNP significantly modifies the local dynamics of the polymer matrix. Analysis of density evolution and relaxation times shows a clear slowdown of polyethylene dynamics in the vicinity of the nanoparticle surface, particularly within the innermost shells adjacent to the grafted ligands. The slower ACF near the nanoparticle implies increased local relaxation times and reduced chain mobility in the interfacial region, consistent with a dynamically constrained interphase. This is a characteristic behavior for systems like fillers incorporated in polymers and is consistent with experimentally observed dielectric interphases in polymer nanocomposites. In other words, the ACF results provide microscopic evidence that the Janus nanoparticle generates a dynamically constrained interphase that governs macroscopic rheological and dielectric behavior in the nanocomposite. In the present work, we study a specific grafting density (∼1.4 chains/nm²) and chain lengths consistent with experimentally studied alkanethiolate-covered gold nanoparticles; the detailed investigation of different densities and/or chain lengths could be the topic of a future work. However, based on previous works, we expect that longer grafted chains would penetrate deeper into the matrix, broadening the interphase and increasing the spatial extent over which chain orientation and mobility deviate from bulk values. Moreover, higher grafting density would form a denser brush, further excluding matrix chains from the NP surface and potentially altering the density profiles of both grafted and free chains. On the other hand, slower MSD near the surface of the nanoparticle indicates a structured low-mobility interphase that could help regulate mass transport through the composite. This interphase behaves as a kinetically constrained region (barrier) that could reduce transport rates: the effect is not due to increased stiffness (Flory’s ratio unchanged), but due to surface-induced mobility suppression. The low-mobility interphase predicted by MSD should be experimentally observable as a distinct NMR relaxation component corresponding to immobilized or slowed polymer chains. This dynamical perturbation gradually decays with distance, converging to bulk-like behavior beyond a few nanometers from the AuJNP surface. The comparison with bulk polymer systems highlights the emergence of an interphase region characterized by altered relaxation behavior rather than a sharp interface, a feature that is central to current understanding of polymer nanocomposites and interfacial soft matter. As to static structural effects, the slowed relaxation of PE chains near the AuJNP is qualitatively consistent with experimental solid-state NMR studies reporting low-mobility interphase regions in polymer nanocomposites containing nanoparticles. ,

From a physical chemistry perspective, these findings underline the importance of explicitly accounting for surface anisotropy when describing nanoparticle–matrix interactions. Unlike isotropic nanoparticles, Janus nanoparticles induce both spatially asymmetric structuring and direction-dependent dynamical responses, which cannot be captured by simple effective-medium or uniform-surface models. The results therefore provide valuable benchmarks for the development of coarse-grained theories and mesoscale simulation approaches aimed at describing anisotropic fillers in polymeric and multiphase systems.

Several directions for further research naturally arise from our work. Extending the simulations to systems containing multiple Janus nanoparticles would allow exploration of collective effects such as nanoparticle-mediated bridging, aggregation, or self-organization at interfaces. Systematic variation of the grafting density, ligand length, and nanoparticle size would help disentangle the relative contributions of surface chemistry and geometric confinement to interfacial structure and dynamics. Finally, coupling atomistic simulations with coarse-grained models could bridge time and length scales, enabling a direct comparison with experimental observations and providing a more complete multiscale description of Janus nanoparticle behavior in complex environments.

Overall, this study contributes to understanding how chemically anisotropic nanoparticles modify local structure and dynamics in soft matter systems, offering insights that are broadly relevant to polymer physics, interfacial science, and the molecular design of functional nanocomposites.

The open access publishing of this article is financially supported by HEAL-Link.

The authors declare no competing financial interest.