Structural Properties and Magnetic Ordering in 2D Polymer

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Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers

Structural Properties and Magnetic Ordering in 2D Polymer Nanocomposites: Existence of Long Magnetic Dipolar Chains in Zero Field Christian Appel, Björn Kuttich, Lukas Stühn, Robert W. Stark, and Bernd Stühn Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b02094 • Publication Date (Web): 20 Aug 2019 Downloaded from pubs.acs.org on August 21, 2019

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Structural Properties and Magnetic Ordering in 2D Polymer Nanocomposites: Existence of Long Magnetic Dipolar Chains in Zero Field Christian Appel,∗,† Björn Kuttich,† Lukas Stühn,‡ Robert W. Stark,‡ and Bernd Stühn† †Institute of Condensed Matter Physics, Technische Universität Darmstadt, Hochschulstr. 8, D-64289 Darmstadt, Germany ‡Physics of Surfaces, Technische Universität Darmstadt, Alarich-Weiss-Str. 16, D-64287 Darmstadt, Germany E-mail: [email protected]

Abstract The existence of magnetic dipolar nanoparticle chains at zero field has been predicted theoretically for decades, but these structures are rarely observed experimentally. A prerequisite is a permanent magnetic moment on the particles forming the chain. Here we report on the observation of magnetic dipolar chains of spherical iron oxide nanoparticles with a diameter of 12.8 nm. The nanoparticles are embedded in an ultrathin polymer film. Due to the high viscosity of the polymer matrix, the dominating aggregation mechanism is driven by dipolar interactions. Smaller iron oxide nanoparticles (8 nm) show no permanent magnetic moment

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and do not form chains but compact aggregates. Mixed monolayers of iron oxide nanoparticles and polymer at the air-water interface are characterised by Langmuir isotherms and in-situ X-ray reflectometry (XRR). The combination of the particles with a polymer leads to a stable polymer nanocomposite film at the air-water interface. XRR experiments show that nanoparticles are immersed in a thin polymer matrix of 2 nm. Using atomic force microscopy (AFM) on Langmuir-Blodgett films, we measure the lateral distribution of particles in the film. An analysis of single structures within transferred films results in fractal dimensions that are in excellent agreement with 2D simulations.

Introduction Due to their perfect combination of well adjustable macroscopic properties, polymer nanocomposites have attracted a lot of research interest in recent years. The polymeric component of the composites serves as a matrix in which a wide range of nanoparticles can be introduced in order to achieve desired thermal, rheological, optical, conductive, magnetic, etc. properties. 1–5 While tuning of these properties is of outstanding interest in most applied research, fundamental studies focused on the precise interactions between polymer and particles as well as on the microstructure of the nanoparticles distributed in the polymer matrix. 6–9 To prevent nanoparticles from aggregation they are stabilised by a shell. 10 The use of carboxyl acids, such as oleic acid, as shell ligands leads to a steric stabilisation of the particles. In a composite the shell type strongly influences the interaction between nanoparticles and polymer matrix. Metal nanoparticles are not only characterized by their high surface to volume ratio but also by quantum confinement. 11 Most prominent among these are gold nanoparticles, especially in combination with polymers, because of their localised surface plasmon resonance, allowing for many sensor applications. 1,12,13 Besides these, nanoparticles composed of iron, nickel and cobalt have come more and more into focus because of their outstanding magnetic

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properties. 14–16 Sufficiently small particles (size ≈ 100 nm) consist only of one single ferromagnetic domain, while for even smaller particles the permanent magnetisation completely vanishes. The material is below its Curie temperature but its magnetic dipole moment flips on a fast timescale. This state is known as superparamagnetic. 14,17,18 Introducing magnetic particles into a non-magnetic solvent yields a so-called ferrofluid, which offers the possibility to investigate fundamental magnetic dipole-dipole interactions. 19–22 A polymer nanocomposite consisting of magnetic nanoparticles can be understood as a ferrofluid-like system with a highly viscous solvent. Due to the high viscosity, Brownian dynamics are significantly slowed down. It may therefore be expected that aggregation of nanoparticles is unlikely to be limited by diffusion (DLA) but dominated by magnetic interactions. This effect is particularly interesting when the system’s degrees of freedom are reduced to two dimensions (2D). Firstly, the probability of direct particle interaction is increased due to the 2D confinement. Secondly, aggregated structures can be visualised directly by real space methods. Thin films of polymers, nanoparticles, and their composites can be prepared at the air-water interface. Their structure is determined by the 2D confinement and the interfacial properties but not by external shear fields as is the case for spin coated layers. In the following we will investigate the influence of particle size on the dipolar interactions and the impact on structure formation of nanoparticles inside a polymer monolayer. We choose to study spherical iron oxide nanoparticles having sizes between 5 nm and 20 nm which are commercially available and should be close to the limit of expected superparamagnetic behaviour. 23–25 They are stabilised by a hydrophobic oleic acid shell. The used polymer is a hydrophilic-hydrophobic diblock copolymer. For the preparation of the nanocomposite film we make use of the air-water interface and the fact that both nanoparticles and the used polymer are surface active and thus form stable monolayers at this interface. The properties of the polymer at the air-water interface have been investigated recently. 26 It forms a thin layer with thickness between 2 nm and 3 nm. Size and shape of the nanoparticles are characterised by small angle X-ray scattering. We investigate their capability to

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form stable monolayers at the air-water interface using Langmuir isotherms and in-situ Xray reflectivity (XRR). Our main interest lies in the properties of the combined films of nanoparticles and polymer. First, the Langmuir isotherms and XRR are used to determine the properties of the composite films. We then turn to the question of the structure of nanoparticle aggregates within the polymer layer. In order to address this question we transfer the films on a solid substrate to obtain Langmuir-Blodgett (LB) films which can then be investigated with atomic force microscopy (AFM). Depending on the aggregation mechanism being diffusion limited (DLA) or dominated by magnetic dipole interaction, the appearance of differently formed clusters is predicted. 27,28

Experimental section Samples Samples used in this work are three different sizes of iron oxide nanoparticles (diameter: 5, 10 and 20 nm) and a diblock copolymer poly(ethylene glycol)-b-poly(n-butyl acrylate) (PEG6 -b-PnBA132 , Mn = 17.2 kg/mol). The polymer was synthesized via controlled living polymerization as described in our previous publication. 26 The iron oxide nanoparticles were purchased from Ocean Nanotech where their size and shape was characterized by TEM experiments. The 5 nm particles (FeNP5) are stated to consist of a mixture of Fe2 O3 and Fe3 O4 , while the 10 nm (FeNP10) and 20 nm (FeNP20) ones are claimed to consist of only Fe3 O4 nanocrystals. Particles were received dissolved in chloroform and stabilized by an oleic acid shell of 2 nm thickness.

Pressure-area isotherms Pressure-area isotherms were performed on a KSV NIMA trough system (KSV NIMA, Langmuir Troughs KN 1006, Biolin Scientific) equipped with two hydrophobic symmetricallymoving barriers and a Wilhelmy platinum plate placed exactly in-between those barriers 4


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(45◦ - tilted with respect to the direction of compression) at room temperature of 22 ± 1 ◦C. The trough dimensions are (58 x 14.5) cm2 . For each measurement both barriers and the trough are first cleaned with ethanol and rinsed with purified water. We use deionized Millipore water (Millipore Direct-Q) with a specific resistivity of 18.2 MΩ cm at 25 ◦C for rinsing and for the subphase. All isotherms were measured on a water surface and the purity of the surface was checked prior to each measurement. The iron oxide nanoparticles and composite samples were dissolved in chloroform (concentration c ≈ 5.0 mg/ml) and mixed before each measurement (Vortex 2000 mixer). For the composite samples, concentrations were cPnBA = 1 mg/ml : cFeNP = 1...3 mg/ml such that area fractions of nanoparticles spread at the interface are comparable for the systems (5 % particle coverage at maximum trough area). A specified volume V was spread in drops of 1.5 µl on the water subphase with a Hamilton syringe (maximum volume 5 µl). The chloroform evaporated quickly and the composite formed stable monolayers. All isotherms were measured after a waiting time of 30 min. The available surface for each object (i.e. nanoparticle or polymer molecule) is given by the mean molecular area mmA, which is defined as:

mmA =

AM cV NA

(1)

A is the actual area enclosed by the two barriers, M the molecular weight of the object and NA the Avogadro constant. In case of the polymer nanocomposites, mmA was calculated for the polymer chain solely in order to compare to the pure polymer isotherm. We subtracted the area occupied by nanoparticles from the area enclosed by the two barriers (A) to get the correct mmA for a polymer chain. The second parameter is the surface pressure Π given by

Π = γ0 − γ

(2)

where γ0 is the surface tension of the subphase (water) and γ the surface tension of the 5


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system after spreading the sample. The layers were compressed with a constant velocity of 435 mm2 min−1 . All isotherms were repeated at least three times to check for the reproducibility of the results.

X-ray reflectometry Experiments were performed on a modified D8 Advance reflectometer (Bruker AXS, Germany) measuring reflectivities in the θ − θ geometry. A conventional X-ray tube with a Cu anode (CuKα , wavelength λ = 1.54 ˚ A) is used to generate a X-ray beam with a line focus. The beam is monochromized by a Goebel mirror (W/Si multilayer mirror) and collimated through two narrow horizontal slits of 0.1 mm with an switchable absorber (calibrated Cu attenuator) in between. Soller slits (∆θx = 25 mrad) are placed after the last horizontal slit and directly in front of the detector. 29 Intensity is detected by a V˚ antec-1 line detector (Bruker AXS, Germany) providing the possibility to measure the specularly reflected intensity and the diffuse intensity simultaneously in an angular range of ∆θf = 2◦ for a given incident angle θi . A variable beam attenuator is used to protect the detector in the high intensity regime of the reflectivity curve. Data are corrected using the known attenuation factors. For each incident angle θi , a single intensity I(θf ) contains the specular and off-specular scattering. Reflectivity is calculated by integrating over a small window around the specular peak, while a constant background was determined as an average over a range of points on each side of the specular peak to account for a diffuse background. Data fitting is restricted to reflection angles above the critical angle.

Langmuir-Blodgett films LB films were produced on a Langmuir trough system (µTrough System Kibron Inc., Helsinki Finland). The silicon wafers were cleaned for 2 minutes in piranha solution prior to the transfer of the monolayers. Film transfer was performed with wafers oriented perpendicular 6


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to the air-water interface and a constant dipping speed of 2 mm/min. The area of the trough was automatically adjusted and monitored to guarantee a constant pressure during the transfer. This data shows that the monolayer was transferred when the silicon wafer was pulled out of the subphase (Z-type LB film) with transfer ratios between 80 − 100 %.

Small angle X-ray scattering SAXS was performed using a laboratory X-ray set-up. The Kα -line of a conventional copper X-ray tube with a wave length of λ = 1.54 ˚ A is used and further monochromated by a X-ray mirror. The point focused beam is collimated by three pinholes. The detectorsample distance is 1.5 m and calibration of the scattering vector q is done by measuring the first peak of silver behenate as a calibration sample. 30 The scattering vector is defined as |~q| = 4π/λ sin θ, with 2θ being the scattering angle. The scattered intensities are measured with a two-dimensional multi-wire gas detector (Molecular Metrology, 1024 x 1024 pixels). The overall instrumental resolution is estimated to be ∆q = 0.01 ˚ A

−1

by the full

width half maximum of the silver behenate peak during calibration. The accessible q range is 0.07 nm−1 ≤ q ≤ 2.5 nm−1 and the data was radially averaged because all scattering was isotropic.

Atomic force microscopy AFM images of polymer nanocomposite LB films were measured with a Cypher S AFM (Asylum Research Oxford Instruments) in amplitude modulation mode. Cantilevers (BudgetSensors and Olympus) with spring constants between 2 N/m and 3 N/m, resonance frequencies between 70 kHz and 75 kHz and tip radii smaller than 10 nm were used. As feedback signal, the ratio of setpoint and free amplitude was kept at 0.6 with a free amplitude of about 30 nm. Scan rates varied between 0.2 and 2 lines per second, depending on the scan size. Measurements were performed in air under ambient conditions. To correct for scanner drift, height images were second order flattened using the free software Gwyddion. 31 For further 7


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analysis, nanoparticles were identified via a height threshold of 4 nm.

Properties of iron oxide nanoparticles Three different sizes of iron oxide nanoparticles were obtained from Ocean Nanotech with diameters specified as 5nm, 10nm, and 20nm: FeNP5, FeNP10, and FeNP20. FeNP5 and FeNP10 form homogeneous Langmuir layer at the air-water interface in pure and nanocomposite films while FeNP20 does not. It has also been reported that particles of similar size tend to form 3D agglomerates at the air-water interface instead of uniform Langmuir layers. 32 In the following we will therefore focus on results obtained from the two smaller nanoparticles. Results referring to the largest particle are presented in the electronic supplementary information (ESI).

Size and shape of the nanoparticles Small angle X-ray scattering was performed on dilute solutions of the nanoparticles. The solvent was changed to toluene in order to avoid the strong X-ray absorbance of chloroform and diluted to low concentrations (c ≈ 1 mg/ml). Resulting scattering profiles with background scattering from pure toluene subtracted are shown in figure 1. For both particle sizes, the scattering at small q displays a plateau indicating single particle scattering. The scattering pattern from FeNP5 is fundamentally different compared to that from FeNP10. For FeNP10, several clear oscillations are visible and intensity decays much faster with q (I ∼ q−4 ) compared to FeNP5 (I ∼ q−2 ). The scattering from the large particles can be described perfectly by a spherical form factor with a polydisperse radius (polydispersity: PDI, details see ESI). This is not possible for FeNP5, due to the weaker decaying intensity. Different models to describe FeNP5 scattering have been tested (details see ESI), eventually a mixture of two particle sizes with narrow size distribution turned out to provide the best description of the scattering profile. This is also further supported by TEM investigation

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sample name FeNP5 FeNP10

fit type two sizes one size

diameter1 7.8 ± 0.1 nm 12.8 ± 0.1 nm

diameter2 4.8 ± 0.1 nm -

PDI 0.10 ± 0.03 0.10 ± 0.03

1

10

0

10 intensity in a.u.

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-1

10

-2

10

x5 -3

10

FeNP5 FeNP10

Figure 1: Small angle scattering data from the two different iron oxide nanoparticle sizes (FeNP5 and FeNP10). Very good agreement between fits and raw data is demonstrated using a polydisperse form factor of a sphere (FeNP10 single size, FeNP5 two sizes). The fit parameters for the models are summarised in the table above the figure.

-4

10

8 9

2

3

4

5 6 7 8 9

0.1

2

3

4

1 -1

q / nm

(image and analysis given in the ESI). The oleic acid shell is not considered in the fitting of both particle sizes, due to the low contrast between oleic acid and toluene. For both particle sizes a very good agreement between model and experimental data is found as demonstrated by the full black lines in the figure. The important parameters obtained from data fitting are summarised in the table above figure 1. The results agree well with the claimed sizes provided by Ocean Nanotech for the large particles. However, for FeNP5 we find a mixture of two sizes, which will introduce additional complexity for the Langmuir, XRR and AFM investigation presented in the following. Since all small angle scattering data could be described by form factors only, the nanoparticles have to be dispersed homogeneously in the solvent. Thus, changing the solvent from chloroform to toluene did not lead to aggregation of the particles.

Nanoparticle layers at the air-water interface Iron oxide nanoparticles with an oleic acid shell are capable of adsorbing at the air-water interface. Their ability to form stable monolayers depends on their size. 32 It is therefore 9


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ρ(z)

Figure 2: Sketch of a core-shell particle at the air-water interface. The scattering length density profile at the interface is illustrated with the particle being immersed into the subphase.

core shell particle

water

z important to determine the structure of the nanoparticle layers at the air-water interface with varying surface pressure before transferring the layer on a solid substrate. We use XRR to measure the scattering length density profile (SLD) ρSLD (z) with z varying perpendicular to the interface. The reflected intensity decays rapidly with scattering angle 2θ or q as soon as q is above the critical qc . For a sharp interface ρ(z) it is a step function and the intensity of the reflected beam is given by the Fresnel formula. For q qc this corresponds to a rapid decay of intensity ∝ q−4 . An existing interface modulates this decay. In order to emphasise this contribution of the interface to the intensity we present reflectivity data as Rq4 with R being the reflected intensity. For a given SLD profile the reflectivity curve may be calculated using the Parratt formalism. For the case of thicker films with a true multilayer structure data may be fitted directly with a multilayer model. 33 For the layer of nanoparticles we can calculate ρSLD (z) based on the model sketched in figure 2. SLD is averaged laterally such that the area densities for core and shell of the nanoparticles contribute to the SLD at a given z. Particles may be partly immersed in the water phase which is taken into account in the position of the water interface. We can then formulate a z-dependent SLD profile for a nanoparticle film at the air-water interface:

ρSLD (z) = Φcore (z) ρIO + Φshell (z) ρOA + (1 − Φcore (z) − Φshell (z)) ρa/w interface (z)

10


(3)

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where the packing fractions of core and shell are defined as

Φcore (z) =

    0    

2 rcore −(R−z)2 R2 

, Φshell (z) =

     0

   R2 −(R−z)2   R2    2 R2 −rcore 2 R 

    2 2   R −(R−z) R2

z ≤ dshell dshell < z ≤ 2rcore + dshell

(4)

z > 2rcore + dshell

with core radius rcore , shell thickness dshell , particle radius R = rcore + dshell and the packing fraction Φ = Φcore + Φshell with a a maximum of 0.91 for monodisperse hard spheres in 2D. Here, the scattering length density of the iron oxide core is ρIO = 44.16 · 10−6 ˚ A

−2

−2

and of the oleic acid shell ρOA = 8.5 · 10−6 ˚ A . The air-water interface is modelled as z−z0 with the error function erf(z), the interfacial 1 + erf √2σ ρa/w interface (z) = ρwater 2 water

roughness σwater and the position of the air-water interface z0 . Although this model already contains a considerable number of parameters, it is not able to describe all details in the experimental data. A realistic scattering length profile of the nanoparticle film is additionally smeared out due to particle size distribution, packing defects, uncertainties in positioning, penetration depth etc. 34 Therefore, we evaluate our experimental data in a two stage process. We start with a multilayer model and fit this to the data using a standard Parratt recursion scheme with incorporated Debye-Wallerlike factors. We then consider the resulting SLD profile within the physical model of the nanoparticle layer as described above. In general a large number of fit parameters bears the risk of receiving results with no physical interpretation. Results may depend on the chosen starting values and parameters may turn out to correlated. It is therefore important to analyse the statistics of the fitting parameters. To account for this, we apply a sophisticated fitting procedure relying on a Python package written for reflectivity analysis (refnx package 0.1.6 in python 3.6.3 ). 35 The program reconstructs the posterior probability distribution of our fitted parameters using Marcov chain Monte Carlo (MCMC) sampling. 35,36 Through this procedure we can obtain

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a probability distribution for each parameter within physically motivated boundaries. This provides us with a confidence interval as a measure for stability and quality of our results.∗ In a second step we then compare the SLD profiles obtained from the fit to the analytical model in equation (3) to see if we can characterise the film in terms of the parameters within the analytical model: particle size 2R, packing density Φ and position of the air-water interface z0 . Langmuir monolayers were prepared for both particles sizes at low surface coverage. The films were characterised in terms of surface pressure - area isotherms and in-situ specular reflectivities. We will first present our results for FeNP10 films. The surface pressure isotherm for FeNP10 is shown in the top left panel of figure 3. The film was prepared for low surface coverage (Π ≈ 1 mN/m) and upon compression surface pressure increases with an approximately linear slope. For small mmA ≤ 100 nm, we can observe a change in the slope with a strong increase in pressure. This change in the isotherm is accompanied by an increase of the elastic modulus E from 12 − 20 mN/m up to 75 mN/m which is in agreement with published data on iron oxide nanoparticles. 38 The modulus compares well with characteristic values of an oleic acid monolayer in the liquid-expanded state, 39 and the change in the slope can be understood as resisting interdigitated shell ligands of a dense nanoparticle layer. The structure of the film is now determined by specular reflectivity measurements for three different positions as highlighted in the isotherm of figure 3. Reflectivity curves are shown in the right panel of figure 3 and compared to the bare water surface. An interference pattern can be clearly identified compared to bare water, which becomes more distinct upon compression of the film. Reflectivity of pure water leads to a surface roughness of σwater = (0.26 ± 0.01) nm. Intensity of the primary beam was determined using the bare water surface and adjusted accordingly for reflectivities of the nanoparticle films. Missing data points in the raw curves are attributed to high intensities in the proximity of absorber ∗

MCMC is getting increasingly popular for statistically challenging problems and a brief introduction is found in this reference. 37

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20mN/m 12mN/m 2mN/m water

P1P

PP$QP2

5T4

6/'10 6Å 2

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Langmuir

GLVWDQFHQP

TQP 1

Figure 3: Langmuir isotherm of FeNP10 is presented in the top left panel as surface pressure Π vs mean molecular area mmA. Film structure was determined at three different positions in the isotherm. X-ray reflectivity curves are shown as Rq4 vs q in the right panel of the figure. The bare water surface is shown for comparison. A very good agreement between model (MCMC sampling) and experimental data is illustrated by the full line. The shaded black line represents the uncertainties while the dashed line is a continuation of the model to regions not considered within the fit. The results for the SLD profile are displayed in the bottom left panel. The dashed lines represent the analytical model from equation (3) fitted to the SLD profiles. switches. The fitting routine was limited to q ≥ 0.22 nm−1 and q ≤ 2.5 nm−1 . Data were fitted according to the procedure described above. We use a model of eight layers with equal thickness. In each layer, the SLD is assumed to be constant. A smooth transition of SLD between the layers is guaranteed by modulating the transition between inner neighbouring layers with the same Debye-Waller-like roughness factor. To account for the top and bottom roughness of the nanoparticle layer we include these as additional independent parameters in the model. The final number of fitting parameters is 12 (slab

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size & roughness, top & bottom roughness and 8 SLDs). MCMC sampling is performed to reconstruct the posterior probability distribution for each fitting parameter allowing us to monitor spread and stability of our results. The excellent agreement between model and experimental data is displayed in the right panel of figure 3. The median of the distribution is presented by the solid line and continued as a dashed line to the regions of the XRR curves not included in the fitting. Corresponding SLD profiles are shown in the bottom left panel. In both cases, the uncertainty interval is obtained from a randomly chosen set of samples from the MCMC and illustrated as shaded lines in black (black shadow around the fit). SLD profiles are now compared to the analytical model from equation (3). The dashed line in the bottom panel of figure 3 represents the best fit of the analytical model to the SLD profiles obtained by XRR. In the fit we assume an oleic acid shell thickness of 2 nm. It is remarkable how well the analytical model matches with our experimental SLD profiles. For all surface pressures we obtain a core diameter of (13.0 ± 0.2) nm for the nanoparticles, which is in excellent agreement with the result of 12.8 nm obtained from our SAXS experiments. Furthermore, particles are found to be immersed into the water subphase by 50 %, slightly increasing to 55 % with increasing packing density. The packing density Φ increases from 0.13 to 0.29 and 0.55. It approximately doubles between each measurement, which is in good agreement with the relative distance between the measurement positions in the isotherm. The left panel in figure 4 highlights the contribution of the core-shell particles and the airwater interface to the analytical model for FeNP10 at Π = 20 mN/m. The contribution of the air-water interface continuously decreases due to the z dependent area fraction of the particles. The sum of both contributions describes almost perfectly our experimentally fitted SLD profiles. A detailed look reveals small differences between the analytical and the fitted profile, especially around the edge of the particle. For small surface pressure, we can clearly observe a slightly increased density close to air (see figure 3), indicating small defects or single particles driven out of the monolayer. The overall picture of a densely packed particle layer is consistent, including the increase of surface pressure for small areas where shell

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FeNP10: = 20mN/m analytical model water surface particle

FeNP5: = 18mN/m analytical model water surface smaller particle larger particle

6/'10 6Å 2

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Langmuir

GLVWDQFHQP

GLVWDQFHQP

Figure 4: In the left panel we see the SLD profile of FeNP10 for Π = 20 mN/m obtained from fitting the XRR data compared to the analytical model of equation (3). We highlight the contribution of particles and the interface by plotting them as shaded areas beneath the fit. The right panel gives us the same plot for the SLD profile of FeNP5 for Π = 18 mN/m, right before the kink in the isotherm. Due to its size distribution, we have two different particles contributing to the analytical model. ligands begin to interpenetrate, which results in the strong increase of surface pressure. For the lower coverage, we attribute the increase in surface pressure to the resistance of loosely connected groups of particles sliding past each other. XRR measurements on FeNP5 films at the air-water interface are presented in figure 5. The isotherm was prepared in the dilute state as indicated by the constant pressure for large areas (mmA ≤ 110 nm). During compression, surface pressure increases again with a linear slope until a small kink appears in the isotherm at mmA ≈ 40 nm2 . The kink is accompanied by a maximum in the elastic modulus of E ≈ 48 mN/m indicating structural changes in the film. Specular reflectivity was measured for two positions before and one after the kink to determine SLD profiles and investigate the transition. Figure 5 presents reflectivity measurements for the three positions in the isotherm. A closer look at the raw data immediately reveals that additional interferences appear upon compressing the film. The biggest change in the data occurs when the kink in the isotherm is crossed (Π = 26 mN/m). We apply the same fitting routine as described for FeNP10 using a freely varying SLD profile with 8 slabs. The solid lines display a good agreement between

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Langmuir

P1P

26mN/m 18mN/m 10mN/m water

PP$QP2

5T4

6/'10 6Å 2

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GLVWDQFHQP

TQP 1

Figure 5: The Langmuir isotherm of FeNP5 is presented in the top left panel as surface pressure Π vs mean molecular area mmA. Film structure was determined at three different positions in the isotherm. X-ray reflectivity curves are shown as Rq4 vs q in the right panel of the figure. The bare water interface is shown for comparison. A very good agreement between model (MCMC sampling) and experimental data is illustrated by the full line. The shaded black line represents the uncertainties while the dashed line is a continuation of the model to regions not considered within the fit. The results are displayed in the bottom left panel. The dashed lines represent the analytical model from equation (3) fitted to the SLD profiles. our model and the experimental data. Again, uncertainties of the results are indicated by the shaded lines in black around the fit. The SLD profiles of our fit are plotted in the bottom left panel. The SLD profile for Π = 26 mN/m immediately reveals an increase in layer thickness when crossing the kink compared to the two measurements before the kink. To derive more detailed properties of the FeNP5 film structure, we try to do a fit of the analytical model from equation (3) to our SLD profiles with fixed shell thickness (dshell = 2 nm). However, it is more difficult to get a good description for FeNP5 than for FeNP10. Size and shape

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characterisation performed with SAXS and TEM (shown in ESI) experiments indicate a bimodal size distribution of the FeNP5 particles in our sample. If we consequently include the full size distribution obtained from SAXS and TEM (d1 = 4.8 nm und d2 = 7.8 nm, see ESI) in equation (3), we get a good match between model and SLD profiles, shown by the dashed lines in the figure. The right panel of figure 4 provides more details of the SLD profile at Π = 18 mN/m, right before the kink in the isotherm. We see the contribution of the two nanoparticle sizes centred to the same position. With the given size distribution, we get a densely packed layer of particles which are almost completely immersed into the water subphase. The combined packing fraction of the two particle sizes is 0.96 (we note: slightly above the limit for monodisperse hard spheres ∼ 0.91), indicating that particles are indeed densely packed in the system. These results clearly indicate that with further compression structural changes need to occur which result in the formation of a second layer. This is shown in the SLD profile at the highest surface pressure (Π = 26 mN/m). Therefore, we can assume that the kink accounts for the formation of a second layer within the film.

Polymer nanocomposite films Our characterisation of both particle sizes revealed that particles are arranged in a monolayer as long as the surface coverage is sufficiently low. Both are good candidates to form polymer nanocomposite films. In a previous publication, we presented a block copolymer PEG6 -bPnBA132 that forms very stable films at the air-water interface. 26 The polymer spreads as a thin layer (thickness around 2 − 3 nm) and has a very characteristic isotherm. A unique feature of the polymer layer is a wide plateau of constant pressure in the isotherm in which polymer chains dewet from the air-water interface. The 2D radius of gyration for one chain is approximately 60 nm2 which is comparable to the core size of the FeNP5 and FeNP10 particles (FeNP5: 18 nm2 or 48 nm2 and FeNP10: 129 nm2 ).

17


Langmuir

30 PEG6-b-PnBA132 NPC5 NPC10

/ mN/m

25 20

Figure 6: Langmuir isotherms presented as surface pressure (Π) vs. mean molecular area (mmA) for NPC5, NPC10 and the pure polymer film.

15

P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

10 5 0 20

40 mmA / nm

60

80

2

Composite solutions were prepared by adding polymer to iron oxide nanoparticles dissolved in chloroform. SAXS measurements of the form factor (data shown in ESI) demonstrate that addition of polymer does not lead to aggregation of the particles. The same particle form factors are observed as in the toluene solutions of the nanoparticles. The particle concentration for both composites was adjusted so that a coverage of 5 % of the area in a film is occupied by particles (before film compression starts). Composite films were prepared for low surface coverage and the mixture of FeNP5 will be referred to as NPC5, and NPC10 for FeNP10.

Nanocomposite film structure at the air-water interface Compression isotherms of the composites are shown in figure 6 and compared with the isotherm of the pure polymer layer. They are presented as surface pressure Π over the mean molecular area mmA for a polymer chain. For the composites, the area covered by nanoparticles is subtracted from mmA so that composite and polymer isotherms can be directly compared. Composite and polymer isotherms closely resemble each other, while we can clearly differentiate them from the pure particle isotherms (figure 3 & 5), thus the compression behaviour is dominated by the polymer. In detail, we can identify differences in the overall shape of the isotherms. For the composite films the surface pressure starts to increase already for larger areas compared to the polymer film. After reaching a critical 18


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surface pressure of ΠC ≈ 22 − 23 mN/m, a kink can be observed for the pure polymer and NPC5 film at almost the same value of mmA. Further compression of both films does not lead to any increase in surface pressure. In case of the pure polymer system, the constant pressure plateau has been identified as polymer dewetting the air-water interface. 26 The overall similarity between the pure polymer and NPC5 isotherms suggests that this dewetting mechanism is also found in this composite. No signs of multilayer formation are found in this isotherm. This is presumably due to the significantly reduced density of nanoparticles in the composite film, in comparison to the pure nanoparticle ones. For NPC10 the similarity to the pure polymer isotherm is not as obvious. In the low pressure regime (Π ≤ 15 mN/m) the shape of the isotherm is very similar to NPC5. However, there is no transition in form of a kink into a constant pressure plateau, instead the slope of the NPC10 isotherm flattens. No clear plateau is visible. The flattening of the surface pressure can also be interpreted in terms of the elastic modulus. The resistance upon compression decreases for NPC10 films for lower surface pressures. We can clearly identify this as an influence of FeNP10 particles, since it is not observed for NPC5. Structural properties of the layer are again determined using in-situ XRR reflectivity measurement at four positions in the isotherms for both composites. We use the same model as described for pure films and get a good agreement between model and experimental data. Details are given in the ESI. Here, we focus our discussion on a single SLD profile for each of the composites at a surface pressure of Π ≈ 20 mN/m. The SLD profiles are shown in figure 7. For an interpretation of the profiles we use the analytical model introduced in equation (3) and add a polymer layer on top of the water surface with a scattering length density of −2

ρPEG-b-PnBA = 10 · 10−6 ˚ A . 26 As in case of the pure nanoparticle layer, we need to assume a shell thickness of dshell = 2 nm and extend the model with the full size distribution of FeNP5 in case of NPC5. We get a nice agreement between model and the experimentally obtained SLD profiles. Thickness of both composite films is in agreement with our results on the pure particle films. For NPC10, we can obtain packing densities of Φ = 0.07, 0.12, 0.16 and 0.26

19


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6/'10 6Å 2

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NPC10: = 20mN/m analytical model water surface polymer particle

NPC5: = 19mN/m analytical model water surface polymer larger particle smaller particle

GLVWDQFHQP

GLVWDQFHQP

Figure 7: In the left panel we see the SLD profile of NPC10 at Π = 20 mN/m obtained from XRR fitting and compared to the analytical model of equation (3). We highlight the contribution of particles, polymer film and the air-water interface by plotting them as shaded areas beneath the fit. The right panel gives us the same plot for the SLD profile of NPC5 at Π = 19 mN/m, right before the kink in the isotherm. Due to its size distribution, we have two different particles contributing to the analytical model. which compare very well with the values calculated from the amount of sample spread at the interface. Furthermore, we can associate the abrupt decay of density visible in all profiles as the air/polymer interface. The thickness of the polymer film varies between 0.8 nm and 2 nm for all surface pressures which is consistent with results on the pure polymer film. Its contribution in the complete SLD profiles is highlighted in figure 7. The particles in both composites are immersed into the polymer film and the water subphase, as highlighted in the SLD profiles. Fully penetrate the polymer film and are immersed into the subphase by roughly 50 %, which is comparable to the one obtained for the pure nanoparticle films. The lateral distribution of nanoparticles within the film cannot be determined with specular XRR. To investigate the lateral distribution, we therefore transfer the composites on silicon wafer using the Langmuir-Blodgett technique. The pure polymer film can be transferred on a silicon wafer, 26 therefore we consider the technique suitable to investigate the lateral distribution of nanoparticles by ex-situ AFM experiments.

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Lateral distribution of nanoparticles in composite layers The film structure of both composites was transferred on silicon wafer using standard LB technique at a surface pressure of 15 mN/m. At this pressure, the compression behaviour of both composites is still very similar. The LB films were then investigated as prepared using atomic force microscopy (AFM). Results of the AFM measurements are shown in figure 8 for NPC5 (left panel) and NPC10 (right panel). Both images show an area of 22 µm x 22 µm while the maximum height of the observed structures is less than 30 nm. The soft polymer film is not visible in the AFM image. After subtraction of a constant background (3 nm to 4 nm), the vast majority of objects is around 8 to 13 nm (see figure 9). Particles are essentially confined in 2D on the silicon wafer. The lateral dimensions of the structures are a convolution of cantilever and object size, thus they appear significantly larger. 40 Therefore, lateral sizes have to be treated carefully and in our discussion we will refrain from using them in absolute units. There is a fundamental difference between the structure of the objects observed between NPC5 and NPC10 films. In case of NPC5, the lateral distribution of FeNP5 in the polymer matrix is given by many almost disk-like domains (left panel). The inset gives a high resolution scan of a selected region where single nanoparticles may be identified within the small clusters (see green circles). For NPC10, we observe chain-like structures in stages of different size (right panel). Obviously, the aggregation behaviour of the two nanoparticles is very different. To quantitatively investigate the different aggregation behaviour of NPC5 and NPC10, we analyse these images further. We first identify each connected domain with a height above 4 nm as one single object consisting of a number of particles. For every object its radius of gyration rg in pixels (1 px = b 30 nm) is calculated and compared to its area (defined by the number of occupied pixels N). Images from three different positions of the LB film were used to get reliable statistics. In total 4644 objects for NPC5 and 266 for NPC10 were considered. In the bottom of figure 8, we present the results from these considerations. There is a power law relation between the number of pixels N and rg (N ∼ rg d ) with an exponent d. 21


Langmuir

4

4

10

10

NPC5

NPC10

3

3

10

N ~ rg

10

2

10

1.13 ± 0.03

1.97 ± 0.01

N / px

N ~ rg N / px

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1

2

10

1

10

10

0

N ~ rg

1.74 ± 0.01

0

10

10 1

10

1

rg / px

10

100

rg / px

Figure 8: Representative atomic force microscopy images (22 µm x 22 µm) for NPC5 and NPC10 LB films. High resolution scans of characteristic domains are shown as inset. The green circles in the high resolution scan of NPC5 highlight a set of single particles within the domains. The bottom panel shows a fractal analysis in which images from three different regions were used for both systems. For every identified object its area (given by the respective number of pixels N ) is plotted versus its radius of gyration rg . The solid lines are power-law fits to the data. We identify a single power law for NPC5 for the full range of observed aggregate sizes while there is a crossover between two different exponents for NPC10. The exponent of the power law is the fractal dimension d of the structures. The upper limit of d for 2D structures is d = 2 (compact object) and d = 1 is the lower limit (straight line). We note that both limits are logical boundaries, however the upper one can be seen in the data on NPC5. The fractal

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dimension of d NPC5 = 1.97 ± 0.01 confirms our initial description of almost disk-like objects for FeNP5. In case of NPC10, we find two different power laws. For large structures the fractal dimension is d NPC10, large = 1.13±0.03, close to the limit of a straight line. This agrees with the observation of chain-like structures in the FeNP10 layer. However, it is interesting that we observe a crossover of the power law for small domains (rg < 9 px). Small domains tend to be less extended with a fractal dimension of d NPC10, small = 1.74 ± 0.01. Particles with dominating magnetic interaction are expected to form chain like structures, with dipole moments aligned parallel along the chain. Although predicted theoretically and found in many simulations, isolated chains in zero field are rarely observed experimentally. 27,32,41–44 So far, clear evidence has only been obtained by vitrified cryo-TEM samples. 21,22 In order to determine the magnetic properties of our nanoparticles we have obtained magnetisation curves on dried powder of FeNP10. Details are given in the ESI. Due to its non-vanishing remanent magnetisation, we identify FeNP10 as ferromagnetic with a magnetic moment of mparticle = 0.13 A nm2 per particle (obtained from its saturation), which is comparable to values measured for iron oxide nanoparticles (diameter of 18.7 nm). 40 Twodimensional simulations of aggregating magnetic particles indeed find a fractal dimension of d = 1.13 ± 0.01 for large cluster sizes (several thousand particles). 27 This result nicely agrees with our finding for large cluster sizes even though the magnetic dipolar interaction energy between two neighbouring particles is about four times larger than thermal energies. In pure iron oxide nanoparticle films, it is not possible to observe nicely separated aggregates of magnetic particles. 44 Therefore, we assume that the presence of the polymer matrix to be important. A particle approaching a large cluster is driven by Brownian motion, however at the same time it is influenced by the overall magnetic dipole interaction of the cluster. The energetically most favourable position for the particle to attach to the cluster is at either end of the chain. Due to the high viscosity of the polymer, the diffusion coefficient of the particle is low. The diffusion coefficient is reduced by at least an order of magnitude as shown in the

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microrheology study on poly(tert-butyl acrylate) monolayers at the air-water interface using particle tracking. 45 As a consequence, the trajectory of the nanoparticle can be affected more efficiently by the dipolar interaction with the cluster, thus it is more likely moving towards the energetically most favourable position. Although this argument applies to all cluster sizes, we observe a different power law for small clusters. The fractal dimension d = 1.74 ± 0.01 is close to the value for diffusion limited aggregation, DLA (d = 1.715 ± 0.004). 27,46,47 Particle dynamics in diffusion limited aggregation are fully driven by Brownian motion. Particles do not interact with each other unless they attach. Once attached, they are permanently bound to the cluster. For large cluster sizes, we see a dominating magnetic dipolar interaction between a nanoparticle and already existing clusters. The magnetic dipolar interaction with a small cluster is weak. Therefore, even the reduced diffusion coefficient is not enough to allow for the nanoparticle to be significantly influenced by the magnetic dipolar interaction. The same effect is reproduced in 2D Monte Carlo simulations of a comparable system. 27 Smaller clusters tend to exhibit a fractal dimension close to the DLA case. Turning to the smaller nanoparticle FeNP5. In measuring the magnetisation curves, we find that FeNP5 has no remanent magnetisation (see ESI), thus it is superparamagnetic. 48 As there is no magnetic driving force, they do not form chain-like aggregates. However, the fractal dimension of FeNP5 clusters is almost two, significantly higher as expected for DLA. An explanation can be given by a restructuring process during aggregate formation. After attaching to the cluster, particles are able to slightly move towards the energetically most favourable position. For small clusters this leads to a significant compactification. 28 Here, the broad size distribution of particles may be an important reason for the restructuring to take place. The observation on aggregation can also explain the different compression behaviour of the two polymer nanocomposite films. The isotherms for both composites are almost identical up to a surface pressure of 15 mN/m, thus the compression behaviour seems to be polymer dominated. The LB films investigated by AFM are prepared precisely at this point.

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0.30 NPC5 NPC10

0.25 0.20 p(height)

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Langmuir

0.15 0.10 0.05 0.00 6

8

10

12

14

16

18

20

height / nm

Figure 9: Height distribution of identified objects for NPC5 and NPC10 LB films. For the anisotropic FeNP5 a bimodal distribution is found, while FeNP10 exhibit a single maximum. Full lines represent guides for the eye. For NPC10 the randomly oriented nanoparticle chains can be deformed thus leading to a weaker increase in surface pressure. The compact aggregates in NPC5 can not be further deformed. Their interaction with the polymer matrix is determined by the interaction of the oleic acid molecules and for the investigated concentration of particles compression behaviour is still dominated by the polymer. We now turn to the height distribution of the aggregates in the film. We calculated the height distribution by extracting the maximum height for each object in both composites. The same three images per particle size as for the fractal analysis were used. Results for the normalized height distributions are shown in figure 9. We observe a single peak for NPC10 with its maximum at 13.3 nm. The great majority of objects in the composite films possess a maximum height identical to the core diameter obtained from SAXS and XRR profiles (see figure 1 and 3). For NPC5, a bimodal distribution is observed with maxima at 8.6 nm and 11.7 nm. The two values match very well with the film thickness obtained for pure FeNP5 films before and after the kink in the isotherm. Although in-situ XRR on composite films does not show the formation of a second layer, the appearance of this

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bimodal distribution is attributed to this. By using the maximum height within each object, a single particle lifted up into a second layer is sufficient to contribute to the higher peak in the height distribution. Therefore, we believe that the higher peak is contributed to a small number of particles being in a second layer. This idea is supported if we consider the mean height for each object. Obviously, the distribution shifts to smaller heights (spherical shaped particles) but additionally we observe that the bimodal distribution is smeared into a single peak. The peak in the mean distribution is exactly in between the two sizes used in our SAXS characterisation.

Conclusion Iron oxide nanoparticles form homogeneous layers at the air-water interface and mixed homogeneous layers with polymer. FeNP10 are shown to be spherical with core diameter of 12.8 nm and narrow size distribution. They are ferromagnetic with a permanent magnetic moment of mparticle = 0.13 A nm2 per particle. For FeNP5, SAXS and TEM results lead to a broader size distribution. A superparmagnetic behaviour is found for FeNP5. The largest particle FeNP20 does not form stable homogeneous monolayer at the air-water interface, 32 and shows the strongest ferromagnetic moment mparticle = 0.63 A nm2 . This may be the reason for its inability to form Langmuir monolayers. Film structure and compression behaviour of FeNP5 and FeNP10 can be nicely characterised with consistent results between Langmuir isotherms, in-situ XRR profiles and SAXS measurements. For both particles sizes, we find dense monolayers with particles immersed by 50 % into the water subphase. With compressing the FeNP5 monolayer even further, multilayer formation is observed. FeNP5 and FeNP10 can be introduced in a thin polymer matrix and form stable polymer nanocomposites at the air-water interface. In-situ XRR profiles reveal packing densities varying between 0.07 and 0.26. We identify a thin polymer layer with thickness ranging

26


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Langmuir

between 0.8 nm and 2 nm. Particles are immersed in the polymer and water interface again with about 50 % of its size. On LB films, the height distribution of aggregates seen by AFM reproduces the particle sizes measured by SAXS and in-situ XRR. The superparamagnetic FeNP5 forms small compact aggregates within transferred structures (LB film). The formation of these aggregates can be explained by diffusion limited aggregation (DLA) in the limit of small cluster sizes with dominating restructuring effects. 28 In contrast, the ferromagnetic FeNP10 forms open chain-like aggregates. We associate the formation of chain-like aggregates with magnetic moments of the particles being aligned along the chain due to magnetic dipole-dipole interaction. The existence of such randomly oriented chains of magnetic particles has been predicted by de Gennes theoretically even in the absence of an external magnetic field. 41 Their existence has also been observed in magnetotactic bacteria as linear chains of magnetic nanoparticles coated with a lipid biomembrane. 49,50 2D simulations find a fractal dimension of d = 1.13 for sufficiently large aggregates for particles with dominating magnetic interaction. Smaller aggregates display a diffusion limited aggregation mechanism (d = 1.74). 27,42 Although thermal energy should be dominating in our system, results nicely agree with these predictions. We suggest that the reason for this is the presence of the polymer matrix providing a highly viscous environment for the nanoparticles. 45 Therefore, nanoparticles are more easily guided towards the energetically most favourable position in the cluster. For small cluster sizes the reduced diffusion coefficient of the particles can not compensate the small magnetic field provided by the aggregate.

Supporting information The supporting information contains form factor calculations, model discussions and TEM measurements of FeNP5, full characterisation of FeNP20 particles and films at the air-water interface, XRR data and scattering length density profiles for both composites (NPC5 &

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NPC10) and magnetisation curves on dried powder from all particle sizes.

Acknowledgements The authors gratefully acknowledge Professor Dr. Oliver Gutfleisch and his group for performing the magnetisation measurements.

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(8) Jiang, N.; Endoh, M. K.; Koga, T.; Masui, T.; Kishimoto, H.; Nagao, M.; Satija, S. K.; Taniguchi, T. Nanostructures and Dynamics of Macromolecules Bound to Attractive Filler Surfaces. ACS Macro Letters 2015, 4, 838–842. (9) Starr, F. W.; Douglas, J. F.; Meng, D.; Kumar, S. K. Bound Layers Cloak Nanoparticles in Strongly Interacting Polymer Nanocomposites. ACS Nano 2016, 10, 10960–10965, PMID: 28024345. (10) Zhou, J.; Ralston, J.; Sedev, R.; Beattie, D. A. Functionalized gold nanoparticles: Synthesis, structure and colloid stability. Journal of Colloid and Interface Science 2009, 331, 251 – 262. (11) White, R. J.; Luque, R.; Budarin, V. L.; Clark, J. H.; Macquarrie, D. J. Supported metal nanoparticles on porous materials. Methods and applications. Chem. Soc. Rev. 2009, 38, 481–494. (12) Jain, P. K.; Huang, X.; El-Sayed, I. H.; El-Sayed, M. A. Noble Metals on the Nanoscale: Optical and Photothermal Properties and Some Applications in Imaging, Sensing, Biology, and Medicine. Accounts of Chemical Research 2008, 41, 1578–1586, PMID: 18447366. (13) Mayer, K. M.; Hafner, J. H. Localized Surface Plasmon Resonance Sensors. Chemical Reviews 2011, 111, 3828–3857, PMID: 21648956. (14) Leslie-Pelecky, D. L.; Rieke, R. D. Magnetic Properties of Nanostructured Materials. Chemistry of Materials 1996, 8, 1770–1783. (15) Skomski, R. Nanomagnetics. Journal of Physics: Condensed Matter 2003, 15, R841– R896. (16) Gao, J.; Gu, H.; Xu, B. Multifunctional Magnetic Nanoparticles: Design, Synthesis,

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and Biomedical Applications. Accounts of Chemical Research 2009, 42, 1097–1107, PMID: 19476332. (17) Bedanta, S.; Kleemann, W. Supermagnetism. Journal of Physics D: Applied Physics 2008, 42, 013001. (18) Koplovitz, G.; Leitus, G.; Ghosh, S.; Bloom, B. P.; Yochelis, S.; Rotem, D.; Vischio, F.; Striccoli, M.; Fanizza, E.; Naaman, R.; Waldeck, D. H.; Porath, D.; Paltiel, Y. Single Domain 10 nm Ferromagnetism Imprinted on Superparamagnetic Nanoparticles Using Chiral Molecules. Small 2019, 15, 1804557. (19) Luo, W.; Nagel, S. R.; Rosenbaum, T. F.; Rosensweig, R. E. Dipole interactions with random anisotropy in a frozen ferrofluid. Phys. Rev. Lett. 1991, 67, 2721–2724. (20) Teixeira, P. I. C.; Tavares, J. M.; da Gama, M. M. T. The effect of dipolar forces on the structure and thermodynamics of classical fluids. Journal of Physics: Condensed Matter 2000, 12, R411–R434. (21) Butter, K.; Bomans, P.; Frederik, P.; Vroege, G.; Philipse, A. Direct observation of dipolar chains in iron ferrofluids by cryogenic electron microscopy. Nature Materials 2003, 2, 88–91. (22) Klokkenburg, M.; Dullens, R. P. A.; Kegel, W. K.; Erné, B. H.; Philipse, A. P. Quantitative Real-Space Analysis of Self-Assembled Structures of Magnetic Dipolar Colloids. Phys. Rev. Lett. 2006, 96, 037203. (23) Santoyo Salazar, J.; Perez, L.; de Abril, O.; Truong Phuoc, L.; Ihiawakrim, D.; Vazquez, M.; Greneche, J.-M.; Begin-Colin, S.; Pourroy, G. Magnetic Iron Oxide Nanoparticles in 1040 nm Range: Composition in Terms of Magnetite/Maghemite Ratio and Effect on the Magnetic Properties. Chemistry of Materials 2011, 23, 1379–1386.

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Magnetic ordering in 2D polymer nanocomposite films 4

10

3

d ~ 1.13 dipolar ordering

10

N / px

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2

10

d ~ 1.74 diffusion limited

1

10

0

10

1

10

100

rg / px

Fractal dimension d perfectly matches with 2D simulation

Figure 10: For Table of Contents Only

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Structural Properties and Magnetic Ordering in 2D Polymer

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