## Abstract

## Objective

The goal of this investigation was to assess validity of predictive models of stress relaxation in dental polymers when applied to extended master curves generated from short time experimental data by WLF time temperature superposition method.

## Methods

The stress relaxation modulus changes with time at three different temperatures near the ambient body temperature were determined for selected mono-methacrylate (PEMA and PMMA) and dimethacrylate (bis-acryl) dental polymers. WLF time–temperature superposition procedure of logarithmic shift of the data from other temperatures to those at 37 °C was used to generate extended master curves of relaxation modulus changes with time. The extended data were analyzed for conformity to three different predictive models of stress relaxation including Maxwell, KWW stretched exponential function and Nutting’s power law equation.

## Results

Maxwell model was found to be a poor fit for the extended data in all polymers tested, but the data showed a much better fit for KWW (0.870 < *R *^{2 }< 0.901) and Nutting’s (0.980 < *R *^{2 }< 0.986) models. The non-exponential factor *β *in the KWW function and the fractional power exponent *n *in Nutting’s equation were both significantly different for PEMA based system when compared to that of PMMA and bis-acryl systems.

## Significance

The mean values of *β *in KWW function and power exponent *n *in Nutting’s equation for PEMA resin is consistent with significant viscoplastic contribution to its deformation under stress unlike in PMMA and bis-acryl resin systems. This may have significant bearing for PEMA use in medium to longer term stress-bearing applications even as a temporization material.

## 1

## Introduction

Hookean springs and Newtonian dashpots are conventionally used to model the elastic, viscoelastic and viscoplastic mechanical behavior of polymers under applied stress or strain . In this approach, the spring under stress represents elastic deformation, the dashpot viscous deformation, Maxwell element (spring and dashpot in series) a combination of elastic and viscoplastic deformation, and a Voigt element (spring and dashpot in parallel) time dependent recoverable (viscoelastic) deformation. Typical real polymers combine elastic, viscoelastic and/or viscoplastic deformation in their mechanical behavior under stress within the linear viscoelastic region, and two or more of these elements are therefore necessary to model the cumulative deformation behavior. Examples of such models for typical polymers are the three element standard linear solid model (which combines a spring and a Voigt element), and the four element Burger’s model in which a Maxwell and Voigt element are combined. Fig. 1 shows the different basic types of mechanical models used to model mechanical behavior.

The time–temperature superposition (TTS) is an important tool to estimate a longer term polymer viscoelastic behavior at a use temperature from short term measurements over a range of temperatures . In this approach, data from short time isochronal measurements at different temperatures are used to extrapolate a much longer term isothermal set of data at a use (or service) temperature. Two equations are used to predict TTS behavior in polymers. One is the William–Landel–Ferry (WLF) equation and the other, the Arrhenius equation. Arrhenius equation is typically used to follow secondary transitions and melt viscosity changes, while, over the years, WLF equation has been successfully applied to follow time–temperature superposition in creep, stress relaxation and dynamic mechanical properties at and near the glass transition temperature in many polymers. We have previously shown that the dental polymers typically follow WLF equation for TTS near the body temperature, and that this is a very powerful approach to extend short time deformation measurements at different temperatures to assess longer term isothermal deformation events at 37 °C .

A valuable mechanical approach to study time dependent mechanical behavior of polymers is through a stress relaxation test. In this test, the polymer specimen is subjected to a step strain perturbation by the instantaneous application of a constant strain and the stress required to maintain the constant strain is monitored as a function of time. Taking Burger’s model as a typical mechanical analog of a polymer mechanical behavior under stress, we can consider the Voigt element and the Maxwell element as potentially contributing to the time dependent stress changes in the polymer. The Burger’s model contains two energy storage elements, namely the Voigt element and the spring in the Maxwell unit. It also contains one energy dissipation unit, namely the Maxwell dashpot. The stress dissipation that occurs in the stress relaxation test is associated only with the Maxwell unit since only an unconstrained dashpot can dissipate stress through energy loss. A constitutive relation between stress relaxation modulus ( *E *( *t *)) and time ( *t *) using the Maxwell model is therefore often used to characterize time dependent changes in stress relaxation modulus. This constitutive relation is given by:

*E *( *t *) = *E *_{o }·e ^{(− t / τ ) }, where *E *( *t *) is the transient relaxation modulus at time *t *, *E *_{o }is the initial relaxation modulus immediately on application of deformation, *τ *the relaxation time. The Maxwell model is only obeyed when there is a single relaxation process. Most real polymers show a broad continuous relaxation spectrum, and this is attributed to the relaxation process occurring by an array of Maxwell units in parallel . The Kohlrausch–Williams–Watt (KWW) stretched exponential relaxation function has been used to explain relaxation phenomena, and many aspects of its application has been highlighted in several theoretical, experimental and review articles . In this approach, the relaxation function *φ *as a function of time ( *φ *( *t *)) is given by *φ *( *t *) = exp−( *t */ *τ *_{κωω }) ^{β }, where *β *is a fractional power exponent known as non-exponential factor and *τ *_{κωω }is the KWW relaxation time. In the case of transient relaxation modulus changes with time, the ratio of KWW relaxation function at time *t *to that at *t *= 0 is given by, *E *( *t *)/ *E *_{o }= exp−( *t */ *τ *_{κωω }) ^{β }. A log{log( *E *_{o }/ *E *( *t *))} vs. log( *t *) plot is expected to be linear with slope *β *and intercept − *β *log *τ *_{κωω }. The non-exponential factor *β *in the KWW stretched relaxation function is considered to control the breadth of the continuous relaxation spectrum responsible for the overall relaxation phenomenon. Its value may vary in the range 0 < *β *≤ 1. When *β *= 1, the model becomes identical to the Maxwell model with a single relaxation process (i.e., a Newtonian fluid), and the smaller the *β *value in the above range, the broader the relaxation spectrum. Typical glassy polymers are considered to have a *β *value near 0.33, and higher *β *value is potentially associated with a greater tendency for viscoplastic deformation responsible for stress decay. The *β *-value is thus a useful material characteristic that can be used to distinguish between materials which are different from each other in their mechanical behavior with significant differences in their viscoplastic deformation under stress.

Nutting had also proposed an empirical power law for creep by fitting experimental creep data to the following relation: *ɛ *( *t *) = *K *· *σ *^{α }· *t *^{n }where *ɛ *( *t *) is the creep strain at time *t *by an applied stress of *σ *while *K *, *α *and *n *are constants. For linear behavior, *α *= 1, and since applied stress is constant, the relationship reduces to *ɛ *( *t *)/ *σ *= *K *. *t *^{n }, or *J *( *t *) = *K *· *t *^{n }, where *J *( *t *) is the compliance at time *t *. Because of the inverse functional relationship between creep compliance and stress relaxation modulus changes with time, the inverse of the above power law is also valid for stress relaxation and we can write *E *( *t *) = *K *′· *t *^{− n }where *E *( *t *) is the stress relaxation modulus at time *t *and the negative exponent − *n *indicates the inverse relationship. A plot of log { *E *( *t *)} vs. log( *t *)} will be linear and *n *can be estimated from its slope. It is to be noted that in the case of stress relaxation, *n *= 0 (no stress decay with time) for ideal elastic materials, and *n *= −1(stress decay is linear with time) for Newtonian fluids. Real polymers show intermediate values between these extremes, and its magnitude is also a material characteristic that can be potentially used to distinguish between materials which are different from each other in their mechanical behavior with significant differences in viscoplastic deformation under stress. Although, this relationship is empirical, it is reasonably well obeyed by many polymers.

In this study, we seek to use short-time stress relaxation experiments to determine a longer term stress relaxation profile of dental polymers at 37 °C using TTS, and to analyze the superposed TTS data to assess the validity of the various predictive models such as the Maxwell model, KWW stretched exponential function and Nutting’s power law equations. There is very little work in applying such predictive creep and stress relaxation models to biomedical polymers near the body ambient temperature, and according to the best of our knowledge, this is the first such analysis in dental polymers. The hypothesis is that all the dental polymers assessed in this study approximate the predictive models after TTS superposition. The additional null hypothesis tested was that there is no difference between different polymers in their viscoplastic contribution to deformation under stress, as indicated by various parameters (such as the non-exponential factor *β *in the KWW function and the power exponent *n *in the Nutting’s model) generated by the models. As pointed out earlier, these parameters are material characteristics which may provide important clues about the type of deformation behavior associated with materials having significant differences in these parameter values.

## 2

## Materials and methods

The polymers evaluated in this study include PMMA, PEMA and bis-acryl resin composite formulations used in dentistry. The materials studied are formulations used in popular commercial provisional crown and bridge materials, but the specific resin materials studied are also used in dental composites and in other dental applications. Table 1 lists the formulations with details of ingredients, brand name, manufacturer name, etc. The materials included popular manual mix and auto-mix commercial formulations of mono-methacrylate resins (Alike and TRIM II, coded as ALK and TRM) and bis-acryl composite resins (LUXATEMP, TEMPHASE and VERSATEMP, coded as LXT, TMP and VRS, respectively).

Brand | Code | Supplier | Compositional Type | Manufacturer listed main ingredients |
---|---|---|---|---|

TRIM II | TRM | Bosworth, Skokie, IL | Poly(ethyl) methacrylate (PEMA) | Polyethyl methacrylate (powder) and isobutyl methacrylate monomer (liquid) |

Alike | ALK | GC Inter-national, Alsip, IL | Poly(methyl) methacrylate (PMMA) | Polymethyl methacrylate (powder) and methyl methacrylate monomer (liquid) |

Luxa-temp | LXT | DMG America, Englewood, NJ | Bis-acryl composite | Base catalyst system, contains urethane dimethacrylate, aromatic dimethacrylate, glycol dimethacrylate, filler, curing agents |

Versa-Temp | VRS | Sultan Chemicals, Englewood, NJ | Bis-acryl Composite | Base-catalyst system, contains BisGMA, filler, curing agents |

Tem-phase | TMP | Kerr Manufacturing, Orange, CA | Bis-acryl composite | Base-catalyst system, contains methacrylate ester monomers, filler and curing ingredients |

The specimens were 45 mm × 20 mm × 10 mm rectangular bars prepared in a stainless steel mold. The powder-liquid mix (ALK and TRM II) and auto mix (LXT, TMP and VRS) pastes were placed in the stainless steel mold, leveled with glass slides and cured. After curing for 30 min, excess flash was trimmed off and the specimens were stored in a humidity chamber at 37 °C before testing after 24 h. The stress relaxation tests were performed isothermally at 32, 37 and 42 °C by applying a constant strain over a pre-optimized time span in a Dynamic Mechanical Analyzer model 2980 (TA Instruments, New Castle, DE). The constant strain of 0.2% was used for all the tests in a dual cantilever clamp. All experiments were done in triplicate to assure reproducibility (precision). As pre-optimized in preliminary studies using stress decay with time to a steady state level, the isothermal relaxation data for each temperature was collected for 600 s by which period the relaxation profiles reached a quasi-steady state with very little further change with time. Tests were performed using TA Advantage instrument Control software. The stress data was collected as a function of time by TA Advantage data acquisition program and processed by Microsoft Excel to generate the transient relaxation modulus { *E *( *t *)} profiles and different predictive model plots.

Log–log plots of isochronal *E *( *t *) data sets for each temperature were generated as a function of time for each material. Time–temperature superposition (TTS) master curves at 37 °C were also generated by logarithmic shift of the data at other temperatures to 37 °C as per the TTS superposition method. The superposed data were tested to check conformity with the Maxwell, KWW and Nutting’s models as described in the introduction. The *R *^{2 }value for linear regression for each method and material was estimated using Microsoft Excel Trendline plot. The mean values of *R *^{2 }and the distribution of residuals for different model plots were compared for different materials and their validity assessed.

The mean value of non-exponential factor *β *in the KWW function and fractional time exponent *n *in Nutting’s equation, respectively, were determined and statistically analyzed by one way ANOVA, LS Means profile plots and Tukey HSD contrast to compare and contrast the behavior of different materials studied. The statistical analysis was done by JMP Statistical program version 10 (SAS Statistical Institute, Corey, NC).

## 2

## Materials and methods

The polymers evaluated in this study include PMMA, PEMA and bis-acryl resin composite formulations used in dentistry. The materials studied are formulations used in popular commercial provisional crown and bridge materials, but the specific resin materials studied are also used in dental composites and in other dental applications. Table 1 lists the formulations with details of ingredients, brand name, manufacturer name, etc. The materials included popular manual mix and auto-mix commercial formulations of mono-methacrylate resins (Alike and TRIM II, coded as ALK and TRM) and bis-acryl composite resins (LUXATEMP, TEMPHASE and VERSATEMP, coded as LXT, TMP and VRS, respectively).

Brand | Code | Supplier | Compositional Type | Manufacturer listed main ingredients |
---|---|---|---|---|

TRIM II | TRM | Bosworth, Skokie, IL | Poly(ethyl) methacrylate (PEMA) | Polyethyl methacrylate (powder) and isobutyl methacrylate monomer (liquid) |

Alike | ALK | GC Inter-national, Alsip, IL | Poly(methyl) methacrylate (PMMA) | Polymethyl methacrylate (powder) and methyl methacrylate monomer (liquid) |

Luxa-temp | LXT | DMG America, Englewood, NJ | Bis-acryl composite | Base catalyst system, contains urethane dimethacrylate, aromatic dimethacrylate, glycol dimethacrylate, filler, curing agents |

Versa-Temp | VRS | Sultan Chemicals, Englewood, NJ | Bis-acryl Composite | Base-catalyst system, contains BisGMA, filler, curing agents |

Tem-phase | TMP | Kerr Manufacturing, Orange, CA | Bis-acryl composite | Base-catalyst system, contains methacrylate ester monomers, filler and curing ingredients |