2.1

Materials

Tetraethyl orthosilicate (TEOS), cyclohexane, absolute ethyl alcohol (CH ₃ CH ₂ OH), n-propylamine and ammonium hydroxide (NH ₃ ·H ₂ O) used in the synthesis of silica nanoparticles were purchased from Sinopharm Chemical Reagent (Beijing, China). 3-methacryloyl trimethoxypropylsilane (γ-MPS) used in silanization of silica nanoparticles was purchased from Alfa Aesar (Haverhill, MA, USA). The monomers, including Bisphenol A glycerolate dimethacrylate (Bis-GMA) and triethylene glycol dimethacrylate (TEGDMA), were purchased from Aladdin Biochemical Technology (Shanghai, China). The photoinitiator systems were ethyl 4-dimethylamino benzoate (4-EDMAB) and camphorquinone (CQ) from Aladdin Biochemical Technology (Shanghai, China). All reagents were used directly without further purification.

2.2

Experiment methods

2.2.1

Synthesis and modification of silica nanoparticles

The silica nanoparticles were synthesized using the Stöber method [ ]. In a typical procedure, deionized water, ethanol and NH ₃ ·H ₂ O solution were first mixed in a three-necked flask and equilibrated to 60 °C. TEOS was added into ethanol and homogenously mixed. Then the obtained solution was added into the flask under vigorous stirring at 60 °C. The whole reaction process lasted for 3 h. The unmodified silica nanopowders were obtained by a freeze-drying process in the freeze-dryer (LGJ-10F, Beijing Songyuanhuaxing Technology Develop Co., Ltd., Beijing, China).

The modified silica nanopowders were obtained by the silanization process [ ]. The cyclohexane, silica nanopowders, γ-MPS, and n -propylamine with a mass ratio of 800:50:5:1 were stirred at 25 °C for 30 min, and then kept at 65 °C for another 30 min. The obtained mixture was placed in a rotary evaporator (RE301, Shanghai Kexing Equipment Co., Ltd., Shanghai, China) at 45 °C to remove the solvent and then heated at 90 °C for 1 h. Finally, the powders were dried at 90 °C in a vacuum oven for 16 h.

2.2.2

Preparation of dental resin composites

CQ and 4-EDMAB with a weight ratio of 1:4 and a total weight of 1 wt% of the resin matrix were added to the resin monomer mixture (Bis-GMA/TEGDMA) to prepare a resin matrix. All formulations of dental resin matrix are consisted of Bis-GMA and TEGDMA with a mass ratio of 1:1. The inorganic filler and resin matrix were manually pre-mixed in a beaker. Then, the composite paste was uniformly mixed using a three-roll extruder (TR50M, Trilos Precision Equipment Co., Ltd., Shenzhen, China) at a speed of 40 rpm. The roll gaps for feeding and exporting were 9 and 5 μm, respectively.

2.2.3

Maximum filler loading (MFL)

The experiment MFL is the maximum mass fraction of inorganic filler which can be filled in the resin matrix. The calculation equation is as follow:

where m _filler is the mass of inorganic filler and m _resin is the mass of resin matrix.

The MFL was evaluated by the condition of resin composites after mixing through the three-roller extruder. The silica nanoparticles were added to the resin matrix, and manually pre-mixed in a beaker. Then the hand-mixed composite was further mixed through a three-roller extruder. The particles were considered to be completely filled in the resin matrix until the product was continual and pasty as shown in Fig. S1a. The content of filler was then increased, and the above process was repeated until the mixture cannot be completely mixed through the three-roller extruder (Fig. S1b). The filler content before the last addition was regarded as the MFL. The experiment was carried out for three times, and the average value was taken as the final result.

2.2.4

Experimental program

In the single-level filling, eight particle sizes (33, 62, 76, 120, 193, 250, 285 and 445 nm) were selected and encoded ( Table 1 ). The gradient of particle size was fully considered in order to better reflect the difference of filler loading. The MFL of both unmodified and modified fillers of single-level filling were studied to explore the influence of modifiers.

In the multi-level filling, three particle sizes of 62, 120, and 285 nm were selected. The reason for choosing these sizes is to make a comparison with the idealized filling model, thereby proving that the filler size ratio of 0.225:0.414:1 may achieve the highest filler loading . In addition, these sizes have an appropriate gradient, which is beneficial to the rapid simulation and maintains accuracy. The data points of multi-level filling are shown by a triangular diagram ( Fig. 1 ), in which three endpoints represent the single-level filling, three edges present the binary-level filling, and the inner region is on behalf of the ternary-level filling. The mass fraction of these three size particles is regarded as the coordinates of triangular diagram. Twenty-five points were selected for the experiments ( Table 2 ). Only the MFL of modified particles was studied in multi-level filling.

Data points of multi-level filling represented using a triangular diagram.

2.2.5

Characterization

The sizes, morphologies and dispersity of silica nanoparticles were observed by a transmission electron microscopy (TEM) (Hitachi HT7700, Japan) at an accelerating voltage of 100 kV. The particle size distributions of silica nanoparticles were obtained by measuring over 200 particles of TEM images using an image analysis and processing software (Image-Pro Plus 6.5, Media Cybernetics Inc., USA). Thermogravimetric (TG) analysis (TGA701, LECO, USA) was used to measure the modifier grafting rate (MGR) of silica nanoparticles. The temperature rose from 50 to 900 °C with a heating rate of 10 °C/min in the N ₂ atmosphere.

2.3

Simulation methods

2.3.1

DEM model

In this work, the DEM was used to simulate the filling of silica nanoparticles in the resin matrix. In the DEM simulations, the particles are treated as a combination of a finite number of discrete elements. In each time step of the simulation process, the force-displacement law and Newton’s law of motion are alternately used to predict the behavior of the particle group [ , ]. The motion of a particle in the DEM simulations is divided into translational and rotational motions. The equation of these motions of particle i can be written as:

where m _i , I _i , v _i and ω _i are the mass, inertia, translational velocities and angular velocities of particle i, respectively. F _i and M _i are the resultant force, and torque acting on particle i, respectively.

In order to effectively compute and appropriately apply, a variety of contact models have been proposed to express the force and torque generated by the contact among particles. Based on the similarity between dental resin composites and sedimentary rocks, the bonded particle model (BPM), which was used to study the mechanical behavior of sedimentary rocks, can be applied in our work [ , ]. In the BPM, the interaction is mainly described as a contact bond and a parallel bond, as shown in Fig. 2 . For contact bonds, only the interaction between particles at the contact point is considered, which is represented by an elastic spring with constant normal stiffnesses ( k _n ) and shear stiffnesses ( k _s ). For parallel bonds, the interaction between particles in a finite-sized section is considered, which is represented by a set of elastic springs with constant normal stiffnesses ( k _n ) and shear stiffnesses ( k _s ) [ , ]. The BPM reproduces the physical behavior of organic resin matrix gluing silica nanoparticles together, which is well suited for the simulation of dental resin composites.

Schematic illustration of the interactions between particles in the BPM: (a) normal stiffness and normal force, (b) shear stiffness and shear force.

2.3.2

DEM simulation conditions and processes

Particle Flow Code 3D V4.0 (PFC3D) developed by Itasca was applied in this study [ ]. The particle size used in the simulation corresponded to the experiment. Fig. 3 shows a specific simulation process. Firstly, spherical particles with the preset sizes were generated in a specific cuboid container (2225 nm × 2225 nm × 6675 nm), and randomly placed in the container one by one ( Fig. 3 a). Then, the dynamic motion and interaction of spherical particles were simulated by the DEM. The particles settled under gravity until the velocity of all particles was basically zero to produce stable filling ( Fig. 3 b). To simulate the compression process of the dental resin composites through the three-roll extruder, servo-control was used to control a wall to compress from top of the particles until the specified stress was reached ( Fig. 3 c). In order to avoid the influence of wall effect on the particles after compression, a spherical area was selected in the compressed particles to calculate the MFL ( Fig. 3 d). To reduce the error of simulation, particles were generated by selecting different initial porosity (0.495, 0.5, 0.505) in step 1 ( Fig. 3 a) for each particle size composition. The MFL is defined as the mass fraction of spherical particles in the selected spherical area, where the densities of silica nanoparticles and resin matrix were considered as the densities of spherical particles and voids among the particles, respectively. The final MFL was obtained by averaging the results of multiple simulations. The calculation equation is as follows:

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