University of Rome Tor Vergata

Transcription

1 University of Rome Tor Vergata THERMAL BEHAVIOUR OF TEETH DURING RECONSTRUCTION: SIMULATION, CURING KYNETICS AND MEASUREMENT OF THERMOPHYSICAL PROPERTIES OF INVOLVED MATERIALS P. Coppa 1, G. Bovesecchi 1, E. Armellin 2, S.G. Condò 2, L. Cerroni 2, G. Pasquantonio 2 1 University of Rome Tor Vergata Department of Industrial Engineering 2 University of Rome Tor Vergata Department of Clinical Science and Translational Medicine PhD of Industrial Engineering - Research activity of interest for medicine

2 Introduction The purpose of conservative dentistry is to restore the teeth compromised to maintain it and preserve the integrity of the remaining healthy tissue. In the past the fillers used were based on amalgam of lead, nowadays the materials used are polymeric resins irradiated by a lamp that initiates the active principle for curing. During treatment, the lamp is a source of radiation heat which propagates by conduction through the dental tissues. Due to the transmittance of the material, a part of the heat directly reaches the pulp chamber. The temperature in the pulp chamber must not exceed 42.5 C to avoid the necrotization of tooth tissue.

3 Introduction Typical temperature trend in a tooth during lamp emission Front Mid Rear

4 Introduction From this the need to study the properties of reconstruction resins derives, and particularly the transmittance of the material and the reaction kinetic. From preliminary measurements a change of transparency of the resin during the curing process was noted H H L S Px H H Ph Photo of the tests: L, lamp; Ph, photodiode; S, sample, Px, plexiglass slabs; H, holders From the transmittance change it is possible to obtain the reaction kinetics.

5 Analytic model for kinetic reaction The rate of a chemical reaction is given by: a t = k 1-a ( ) x α the extent of reaction (0 α 1); x reaction order; k reaction rate coefficient. Assuming a single species present (x = 1) and integrating: a = 1- e -kt Experimentally α is expressed as a function of transmittance: a = t -t 0 t - t 0 τ 0 is initial transmittance for t=0 and so α=0; τ is asimptotic transmittance for t= and so α=1. Combining the two expressions we obtain the final equation: ( ) t = t 0 + ( t -t ) 0 1- e -kt

6 Experimental Lamp tests: Spectral analysis; Lamp spectral intensity measurements. Irradiance P 1 (20s) P 2 (15s) P3 (3s) Nominal (Wm -2 ) Measured (Wm -2 )

7 Experimental Resin tests: calibration gray filters; transparency measurements. Typical transparency measurement during curing: The rate coefficient k is obtained by a least-squares regression on the exponential trends detected.

8 Experimental results I R (W/m 2 K) k (s -1 ) s k (s -1 ) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± The intensity of the lamp was changed applying gray filters between the sample and the lamp, and different values of k were achieved

9 Experimental results The values of k were interpolated with a phenomenological model of equation: k = b 1 b 2 + b 3 I R - From the graph: If I r =0 W m -2 there is no polymerization (k = 0); b 1 ( ) I R + b 2 k increases with I R increasing ; for 1000 < I R < 10.5 kw m -2 k becomes asymptotic; for I R > 10.5 kw m -2 there is an abrupt increase of k and the trend is similar to the first part.

10 Thermophysical properties Beside the evaluation of reaction kinetics, it is important to know the thermophysical properties of the resins and teeth. Typical temperature trends in teeth during and after curing.

11 Thermophysical properties Comparison between different temperature trends with and without heat generation due to polimerization.

12 Theoretical model for thermophysical properties The tooth was modeled as a cylinder, with a coaxial cavity representative of the filling chamber. Cylindrical propagation of temperature is described by the basic conduction partial derivative equation: Where: is the specific thermal power generated by the exothermic curing reaction and the propagation of light in the material when it is partially transparent (in W m -3 ) Boundary conditions are set in order to take into account convection at the sample surface (h), irradiation at the lightened surface ( )

13 Theoretical model for thermophysical properties The resulting temperature difference is a function of the following quantities: the partial derivative differential equation cannot be solved analytically, so the numerical method of finite differences was adopted. The final equation describing the temperature increment from a former step is:

14 Theoretical model for thermophysical properties Experimental data were processed with a non-linear least square regression in order to find the best estimate of the unknown parameters of: Due to the nature of the LS analysis, when the number of parameters to be estimated is relevant, it is likely that some of them would be linear combination of others, so irradiance and transmittance were measured separately. So only α, λ and h were calculated by the LS-algorithm.

15 Results for thermophysical properties Appling the theoretical model to a cylinder made of resin, thermal conductivity, thermal diffusivity and convection heat transfer coefficient were calculated. Property units value h W m -2 K-1 65 ± 7 α m 2 s-1 (0,215 ± 0,007)10-7 λ W m -1 K-1 0,582 ±0,005

16 Results for thermophysical properties Presently the reaction heat is being evaluated by LS. In this case, measurements were performed on a cylinder with a cavity filled with resin and a thermocouple located at the bottom of the cavity. The first results show a good agreement between experimental data and theoretical model.

17 Results for thermophysical properties A problem occurs in the modeling of cooling, whose temperature presents lower rate compared to the measured data.

18 Conclusions Reaction kinetics of resins can be evaluated through transparency measurements. There are two different kinetics, one for I r <10 kw/m 2 and another for I r >10 kw/m 2 From the data, it is also possible to derive a time-to-lamp intensity to get the best performance of the resin. Comparison with other indexes of reaction kinetics (hardness, FTIR, etc) are in progress.