Modelling of Aggregation of Nanoparticles and its effect on their Structural and Biological Functions

: Nanoparticles have applications such as drug delivery and cancer treatments, reinforcement of the polymer or metal matrix, consumer products and environment. This work concentrates on how aggregated nanoparticles might realistically effect performance of the intended structural or biological function. As a conceptual basis, primary aggregation is assumed to produce the backbone of micro-structures which then cluster, covering a large portion of the material. This process is assumed to be chaotic and to occur rapidly. Molecular dynamic analysis of this aggregated model is difficult because the problem is not clearly bound and regions not spatially defined. Moreover the modulus of the micron-sized aggregate within the cluster is also difficult to measure directly. Instead an indirect method is developed of the polymer/particle interface in the aggregate which can be verified by bulk modulus experiments on nano-composite samples produced specifically for this work. A computer program equates minimum free-energy of the absorbed polymer molecule to dipolar interaction energies having a Boltzmann’s Distribution. Fractal numbers are used to characterise the molecular/particle interface and configuration of the aggregate backbone. After the principle has been established it is extended to other applications for example how aggregation might effect the probability of release of artificial DNA from silica nano-particles within the body


Introduction
In this work two different applications of nanoparticles are examined for the effect of agglomeration. One application is reinforcement filler in a polymer nanocomposite and the other is as means of delivering artificial DNA (CpG) to stimulate the immune system in diseased target cells. It may be conceptually easier to consider the nanoparticles as separate entities however this condition does not normally exist during processes such as inculcation or injection in the body. Once any electrostatic repulsion between the particles, if it exists, has been overcome then van der Waal's forces dominate often irreversibly. In both examples experimental data are compared to results from a theoretical analysis. The purpose of the analysis is to determine the effect of agglomeration on performance. Molecular dynamic modeling is a powerful tool in simulations of material interaction [1,2,3], however, full scale molecular dynamic modelling of the aggregated model is difficult because the problem is not clearly bounded, and regions are not spatially defined. Also such simulation would also be computationally expensive [4].The method of analysis in this work depends on the application.
In the case of the nanocomposite filler, a Cluster-Cluster Aggregation (CAA) model is assumed, in which primary clustering produces short fibre-like groupings. These have a backbone of nanoparticles connected by a flexible nanoscopic bridge of glassy polymer between the nanoparticles. A computer simulation based on fractal properties of the particle estimates the modulus of this primary grouping. Then, because nanoparticle content is high (40% by weight) the composite consist of secondary clustering. Linear approximations are used to estimate the secondary modulus and hence elastic modulus of the entire nanocomposite. This value is compared to elastic modulus obtained from particulate (separated) and fibre filler models.
For the nanoparticle application where they are required to deliver artificial DNA (CpG), the measure of performance is as probability of release for theoretical separated particles, compared to experimentally derived agglomerated particles. The rate of release must be sufficient to ensure that CpG is released in the vicinity of the target cell but not so great that all the CpG load is released before it reaches the target.
2 Agglomeration of the Nano-filler in the Polymer Matrix This section describes the preparation of a nanocomposite with agglomerated filler, experimental estimates of its composite strength and comparison of values with those from a theoretical analysis.

Preparation of Nanocomposite Samples
It was important to devise a method for producing the nanocomposite that ensured agglomeration of the nano-filler after inculcation in the polymer matrix. Basically this was done by directly mixing a colloid suspension of nanoparticles with a polymer solution. The method is fully described elsewhere [5] but is now explained briefly: Powdered PVC was added to DMF in a mass ratio of 25:75. The mixture was manually stirred, at 2 hourly intervals, over a period of 10 hours until there was no discernible solid in the mixture. A known weight of the polymeric mixture was placed in a glass-dish. The colloid suspension with approximately 30% mass-ratio of silica nanoparticles was then poured into the container to immerse the sample. The sample remained immersed for 11 hours during which time the DMF solvent diffused out of the polymeric mixture and the aqueous suspension of Silica Nano-Particles diffused in. The sample was then removed dried in a warm air-flow and weighed. The micrograph in Fig 1 shows a porous microstructure. This porosity is also described in the literature [6,7]. The large internal surface area due to porosity is likely to increase uptake of nanoparticles, which in these experiments reached a maximum of 40% by weight. For compression testing purposes the sample porosity was later removed by re-dissolving in DMF and allowing it to solidify by evaporation. Table 1 shows the measured weights of nanoparticles in three samples produced with different concentrations (8%, 18% and 30%) in the precipitation bath.

Compression Tests and Results
The samples in Table 1 were subjected to compression testing which was repeated until Young's Modulus had the same value following two consecutive tests. The Young's Modulus was estimated from the linear part of the stress/strain curve. Composite moduli of the samples are shown in Table 2 together with percentage weight and volume of nanoparticles. The volumetric percentage of nanoparticles in the sample was calculated using a Silica density of 2.65 g/cm 3 and a PVC density of 1.3 g/cm 3 . Fig. 2 shows composite modulus (Ec) plotted against filler fraction (by volume). Also plotted are the theoretically derived values for spherical particulate and fibrous composites based on the Halpin-Tsai equations [8]. Trial values of filler strengths (Ef) were substituted into the formulae; 1000 MPa and 700 MPa respectively for spherical and fibrous filler. Fig. 4 shows that the experimental sample modulus is greater than that of the spherical particulate composite over the  [9,10] for dry mixed silica nanoparticles with polyimide and nylon matrix; composite strengths were much greater than that obtained with dispersed spherical-particulate filler.
Using the Halpin-Tsai equations, neither the spherical model nor the fibrous models can predict the effects of the filler volume fraction on composite modulus. A new approach based on a Cluster-Cluster Aggregation (CCA) model is presented.

Analysis of the elastic properties based on a Cluster-Cluster Aggregation (CAA) model
Nanofiller aggregation has been proposed as a reinforcing mechanism in nanocomposites [11]. The notion is that aggregated particles form fillers that are geometrically similar to a micro-fibre; hence greater reinforcement. A Cluster-Cluster Aggregation (CCA) model of the filler microstructure [12] is used as a conceptual basis and is illustrated in Fig. 3. According to the CCA model, clusters are formed from strongly bonded primary aggregates of nanoparticles as shown in Fig. 3a. In this work the modulus of the primary aggregate bond is estimated using a computer-based model.

Primary Aggregation of Filler Reinforcement
The basic concept for primary bonding (shown in Fig. 3a) is illustrated in Fig. 4 [14]. The strength of a polymer/nanoparticle bond depends on its location within the cluster. The bond strength within a confined region between the nanoparticles is greater than that at the cluster/matrix boundary. A flexible nano-scopic bridge of glassy polymer exists between the nanoparticles. This bridge effectively provides a backbone to the primary aggregate structure.
A MatLab computer model was used to compute the filler-filler bond modulus between nanoparticles i.e. modulus of the nanoscopic bridge described above. The model equates dipole interaction energy using two equations, each having a different theoretical basis. One of the equations is derived from minimization of free-energy of the absorbed polymer molecule which is treated as a Gaussian Chain [15]: Where: Gp is the modulus of the filler-filler bonds (immobilised polymer) (GPa) Rv is the mean height of the bound polymer molecule above the particle surface (nm) ds is surface fractal dimension of nanosilica b is the effective length of the Carbon-Carbon bond in the PVC backbone≈ 0.126 (nm) Nb/N is the ratio of bound monomers to unbound monomers The other equation is the so-called Keeson Equation for the interaction energy between dipoles having configurations with a Boltzmann's Distribution [16]. The computer algorithm, iteratively calculates the interaction energy using the Keeson Equation for each dipole interaction between the absorbed polymer-molecule and nanoparticle. The simulation stops when the iterative total equals the interaction energy obtained from the free-energy equation. Convergence is achieved when the number of iterations equals the guessed number of bound to unbound monomers Nb/N.
The value of surface fractal dimension ds has a profound effect on bond modulus due to the exponential function in Equation 1. For a flat surface ds = 2 and for a Brownian surface ds = 2.5. The glassy-polymer bond between particles is assumed to be a space-filling surface where both particle surfaces are in contact with the same polymer molecule. This is represented by a limiting value of ds ->3; for simulation purposes ds≈2.95. The thickness of the immobilised polymer layer Δ (bond gap size) is most commonly taken to be ≈2nm [17]. At this thickness and a value of ds =2.95 the calculated filler-filler modulus is 11.5 GPa. This modulus is near the upper limit of modulii for crystalline polymers [18]. Results from the computer simulation for filler-filler bond modulus are shown in Table 3.

Composite Modulus for Filler Fractions greater than the Mechanical Gel-Point (φ > φG)
For composites with filler fractions greater than the gel-point, it is assumed that the clusters are connected and start to form the network as shown in Fig. 3(b). The cluster backbone then becomes an essential factor in the estimation of cluster modulus. The formula used to calculate composite modulus (Gc) takes the form of filler-filler bond modulus (Gp) multiplied by filler fraction (φ) with an exponent that involves the size and geometrical structure of the cluster and cluster backbone [19,20,21]: Where: Gp is the modulus of the filler-filler bond d is the particle diameter Δ is the thickness of the immobilised polymer layer shown in Fig. 4 φ is filler fraction dfc is the fractal dimension of the cluster dfb is the fractal dimension of the cluster backbone Trial values of dfb and dfc were substituted into Equation (2). For example a straight backbone would have a theoretical (dfb) ≈ 1 corresponding to a cluster value (dfc) < 2 [22].

Model for Filler Fractions less than the Mechanical Gel-Point (φ < φG)
For composites with filler fractions less than the gel-point it is assumed that the clusters are unconnected as shown in Fig. 3 Radius of gyration Rg for the cluster [23] is scaled from Small Angle Neutron Scattering (SANS) data for 9.8 nm radius silica particles using the fractal dimension for the data set corresponding to the filler fractions φ used in the experimentation. Table 4 lists the results from calculations using Equations (2) and (3). Results from calculations using Equation (2) are in good agreement with experimental data for the highest filler fraction φ = 0.2. To obtain this agreement, it was necessary to assume a value for the exponent = 2.97 which is possible with dfb = 1 and dfc = 1.65. These are minimal values that imply that the material has a very simple cluster structure with a straight backbone. Results from calculations using Equation (3) are in good agreement with experimental data at most filler fractions. There is a discrepancy at the lowest filler-fraction (=0.054) with a +10% difference.

Results from theoretical calculations and comparisons with experimental data
The results in Table 4 and Fig. 2 suggest that lower filler fractions (0.054 and 0.1) are likely to be below the mechanical gel-point. The highest filler fraction (φ = 0.2) is likely to be a transition point; at or near the gel-point. This is inferred from the fact that both Equation (2) (for φ > φG) and Equation (3) Table 4 Comparisons between predicted composite modulus (Gp) using equations 2-3 and experimental data.
body's immune mechanisms. It has been shown that when CpG ODN is attached to the surface of porous nanoparticles MSN, they are protected from this attack and can thus be delivered to targets cells. A CpG ODN delivery system has been developed [24] by binding CpG ODN noncovalently onto the modified surface of the MSNs. However, to optimise the induction rate the CpG ODN/MSN system must be designed with binding energy which is sufficient to carry the CpG load to the target cell, yet not so strongly bound that it cannot release the CpG in the vicinity of the TLR9 receptor. In this case it is assumed that Silica Nanoparticles are initially agglomerated based on micrographic evidence in Fig. 5

Analysis of Experimental Data for agglomerated CpG separation from the MSN
The data was expressed as a probability density distribution. Basically, this involved obtaining a mathematical function by curve fitting for each set of data. This mathematical function was then differentiated to determine the release rate over each time interval. Multiplying the release rate by time interval gave the mass released. The probability density of free dissociation was then obtained from the ratio of the mass released and the mass remaining on the aminated MSNs. This was then expressed as a cumulative probability versus time. The results are shown as bluediamonds on the graph in Fig. 6.

Theoretical Simulation of the Non-agglomerated CpG separation from the MSN
In order to compare the agglomerated experimental data with a non-agglomerated state, the CpG is assumed to be connected to the MSN by uninhibited single bonds with MSNs nonagglomerated. The probability density of release is then equal to the failure probability of a single bond with no force acting. Bell's Reliability Theory [25] provides Equation 5:   The graph shows that the hypothetical non-agglomerated bonding has a greater probability of dissociation. This is to be expected since the release would be less inhibited. However, despite the large conceptual difference between the two release mechanisms the variation is relatively small; 5% after 5 hours, 15% after 15 hours and 17% after 24 hours.

Conclusions
Two applications of silica nanoparticles were considered: a nanocomposite and a delivery system for artificial DNA (CpG). In both applications experimental data was available for functioning of the agglomerated nanoparticle state. In this work a theoretical analysis is made of each case in order to gain insight into how agglomeration effects function.
For the nanocomposite, experimental results are not explained by assuming a separated nanofiller.
Instead an analytical methodology based on a cluster-cluster model together with a model of the polymer chain-nanoparticle interface is successfully used to estimate composite modulus when compared to experimental data. The nanocomposite with agglomerated filler has a greater modulus than that of the separated filler. A method is presented to achieve agglomeration with very high particulate content (≈ 40% by weight); without aggregation.
In the case of the agglomerated CpG/MSN delivery system, a relatively simple analysis compares probability of dissociation of CpG from MSN derived from experimental results to failure probability of an non-agglomerated bond. As expected the nom-agglomerated bond has a higher probability of dissociation (failure) but the difference is not great, reaching a maximum of 17% after 24 hours. This would suggest that even if it was possible a non-agglomerated system would have marginal advantages.