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Introduction

The polymerization of dimethacrylates produces densely crosslinked networks, which find wide applications in dentistry such as dental composites, pit and fissure sealants, dentin bonding agents and cements, dental adhesives .

The most commonly used dental composite resins are: the highly viscous (2,2-bis-[4-(2-hydroxy-3-methacryloyloxypropoxy)phenyl]propane) (Bis-GMA) and triethylene glycol dimethacrylate (TEGDMA), used as diluent. Other dental composite resins are urethane-dimethacrylates (UDMA), especially 1,6-bis-(methacryloyloxy-2-ethoxycarbonylamino)-2,4,4-trimethylhexane ( Scheme 1 ). The advantages of UDMA are lower viscosity when compared to Bis-GMA and high flexibility of the urethane linkage. UDMA is used alone or in combination with other monomers such as Bis-GMA and TEGDMA .

Structures of monomers used.

The free radical polymerization of these multifunctional monomers exhibits anomalous reaction behaviors, including autoacceleration and autodeceleration, unequal functional group reactivity, reaction-diffusion controlled termination and limited functional group conversion due to hindered mobility of vinyl groups . Additionally, one of the most important characteristics of this process is the formation of microgels, their agglomeration into clusters and their connections . They arise due to a varied reactivity of reactive groups involved in the process of polymerization: the monomer double bonds and side bonds (the remaining unreacted double bonds of multifunctional monomer molecules which are already incorporated into the polymer chain). At the beginning of the process, at low conversion, the reactivity of side bonds far exceeds the reactivity of monomeric bonds, mainly due to the close proximity of side bonds and the radical chain end. This leads to the formation of loops in the chains and the formation of microgel particles (highly cross-linked polymer regions with a high degree of cyclization, suspended in a less cross-linked matrix). As the conversion increases, side bonds inside the microgel particles are readily available and less reactive, due to the effect of screening. The monomeric double bond reactivity equals the side bond reactivity or even exceeds it, and a less crosslinked matrix is formed. In the final network there are strongly crosslinked microgel agglomerates and, later formed, less cross-linked areas, giving rise to spatial heterogeneity of the polymer.

Recent research on such materials identified that the strongly heterogeneous morphology of three-dimensional polymeric structures affects their mechanical properties. Mainly, the impact resistance is lower than expected. Due to the fact, that the crack propagates through the routes of the lowest crosslink density or the highest stress intensity , it was suggested that the more homogeneous the network, the more uniform and better its mechanical properties .

The problem with defining the role of morphology in the development of mechanical properties of dimethacrylate polymer networks is the lack of established and widely accepted methods for its quantitative description.

The morphology of dimethacrylate systems, considered to be amorphous , is commonly visualized by means of atomic force microscopy (AFM) or scanning electron microscopy (SEM) . The experimental technique, which is widely used for quantitative description of the polymer network heterogeneity, is dynamic-mechanical analysis (DMA) . However, the method providing information about dimensions of heterogeneous domains, present in the dimethacrylate polymer networks, is still lacking.

In the present work the fractal analysis of AFM images has been used to study the morphology of polymer networks obtained from three popular dental, structurally different, dimethacrylate monomers: Bis-GMA, TEGDMA and UDMA ( Scheme 1 ). The surface topography of dimethacrylate homo- and copolymer networks was described with a variety of fractal dimensions, which were correlated with selected physico-mechanical properties of these polymers.

The concept of fractal geometry, used to describe irregular surfaces with a self-similar nature, provides a basis for the quantitative characterization of the tortuosity of fracture surfaces and is the potential linkage between the fractal dimension and material properties. Mandelbrot et al. first introduced this concept to materials science . Since then much research for applications in material science, medicine and biophysics were reported in the literature .

Surface and its profile lines with apparent irregularities are not necessarily fractal objects with a given fractal dimension. The necessary requirement is self-similarity, which means that the object should appear exactly or approximately similar upon being scaled larger or smaller. In real objects the self-similarity is not extended over all ranges of magnification: below and above certain levels the object may not have a fractal character because the nature of the objects recognized at different scales may be completely different.

In classical geometry, a chain of locally linear segments is taken as one-dimensional. Accordingly, the smooth surface is two-dimensional and a cube is three-dimensional. In reality though, the material does not lead to such regular structures. The elements of the texture tend to form irregularly distributed clusters of different sizes. When the complexity of structure increases with the magnification, it may be useful to employ fractal dimension ( D _F ) for describing the surface morphology. The more irregular the object is, the higher its D _F is, varying between 2 and 3 .

The determination of D _F for stochastic fractals is insufficient for their full characterization. For the complete representation the whole range of parameters has to be estimated, which is defined as the “ D _q spectrum”. One of the most common methods for describing the fractal characteristics of self-similar objects is The Box Counting Method (BCM) . BCM allows for calculating the D _F and its extension–the generalized fractal dimension ( D _q ). The D _q can be expressed in two forms: the “ D _q spectrum” or the “multifractal spectrum” ( f ( α )) . Both methods generate an infinite set of numbers (dimensions) representing details (complexity, self–similarity, irregularity etc.) of the structure being analyzed and both are needed for a comprehensive understanding of surface morphology. The f ( α ), which is a regular graph with a single maximum, obtained by the Legendre transform of the “ D _q spectrum”, more clearly shows the multiplicity of extreme values of probability, thus the range of probability is more defined. From the difference between the extreme values of D _q obtained from the f ( α ) spectrum, corresponding to areas with the smallest ( D _{– ∞} ) and the largest ( D _∞ ) probability of finding the heterogeneous domains in the polymer network, the parameter ΔD can be determined. The lower the Δ D value the more self-similar the set morphology. In extreme cases, for ideal deterministic fractals, this value equals 0 . Fractal analysis based on D _F and D _q is a useful and popular tool for quantifying the surface morphology .The fracture topography of a polymer may also be analyzed by observing its surface shape, as a fracture profile. A profile line can be extracted as a line through an area and expressed as a single numeric fractal parameter–the modified fractal dimension ( D _β ). Its value varies from 1, corresponding to a straight line, to 2, corresponding to an irregular line .

Correlating the morphological studies with the mechanical tests would be beneficial in defining the role of morphology in the mechanical behavior of dimethacrylate networks and consequently, lead to the development of a reliable method for identifying the cause of dimethacrylate-based dental material failures under stress. Thus, fractographic analysis could become one of the key elements in designing and developing dental materials.