## Highlights

- •
New fractional derivative type of viscoelastic model is developed for four different uncured composite resins.

- •
Experiments using rheometer with parallel plates are performed and the model parameters are determined.

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From the obtained results conclusions regarding handling characteristics in clinical conditions are presented.

## Abstract

## Objective

In this study we analyze viscoelastic properties of three flowable (Wave, Wave MV, Wave HV) and one universal hybrid resin (Ice) composites, prior to setting. We developed a mathematical model containing fractional derivatives in order to describe their properties.

## Methods

Isothermal experimental study was conducted on a rheometer with parallel plates. In dynamic oscillatory shear test, storage and loss modulus, as well as the complex viscosity where determined. We assumed four different fractional viscoelastic models, each belonging to one particular class, derivable from distributed-order fractional constitutive equation. The restrictions following from the Second law of thermodynamics are imposed on each model. The optimal parameters corresponding to each model are obtained by minimizing the error function that takes into account storage and loss modulus, thus obtaining the best fit to the experimental data.

## Results

In the frequency range considered, we obtained that for Wave HV and Wave MV there exist a critical frequency for which loss and storage modulus curves intersect, defining a boundary between two different types of behavior: one in which storage modulus is larger than loss modulus and the other in which the situation is opposite. Loss and storage modulus curves for Ice and Wave do not show this type of behavior, having either elastic, or viscous effects dominating in entire frequency range considered.

## Significance

The developed models may be used to predict behavior of four tested composites in different flow conditions (different deformation speed), thus helping to estimate optimal handling characteristics for specific clinical applications.

## 1

## Introduction

Contemporary restorative dentistry relies greatly on the use of resin composite materials for the restoration of lost tooth structure. However, placing dental composite is not an easy task; it requires skilful and informed clinician who is able to relate the characteristics of commercially available resin composites with specific requirements for each indication in the field of restorative dentistry. With the wide range of currently existing types of composites with different characteristic it becomes difficult for the dentist to make appropriate choice. First and widely accepted classification of composites has been based on filler particle size . Resin composites can also be classified to give more information about their: clinical indications (anterior, posterior or universal), curing mode (self curing, light curing or dual curing), or viscosity (packable, universal, flowable) . Viscosity is directly related to the handling characteristic of the resin composite material like mode and ease of placement, operation time, anatomical shaping, as well as the final quality of a restoration . From a clinical point of view, uncured resin composites should flow easily into the cavities in order to achieve better adaptation to the cavity walls and yet, have high viscosity characteristics for facilitating shaping in order to recreate the anatomy of the restored teeth. Moreover, in large cavities resin composites should be placed in successive increments to ensure light-curing and prevent excessive polymerization stresses particularly on cavo-restoration interface .

All these properties are currently hard to find in one single composite, so manufacturers have developed composite materials with different viscosities which help dentists to choose the most appropriate form. Packable or condensable composites are highly viscous materials designed to imitate the handling behavior of a dental amalgam. Increased viscosity and high filler particles content of packable composites require more force needed to adapt the material in the cavity. Commercially available light-cured resin composites mainly have a generically useful viscosity which is suitable for most clinical conditions. However, they are not indicated for very small cavities or fissures and pits, because they are not able to flow in the cavity. Flowable composites are low viscosity resin-based restorative materials with reduced filler content, compared to non-flowable composites. The viscosity and flow characteristic of flowable resin composites are obvious since their handling is based on the flow of the material in the cavity. In addition, resin-composites are pseudoplastic materials meaning that they become more fluid during placement with a syringe.

Generally, paste type dental materials like resin composites are viscoelastic materials positioned intermediately between elastic, such as metals, and viscous materials like oils. They show non-Newtonian, shear-thinning flow behavior.

Lee et al. using a dynamic oscillatory shear test, reported significant differences in the viscoelastic properties between flowable, universal hybrid and packable composites and even more, significant differences in the viscoelastic properties between composites in the same class. Beun et al. studied viscoelastic properties of unset flowable resin composites and reported huge differences in the viscosity and flow characteristic that can have a potential influence on their handling behavior and thus on their clinical indications. Recently, Watts et al. used a dynamic oscillatory rheometer to evaluate the influence of resin mixture, filler content and temperature on the viscosity of resin composites.

The aim of this study was to develop a new fractional derivative model for the assessment of viscoelastic properties related to handling characteristics of three flowable and one universal hybrid resin composite. We shall use distributed-order fractional isothermal viscoelastic model to derive several specific models, each capable to describe obtained experimental data.

## 2

## Material and methods

Three commercial flowable composites (with three different viscosities) and one universal composite (designed for anterior and posterior restorations) were used in this study. They are listed in Table 1 . It is to be noted that all tested composites were from the same manufacturer, as it was done in .

Material | Type | Batch no. | Manufacturer |
---|---|---|---|

Wave | Flowable nanocomposite—low viscosity | 130110N | SDI (Bayswater, Victoria, Australia) |

Wave MV | Flowable nanocomposite—medium viscosity | 120547 | SDI (Bayswater, Victoria, Australia) |

Wave HV | Flowable nanocomposite—high viscosity | 120107 | SDI (Bayswater, Victoria, Australia) |

Ice | Anterior/posterior nanohybrid composite | 131016T | SDI (Bayswater, Victoria, Australia) |

## 2.1

## Rheological measurements

Rheological measurements on all four materials tested in this study were performed using an HAAKE Mars Rheometer (Thermo Fisher Scientific Process Instruments, Karlsruhe, Germany) at a temperature of 23 ± 0.1 °C, as previously reported in .

A parallel plates viscometer module with a diameter of 20 mm was used to measure the rheological properties of the flowable resin composites tested. The gap between the plates was 1 mm. In amplitude sweep tests linear viscoelastic region was determined by recording the storage ( *G *′) and loss ( *G *″) moduli versus shear stress (0.01–100 Pa) at constant angular velocity (6.28 rad/s). Appropriate shear stress were selected and kept constant in frequency sweep tests. Frequency sweep test was performed from 0.628 to 62.8 rad/s in order to determine the variation of the complex viscosity ( *η *^{* }), storage (elastic) shear modulus ( *G *′) and loss (viscous) shear modulus ( *G *″). Three measurements were performed for each material tested.

## 2.2

## Material modeling and optimization procedure

Most materials used in dentistry have viscoelastic properties. Viscoelastic properties of materials are lately successfully modeled by the use of fractional derivatives, see . In general, a viscoelastic body may be fluid or solid like. In the first case it has infinite creep strain, while in the second case the creep strain is finite. Creep strain is obtained in a creep test, when material is subject to a sudden, but later constant force on its free end.

Recall, the Riemann-Liouville fractional derivative of order *α *∈ (0,1) is defined by

f ( α ) ( t ) = 1 Γ ( 1 − α ) d d t ∫ 0 t f ( τ ) ( t − τ ) α d τ .

Note *f *^{(0) }( *t *) = *f *( *t *) and limx→1f(α)(t)=ddtf(t)

lim x → 1 f ( α ) ( t ) = d d t f ( t )

. The first property is obvious, while the second one holds true for a broad class of functions, see . The Fourier transform F[f(t)](ω)=ˆf(ω)=∫∞−∞f(t)e−itωdt

F [ f ( t ) ] ( ω ) = f ˆ ( ω ) = ∫ − ∞ ∞ f ( t ) e − i t ω d t

of Riemann-Liouville fractional derivative reads F[f(α)(t)](ω)=(iω)ˆf(ω)

F [ f ( α ) ( t ) ] ( ω ) = ( i ω ) f ˆ ( ω )

, see .

Distributed-order fractional linear constitutive model, that can predict both fluid and solid like behavior of a viscoelastic body, may be assumed in the form

∫ 0 1 ϕ σ ( γ ) σ ( γ ) ( t ) d γ = E ∫ 0 1 ϕ ε ( γ ) ε ( γ ) ( t ) d γ ,

where *σ *, *ɛ *and *E *, denote stress, strain and generalized Young modulus, while *ϕ *_{σ }and *ϕ *_{ɛ }*ϕ *_{σ }and *ϕ *_{ɛ }are constitutive functions, or distributions. If constitutive distributions *ϕ *_{σ }and *ϕ *_{ɛ }are chosen as the sums of Dirac delta distributions, constitutive stress–strain relation (1) becomes

where an,bm,E≥e=0,0≤α1≤…≤1,0≤β1≤1,N,M∈ℕ

a n , b m , E ≥ e = 0,0 ≤ α 1 ≤ … ≤ 1,0 ≤ β 1 ≤ 1 , N , M ∈ ℕ

see . In order to obtain thermodynamic restrictions on model parameters: *a *_{n }, *b *_{m }, *E *, *α *_{n }, *β *_{m }, constitutive model (2) is analyzed for four different cases, see . In the sequel, one sub-model, corresponding to each case, will be chosen for fitting the experimental data.

Case 1. There are *N *+ 1 and *M *+ 1 terms containing different orders of fractional derivatives in (2). We chose

where *a *, *b *_{0 }, *b *_{1 }, *b *_{2 }≥ 0, 0 ≤ *α *< *β *_{0 }< *β *_{1 }< *β *_{2 }≤ 1, as the sub-model corresponding to this case, see .

Case 2. There are *N *+ 1 first terms on the right-hand-side of (2) of the same order of fractional differentiation as on the left-hand-side of (2) and additional *M *− *N *terms left on the right-hand-side of (2). We chose

where *E *, *a *, *b *_{0 }, *b *_{1 }≥ 0, *a *≤ *b *_{0 }, 0 ≤ *α *< *β *≤ 1, as the sub-model corresponding to this case, see .

Case 3. There are *M *+ 1 last terms on the left-hand-side of (2) of the same order of fractional differentiation as on the right-hand-side of (2) and additional *N-M *terms left on the left-hand-side of (2). We chose

where *a *, *a *_{1 }, *a *_{2 }, *b *_{0 }, *b *_{1 }≥ 0, *b *_{0 }*a *_{2 }− *a *_{1 }*b *_{1 }≥ 0, 0 ≤ *α *< *β *_{0 }< *β *_{1 }≤ 1, as the sub-model corresponding to this case, see .

Case 4. There is the same number of terms having the same orders of fractional differentiation in (2). We chose the fractional Zener model

where *E *, *a *, *b *, ≥ 0, *a *≤ *b *, 0 ≤ *α *≤ 1, as the sub-model corresponding to this case, see .

In the optimization procedure, the error function

L ( { α n } n = 1 N , { β m } m = 1 M , { a n } n = 1 N , { b m } m = 1 M , E ) = 1 2 P ∑ p = 1 P ( G ′ ( ω p ) − G ¯ ′ ( ω p ) ) 2 + ( G ″ ( ω p ) − G ¯ ″ ( ω p ) ) 2

will be minimized for each sub-model (4)–(6), in order to obtain the optimal model parameters, i.e., to obtain the best fit between the measured moduli and moduli corresponding to each sub-model (4)–(6). The storage and loss moduli *G *′ and *G *″ are real and imaginary part of the complex modulus ( *σ *− *ɛ *transfer function)

G ( ω ) = σ ˆ ( ω ) ε ˆ ( ω ) = G ′ ( ω ) + i G ″ ( ω ) , ω ∈ [ 0 , ∞ ) ,