A new method for predicting the maximum filler loading of dental resin composites based on DEM simulations and experiments



The inorganic fillers in dental resin composites can enhance their mechanical properties and reduce polymerization shrinkage. When the usage amount of inorganic fillers is closed to maximum filler loading (MFL), the composites will usually achieve optimal performances. This study aims to develop a method that can predict the MFL of dental resin composites for the optimization of filler formulations.


A method based on discrete element method (DEM) simulations and experiments was firstly developed to predict the MFL of spherical silica particles for single-level and multi-level filling.


The results indicate that the presence of modifier can increase the MFL, and the MFL increment can be exponentially changed with the content of the modifier. Compared with the single-level filling, the addition of secondary fillers is beneficial to increase the MFL, and the increment can be affected by the particle size and size ratio. The prediction results show a good agreement with the experiment results.


The accuracy of prediction results indicates a great potential of DEM simulations as a numerical experimental method in studying the MFL, and provides an effective method for the optimization of filler formulations.


Dental resin composites have been researched and used in recent decades due to their good aesthetics, biocompatibility and convenient clinical operating conditions [ ]. Typical dental resin composites can be considered as a mixture of inorganic fillers and organic resin matrix [ , ]. Although the organic resin components are called a matrix, the high loading of inorganic fillers are the major contributors to mechanical properties in the dental resin composites [ , ]. Recent studies have shown that dental resin composites can achieve excellent mechanical performances and lower polymerization shrinkage, when the amount of inorganic fillers is closed to the maximum filler loading (MFL) [ ]. In addition, the organic resin matrix is the most expensive component of dental resin composites [ ]. If the inorganic fillers can reach their MFL, the cost of composite materials can be reduced and their performances can be improved as well [ , ]. Consequently, the exploration of MFL of inorganic fillers in composites is of great significance for optimizing dental resin composites. However, there are few researches reporting how to obtain MFL with different filler formulations. Wang et al. [ , ] reported two theoretical filling models for calculating the MFL, close-packed structure and random-packed structure. However, because the close-packed model was idealized and over-simplified [ ] without considering the influence of resin, modifier and interaction among particles, this result was much higher (more than 14 wt%) than the experimental MFL. By comparison, the random loose packing model more closely matched the real packing arrangement, however, only the single-level filling data were calculated in their work [ ]. Usually, the sizes of fillers in different resin composites are quite different. Nevertheless, in their work [ , ], only the MFL with specific filler formulations was investigated, and the proposed theoretical models cannot calculate the MFL of various filler formulations in dental resin composites. Moreover, they mentioned that adding secondary particles could increase the filler content [ ], but the specific increment was not confirmed. Therefore, it is very valuable and desirable to develop a universal calculation method which can comprehensively consider various factors affecting MFL in dental resin composites for accurately predicting the MFL of complex filler formulations.

The MFL of a particle system usually depends on the filler packing structure which may be affected by the size, grading, compaction of particles and medium existing among particles [ ]. In the past couple of decades, different studies, including experiments [ , ], numerical models [ , , ] and computer simulations [ , ], have been conducted to explore the relationship between filler packing structures and loading. Among them, the discrete element method (DEM) proposed by Cundall and Strack can provide an effective way to simulate the interaction between particles, and also can predict the behavior of particle groups [ ]. DEM simulations has a wide range of applicability, which can simulate particles of various shapes, such as spheres, rods, cubes, and irregularly shaped particles [ , ]. In addition, it can also provide the movement parameters of each particle during the movement, such as positions, orientations, and velocities, and these parameters are conducive to in-depth research on the particle system [ , ]. In previous researches, the DEM simulations has been used to study the filler loading and mechanical behavior of concretes and sedimentary rocks, which shows a good agreement with the experiment results [ , , ]. For the dental resin composite systems, the resin matrix fills the voids between inorganic particles and also binds these particles. This situation is analogous to the cement paste and aggregate particles in the concrete mixes or sedimentary rocks [ , , ]. Therefore, DEM simulations can be applied to study the MFL in dental resin composites. However, the fillers used in dental resin composites are usually modified. The modifiers can improve the binding force between fillers and resin matrix, and facilitate the uniform dispersion of fillers, which may affect the filler loading [ ]. Therefore, the DEM simulations is appropriate to explore the MFL of unmodified fillers. The effect of modifiers on MFL needs to be further studied.

In this study, the DEM simulations and experiments were firstly used to explore the MFL of dental resin composites. Spherical particles were used in this study, since it was found that they had lower shrinkage-strain and shrinkage-stress than irregular fillers [ , ]. Through the study of single-level and multi-level filling MFL of modified and unmodified spherical silica fillers in dental resin composites, the effect of modifier on the MFL was determined, and the calculation method, which was suitable for MFL of inorganic filler, was also explored.

Materials and methods


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.

Experiment methods

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.

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.

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:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='MFL=mfillermfiller+mresin’>𝑀𝐹𝐿=𝑚𝑓𝑖𝑙𝑙𝑒𝑟𝑚𝑓𝑖𝑙𝑙𝑒𝑟+𝑚𝑟𝑒𝑠𝑖𝑛MFL=mfillermfiller+mresin

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.

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.

Table 1
Filler sizes and codes of inorganic filler used in single-level filling.
Code Filler size (nm)
S1 33
S2 62
S3 76
S4 120
S5 193
S6 250
S7 285
S8 445

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.

Fig. 1
Data points of multi-level filling represented using a triangular diagram.

Table 2
Filler formulations and codes of inorganic filler used in multi-level filling.
Filling type Code Mass fraction
S2 S4 S7
Single-level filling S7 0 0 100
S4 0 100 0
S2 100 0 0
Binary-level filling D1 20 80 0
D2 40 60 0
D3 60 40 0
D4 80 20 0
D5 0 20 80
D6 0 40 60
D7 0 60 40
D8 0 80 20
D9 20 0 80
D10 40 0 60
D11 60 0 40
D12 80 0 20
Ternary-level filling T1 10 10 80
T2 20 20 60
T3 40 40 20
T4 10 80 10
T5 20 60 20
T6 40 20 40
T7 80 10 10
T8 60 20 20
T9 20 40 40
T10 33 33 33


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.

Simulation methods

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:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-2-Frame class=MathJax style="POSITION: relative" data-mathml='midvidt=Fi’>𝑚𝑖𝑑𝑣𝑖𝑑𝑡=𝐹𝑖midvidt=Fi
<SPAN role=presentation tabIndex=0 id=MathJax-Element-3-Frame class=MathJax style="POSITION: relative" data-mathml='Iidωidt=Mi’>𝐼𝑖𝑑𝜔𝑖𝑑𝑡=𝑀𝑖Iidωidt=Mi

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.

Fig. 2
Schematic illustration of the interactions between particles in the BPM: (a) normal stiffness and normal force, (b) shear stiffness and shear force.

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:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='MFL=PD⋅ρsPD⋅ρs+(1−PD)⋅ρr’>𝑀𝐹𝐿=𝑃𝐷𝜌𝑠𝑃𝐷𝜌𝑠+(1𝑃𝐷)𝜌𝑟MFL=PD⋅ρsPD⋅ρs+(1−PD)⋅ρr
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Jan 10, 2021 | Posted by in Dental Materials | Comments Off on A new method for predicting the maximum filler loading of dental resin composites based on DEM simulations and experiments
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