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New Field Model of Polymer/Nanoparticle Mixture-Realizing Discreteness in the Continuous Description

Field-theoretical method is efficient in predicting the assembling structures of polymeric systems. However, for the polymer/nanoparticle mixture, the continuous density description is not suitable to capture the realistic assembly of particles, especially when the size of particle is much larger than the polymer segment. Here, we developed a field-based model, in which the particles are eventually discrete and hence it can overcome the drawbacks of the conventional field descriptions, e.g., inadequate and crude treatment on the polymer-particle interface and the excluded-volume interaction. We applied the model to study the simplest system of nanoparticles immersed in dense homopolymer solution. Our model can address the depletion effect and interfacial interaction in a more delicate way. Insights into the enthalpic and/or entropic origin of the structural variation due to the competition between depletion and interfacial interaction are obtained. New phenomena such as depletion-enhanced bridging aggregation are observed in the case of strong interfacial attraction and large depletion length. This approach is readily extendable to studying more complex polymer-based nanocomposites or biology-related systems, such as dendrimer/drug encapsulation and membrane/particle assembly.


INTRODUCTION
Addition of nanofillers to polymer materials has long been a practical approach to fabricate flexible composites with enhanced mechanical, electrical or optical properties. [1][2][3][4][5][6] Understanding and controlling the formation of the assembling structures of polymer nanocomposites (PNCs) are the keys to designing and realizing the desired macroscopic performance. Many theoretical and simulation methods have been developed  to study the assembling behaviors of PNCs. The self-consistent field theory (SCFT) is one of the powerful approaches to investigate the mesoscopic structures of multicomponent polymeric systems. [32][33][34] Hence, efforts have been made to establish a field-based model for PNCs. 18,22,29,30 The challenges facing the incorporation of particles into the field-based model of polymers are how to treat the interface between the polymers and the particles and how to address the strong excluded-volume (EV) interactions both between the polymers and the particles and between the particles. The hybrid theory proposed by Thompson and coworkers couples the SCFT for the polymers with the density functional theory for the particles. This hybrid method has obtained many successes in predicting the ordered structures of diblock copolymer/nanoparticle composites. 18,30,[35][36][37] But it's well understood that the EV and the interfacial interactions between the polymers and the particles are not appropriately considered in this approach. Another type of hybrid method combines the field description of the polymers and particle-based Brownian dynamics (BD) simulation of the particles. 22,38,39 The explicit particle coordinates potentially enable the method to overcome the challenges mentioned above. However, the coupling between the SCFT iterations and BD motions becomes a tricky problem. One major difficulty is how to determine the force on the particles exerted by surrounding polymers. Sides and coworkers 22 use the explicit partial derivative of the Hamiltonian to the particle positions as the coupling force on the particles, which is questionable in the sense of the quasi-static approximation, i.e. the polymer matrix is in "equilibrium" between two successive steps of BD motion of the particles. Note that the auxiliary fields (ω 's) in SCFT are implicit functions of the particle positions. Total derivative of the Hamiltonian to the particle positions is more appropriate instead, while the expression of which is not available. This issue may not affect the result seriously since the most concerned aspect is the equilibrium assembling structure, not the dynamics. Another issue of the SCFT/BD method is that the alternating SCFT iterations and small-step BD motions make it very computational. To avoid the complication of considering the EV interactions in the polymer/multi-nanoparticle assembling system, some works focused on the dilute limit (one or two fixed particles) 28,40 or made a simplification by invoking the concept of effective polymer concentration. 41 Here, we developed a field-based model of PNCs. Unlike the conventional field models 18,29,30 in which particles are described by continuous density distribution, the particles in our model are eventually discrete as in the particle-based description, i.e. the particle density in our model no longer represents the ensemble-averaged density distribution of particles. The goal of our model is to obtain the ensemble-averaged density distributions of the polymers with discrete nanoparticles located at the "most probable" positions, hence the structure we predict is more like an instantaneous pattern of the system. The motivations behind are three folds: 1) patterns with instantaneous locations of particles can better exhibit the delicate structures (especially when the particles are packed); 2) discrete description of particles allows delicate treatment on the EV interactions and the polymer-particle interface; 3) due to 2), the evaluation on the polymer-particle interfacial energy is more appropriate and the contribution of the depletion effect can also be involved. We applied this model to study the simplest system of nanoparticles immersed in dense homopolymer solution or melt. We focused on the structural variations under diverse polymer-particle interfacial interactions and depletion lengths.
The paper is organized as follows: we describe the basics of our model and method in Sec. II and more detailed equations are given in the Appendix. Results are analyzed and discussed in Sec. III. We briefly conclude in Sec IV.

II. MODEL AND METHOD
We consider a system consisting of a mixture of p n (variable) homopolymers and c n spherical solid particles. The radius of all particles is c R . The coarse-grained polymer chain is composed of N segments of size σ . The bonded interactions of Gaussian chain are quantified by an elastic potential energy: is the number density of particles whose surface profile is smoothed by the hyperbolic tangent function for numerical efficiency. Note that, principally, the concentration and density quantities in the above and the following energy expressions, etc. should be operators or instantaneous ones. Instead, we directly use the "ensemble-averaged" ones for simplicity since they are interchangeable with instantaneous ones in the mean field approximation.
( ) eff c r φ is introduced to take into account the depletion layer surrounding the surface of each particle and its capability of overlapping with each other and with the entity of other particles, which causes the so-called depletion effect (see Fig.1). We approximate that the density profile of polymers surrounding a particle is described by a hyperbolic tangent function in case of neutral polymer-particle interfacial interaction. Then, the particle at position r′ generates an effective particle concentration which is felt by polymers at position r : where l is the grid size of the simulation box. The width of the depletion layer, D ξ , is related to the rigidity of the polymer chain. The Heaviside function in Eq. (3) guarantees the position r is outside the entity of the particle at position r′ . The effective particle concentration is then: The first max function addresses the overlap between the depletion layer and the entity of (2) is introduced as a threshold quantity to decide whether a location of the system is in the hardcore or non-hardcore regions of particles.
We set this threshold quantity The strong polymer-particle EV interaction is triggered where the total concentration in the hardcore region of particles is larger than one (the first term of Eq. (5)): The second term of Eq. (5) represents particle-particle hardcore repulsion which is set to be proportional to the overlap "volume", op V , between two nearby particles.
1 h κ − is chosen to be large enough to avoid overlap between particles and between polymer and particle. Note that even though there is no problem to express the strong EV repulsions as in Eq. (5) in terms of instantaneous concentrations, it is not appropriate to replace them by the "ensemble-averaged" ones in the conventional mean field approach. As mentioned in the introduction, the particle density in our model is of instantaneous nature instead of ensemble averaged which allows us to use Eq. (5) to address the strong EV interactions.
The chemical nature of the polymers and the particles is encoded in an interfacial energy of exponential form: 25 ε is the dimensionless strength of the interfacial interaction or relative affinity between polymer and particle (positive for repulsion and negative for attraction). Δ denotes the spatial range. Besides the above potentials of real origins, an artificial double-well like potential is introduced to force the formation of discrete particles: λ is the strength of the two harmonic potentials which drive the particle number density at each grid in the simulation box toward 0 or d l − . This artificial potential embodies the inseparability of a real particle. Note that this potential is position-unbiased, i.e., it does not directly influence the spatial arrangements of particles.
We fix 0 4.08 = g R σ . The chemical potential of polymers, μ , is chosen that the concentration of bulk polymers is 1. We set

B. Phase diagram and thermodynamic analysis.
To quantitatively analyze the structures of polymer/nanoparticle composites in Fig. 2, we introduce two useful quantities: the number of pairs of bridging-connected particles, n B and the mean distance between a particle and its six nearest neighbors, 6  To thermodynamically understand the variation of the structures, we took the particle distribution in RD phases, specifically the distribution of particles in Fig. 2(e), as the reference. We calculated the differences of grand potential ( ΔΩ ), potential energy ( E Δ ) and entropy per polymer chain ( s S Δ ) between the "equilibrium" and the reference structure at various ε ( Fig. 3(b)).  S k Δ =− ± for the formation of the structure in Fig. 5(b). These results imply that the closely bridging aggregation of large particles in Fig. 5(b) is enthalpically driven but entropically unfavorable, although it is triggered by the existence of a wide depletion layer around particles.

IV. Conclusion
The field model of PNCs introduced in this work realizes the discrete description of particles which can predict the (more realistic) collective assembly of particles in the polymer matrix with careful consideration of the EV interactions, depletion effect and interfacial interaction. This model allows the investigation into the bridging and depletion effects on the multi-particle collective level instead of calculations based on two-particle correlations. 25,26 It can reveal the entropic and enthalpic contributions in the variation of morphologies. Overall, it is a valuable approach to exploring and analyzing the rich mesostructures in polymer-based nanocomposites and is readily extendable to biology-related systems, such as dendrimer/drug encapsulation 46 and membrane/particle assembly. 47

APPENDIX: SCFT equations and implementation
In our modle, the system of simple nanoparticle/polymer mixture is specified by nine parameters: σ (segment size), ,0 g R (unperturbed radius of gyration of polymer chain), c R (radius of particles), D ξ (width of depletion layer), Δ (spatial range of polymer-particle interfacial interaction), c n (number of particles), μ (chemical potential of polymers), 1 κ − (dimensionless parameter proportional to the compressibility of the polymer matrix) and ε (strength of polymer-particle interfacial interaction). We set σ as the unit length and fix ,0 4.08 g R = (N=100). μ is chosen that the volume/area fraction of bulk polymers is 1.