Dent Mater 8:140-i 44, March, 1992

Determination of thermal properties of impression materials M. Pamenius, N.G. Ohlson Department of Dental Materials and Technology, The Karolinska Institute, Huddinge, Sweden

Abstract. Residual strains are important for dimensional accuracy of impression materials. When thesestrainsare being calculated, knowledge of thermophysical constants is needed. This paper describes determination of thermal diffusivity, thermal conductivity, specific heat, heat of polymerization rate, and total heat of polymerization. The first two constants mentioned were obtained from an experiment in which transient heat conductivity was studied. The evaluation was based on an analytical solution of this problem. The remaining constants were derived by means of standard differential scanning calorimetry, DSC. Results from accurate, cost-efficient methods are given for three silicones and one polyether. Certain physical properties ofimpressionmaterials are available in literature (McCabe and Wilson, 1978; Pamenius and Ohlson, 1986). Thermal properties, however, are not completely covered by the literature. In particular, this is the case when the material is not yet polymerized and, hence, is not stable. Elastomers provide an example of such materials in their initial phase of life. After the base and the catalyst have been mixed together, a substance is obtained which remains completely plastic for a limited amount of time, sometimes referred to as setting time. The thermal history of the material is important to the final accuracy of the impression (Jorgensen, 1982). The question may be raised as to the necessity ofknowing thermal constants. As will be shown by use of thermodynamics, they are needed for the evaluation of the temperature distribution at the time of setting. This distribution is “memorized” and will result in residual deformation, which can affect accuracy. The aim of this paper is to propose a way of determining thermal constants in the unstable as well as in the stable, polymerized state.

ANALYTICAL BACKGROUND Equation.

Heat conductivity in a continuous medium is gov.erned by the differential equation (Boley and Weiner, 1960): zI’ k VzT -=-. at p.c where V 2 denotes the Laplace operator. coordinates, this operator equals:

- iI2 dX2

- d2

+ dy2

(1) In Cartesian

a2

+a22

(24

In spherical coordinates, for the case offull spherical symmetry, it may be written as: G’b)

140 Pamenius & OhlsorvThermal

properties of impression materials

where T is the temperature distribution in the medium. It is a function of time (t) and spatial coordinates. In the case of symmetry mentioned, one has: T = T(t,r)

(3) The constants in Eq. 1 are interpreted as follows: k equals the thermal conductivity, which is the amount of heat passing through a body of unit thickness and unit cross-section when a temperature difference of unity is maintained across the body; C is the specific heat of the material, i.e., the amount of heat necessary for increasing the temperature of a unit mass by one degree; and p is the density of the material. Sometimes the thermal diffusivity is introduced, defined as (O’Brien and Ryge, 1978): (4) D=k C.P

Ifa chemical reaction occurs somewhere in the volume where Eq. 1 is valid, this equation must be modified by an extra term which accounts for the presence of heat sources and sinks, depending on whether it is an exothermal or endothermal reaction. The equation becomes, after modification:

i%++=k.V2T dt

pc

pc

(la)

where q denotes the heat developed in the material per unit volume and unit time. Boundary and Initial Temperatures. A unique solution of Eq. 1 exists for each completely specified set of boundary conditions and initial values. Steady-state problems in which T is independent oft are solved by putting:

dT 0

-at

The solutions of such problems are independent of the thermal properties of the material. For the solution of transient heat problems, however, knowledge of D is required. Two ways of determining D exist: 1) experimentally under steady-state conditions measuring the heat flux, or 2) experimentally under transient conditions measuring the temperature only. The first approach is based on the equation for heat flux: I=-D.C!.p.gradT=-k.gradT

(6)

where I denotes the flux and gradT the gradient operator: gradT = g

(7)

T 3 (t:+, t)

f o

\

y

tI

t2

t~

t4 t

Fig. 1. Curve showing the temperature of a metal sphere as a function of time,

if spherical symmetry is assumed. Measurement of heat flux is usually made by calorimetric methods, for which a certain minimum time is required, in order for sufficient precision to be obtained. When the determination is concerned with an unpolymerized elastomer and the time available for carrying out the experiment is limited, the transient approach is the only one possible. As will be shown below, the effect of the heat development due to polymerization, represented by the second term of Eq. la, may be neglected since q is sufficiently small for the elastomers being investigated. Therefore, the use ofEq. 1 is justified. For the evaluation of k, the temperature of a metal sphere as a function of time was registered. From this curve (Fig. 1), it is seen that the time interval used for evaluation displays a cooling rate,OT~ t, which is of the order of 0.1°C/s, corresponding to the first term ofEq. la. The second term depends on the material but is approximately 0.02°C/s. The solution of Eq. 1 then sufficiently accurately approximates that ofEq. la. AnalyticalSolution.A metal sphere in an infinite space of unpolymerized elastomer is considered. The initial temperatures of the sphere and the polymer are T oand 0, respectively. Since the heat conductivity of the metal is about two orders of magnitude greater that of the elastomer, thermal equilibrium of the sphere is maintained approximately throughout the experiment. The temperature of the sphere as a function of time, T(ro,t), may be obtained by considering the heat flux through a control surface at r = ro from t = 0 to an arbitrary time, t =t. The amount of heat, Q, which has passed through the control surface during this time becomes, according to Eq. 6 and Eq. 7:

Q=ii(ro).4~r~dt=i_k(~TI .4~r~dt 0

0

Fig. 2. Distribution of temperature in the material under test as a function of time and distance from the center of the sphere, ro denotes the radius of the metal sphere and T o its initial temperature.

Inserting Q according to Eq. 8 into Eq. 9 gives:

)

T(ro,t)=T o+Ik -.--dt o ~t r CoPoro

(10)

0

Since the gradient aT/at is always negative, it is realized that: T(ro,t) < To By accurate measurement of the temperature of the sphere, T(ro, t), k can be evaluated by means of Eq. 10. The analytical solution of Eq. 1 in the case of spherical symmetry is derived by the method of separation of variables. Putting: W(r,t) = T(t). R(r) (11) gives: AR 1 1 dv - v 2 (12) R D v dt Here, ~ denotes the time-dependent part of the temperature function, R the part that depends on space coordinates, and v2 the so-called separation constant. The general solution is: T ( r ' t ) = ~e-Dv2t ( Av "v-----~r sinr + BY COSrv r )

(13)

where the sum is taken over all possible values of v. Fig. 2 shows the principal behavior of this function. In our special application, where only the temperature of the sphere is wanted, the following equation for v is valid:

(8)

~-~-r )r

v2sinvro - CoPoro3C----~P .[sinLrovr°

vcos Vrol

(14a)

0

Since Q is the amount of heat lost by the sphere, its temperature has decreased by: T O-T(r o,t) = Q / C o p o 4 u r 3

(9)

where C Oand po denote specific heat and density for copper.

Re-arranging terms and putting vr ° = O, one obtains: tan O =

O 3Cp / CoP o -

(14b) 0

2

Dental Materials~March 1992 141

,'y

2

iiii/

-.02

l e f f hond side r , g h f hond side

0

-.04

F-c-

4

6

8

t (s)

\

\\

-.06 -.08

i Fig. 4. Plot of eq. (18), valid for BaysUex, for determination of the slope of the straight line which approximates the experimental relation between time and the logarithm of the dimensionless temperature of the metal sphere.

to:

-1

H (3)

LOI H

ol

// I

-2

/ /

(D

I

I

Fig. 3. Graphical solution of eq. (14b), with Baysilex used as a testing material. Inserting numerical values of constants yields: tan 0 = 0/(1-0.8702). The solution becomes 0 = O] = 2,70.

This equation is solved graphically. In Fig. 3, two curves were plotted, representing the lefthand and right-hand sides, respectively, of Eq. 14b as functions of O. The intersections are sought. Apart from the trivial solution O = 0, an infinite number of solutions exists, each corresponding to a certain value of O and hence ofv. Summation should be made over all these values, according to Eq.. 13. . Since there is an exponential 2 . . factor, e D v t m each term m this equatmn, the terms corresponding to higher values of v vanish rapidly, a fact which justifies their omission. The only solution retained is thus: ® = ®1

(see Fig. 3)

(14c)

After this has been done, the temperature of the sphere may be written explicitly: T(r o,t) = T o • e -Dv2t

(15)

Registration of T(ro,t) provides a means of determining D by simple curve-fitting procedures.

EXPERIMENTALPROCEDURES Determination of Specific Heat, C. An adiabatic system was established in a thermos flask, containing a water bath. The water temperature was measured with a thermometer, its reading precision being 0.1°C. Calibration of the thermometer was performed with 0 and 100°C used as fLxed points. A piece of the material under investigation was heated to a known temperature and then immersed in the bath and its temperature rise registered. This was achieved both for the

142 Pamenius & Ohlson/-I-hermal properties of impression materials

polymerized and the unpolymerized material. Since the determination of C requires a temperature change of the material from T 2to T r (see below for explanation), the result obtained is the mean value for the temperature interval (T2, T~). This temperature was chosen so as to be clinically appropriate. For unpolymerized material, the determination must be made on the base and catalyst separately and a weighed mean for the mixture calculated. The following notations are used: T 1 = temperature of water before test; T 2 = temperature ofelastomer before test; Tf = temperature of water after test; C1 = specific heat of water; C2 = specific heat ofelastomer; m 1= amount of water; and m 2 = amount of elastomer. As heat is conserved in an adiabatic system, one has: (T1 - T f ) . C I . E 1

+ ( T 2 - Tf).C2 .m2 = 0

(16)

Eq. 16 is solved with respect to C2.

Determination of Thermal Conductivity, k. Taking the natural logarithm of both sides of Eq. 15, one obtains: l n T ( r o,t) = -Dv2t + lnTo

(17)

Re-writing Eq. 17, for the purpose of obtaining a nondimensional quantity after the In operator, yields:

ln[T(ro,t)/To]=-Dv2t

(18)

Plotting ln[T(ro,t)fr o] vs. t should give a straight line from whose slope, -Dv 2, the thermal diffusivity D and thermal conductivity k are evaluated, Fig. 4. According to Eq. 18, the straight line should pass through the origin. In practice, a straight line may be fitted to the experimental points of Fig. 4 for only intermediate values oft. For small values of t, the discrepancy between model and experiments is due to the finite value of the coefficient of heat transfer across the interface between the metal sphere and the impression material. For large values of t, on the other hand, the deviation is due to the reflection of heat in the outer surface of the impression material since the space it fills is not infinite, as was assumed in the analytical solution. The off-set of the fitted line toward a lower temperature, causing its intersection with the vertical axis to occur beneath the origin, depends, finally, on the finite heat conduc-

Fig. 5. Photo of the experimental set-up used when the themlal conductivity, k, is being determined. The recentty mixed elastomer is inserted into the Rexiglass@halves,which are then put together. The thermocouple senses the temperature of the sphereand isconnected to a recorder.

tivity of the metal sphere. Needless to say, the off-set does not affect the constant D since D is determined solely by the slope, A, of the curve. Using the straight-line approximation for intermediate values oft should reduce the effects of the other sources of error mentioned. This evaluation yields h = -Dr 2. Hence: v 1 = 01 / r o and D =

Ar°2

o',

(19)

From Eq. 4, one finally obtains the thermal conductivity, k: k = D. C . p -

Ar~Cp

(20)

MATERIALSAND METHODS The determination of specific heat, C, follows an often-used

experimental method (Allen and Moore, 1949) and requires no further explanation. The determination of k, on the other hand, is somewhat more elaborate and will be outlined below. A sphere was turned, starting from a copper rod, its final diameter being 2r o = 30 mm. In its center, a thermocouple was installed. The sphere was enclosed in a spherical cavity of 70 mm diameter, excavated by the turning of a Plexiglass® rod 90 mm in diameter. The Plexiglass® was cut across a diameter so that the two halves could be separated and replaced at will. The experimental set-up is shown in Fig. 5. Having heated the sphere to the desired temperature, To, one fills the cavity with unpolymerized impression material, re-assembles the Plexiglass® halves and connects the analogue output signal from the thermocouple to a recorder so that the temperature of the sphere, T(ro,t), may be registered during polymerization. The materials tested are listed in the Table.

RESULTS AND DISCUSSION

Specific Heat. Determination was made for the clinically appropriate temperature of about 30°C. Results are shown in the Table. It is concluded that the three vinyl polysiloxanes investigated display approximately the same specific heat; whereas, for the polyether, the value is considerably higher. The specific heat for any material is directly dependent on the number of degrees of freedom for the molecules (or equivalent). In most materials, this number increases slightly with temperature and so does C (Grimvall, 1986). For small changes around 30°C, this temperature dependence may be neglected. A small decrease in specific heat is connected with polymerization, as seen from comparison of values obtained before and after this reaction occurred. Solving Eq. 16 with respect to the specific heat of the elastomer and differentiating the logarithm ofboth sides yields for the relative error:

~

< A(T 1 - T f )

A(T2 - T f )

Am1

Am 2

AC, (21)

TABLE:THERMAL CONSTANTSOF IMPRESSIONMATERIALS Coefficient of Linear

Density Impression Material

p (kg/m3)

Thermal Expansion (1/°C)

Heat of Specific Heat

Thermal Diffusivity

Thermal Conductivity

C (J/kg°C)

D (@/s)

k (W/m°C)

1700

3.0 " 10.7

0.70

960

3.2" 10.7

0.42

1750

3.2" 10,7

0.78

1070

3.8" 10.7

0.57

Polymerization Rate

Total Heat of Polymerization

z(J/kg.s)

Zo(J/kg)

16

1050

Baysilex

unpolym. polym.

1370

165' 10.6

Provil M

unpolym. polym.

1400

190' 10 .6

1430

175' 10 .6

1060

205' 10.6

3000* 44

2000

Provil H

unpolym. potym.

1470

3.9' 10.7

0.82

950

2.2' 10.7

0.30

2240

3.6" 10,7

0.85

2100

4.3 • 10-7

0.96

3100' 45

1900

40

6000

Impregum

unpolym. polym.

* The endothermal reaction is excluded.

Dental Materials~March 1992 143

Provll

H

merization was determined by use of a differential scanning calorimeter (DSC). The heat flow as a function of time is shown in Fig. 6. The exothermal reaction dominates until the majority of the cross-links are formed and the material becomes stable. An endothermal reaction follows. The results presented in the Table were based on this first reaction stage which is the relevant part for the evaluation of heat conductivity. With reference to Eq. la, the DSC results presented in the Table are defined as: QIp =i

which is the heat-of-polymerization rate in the first reaction stage and: $$q dt=z, (251

Fig. 6. Heat flow as a function

of time.

Time (mlnutas)

The first term of the right-hand side is dominant. relevant experimental values in Eq. 21 gives:

Inserting

which may be termed the total heat of polymerization. In summary, computing residual strains in an impression material requires a knowledge of thermophysical constants (Ohlson and Pamenius, 1991). The precision need not be the highest possible, however, since these strains are also affected by boundary conditions which are not accurately known. Therefore, the use of simple methods for determining these constants may be justified. The methods proposed in this paper seem to be cost-efficient compared with methods that use the best equipment available; yet, they should provide results which are sufficiently accurate for use in the calculation of residual strains.

ACKNOWLEDGMENT The help ofP. Lockowandt, Professor emeritus, for computeraided completion of Fig. 2 is gratefully acknowledged.

$ ~0.03+0.004+0.005+0.005+0.001=0.04 I I Thermal Conductivity. The results from the transient

heat conductivity experiments are shown in the Table. Primary values in the form of thermal diffusivity coefficient D are shown first. The scatter in D is due to errors in the present experiment only. The scatter in Kdepends also on experimental errors in C and p, according to Eq. 20. The relative error

Received November Address

correspondence

Department

of Dental Materials

(23) Eq. 12 is used for calculating D. Numerical values inserted yield: AD 5 0.1+2 l-lD and $ I0.14+0.04+0.001 L- 0.2 1

I

It can be seen in the Table that the values for unpolymerized material are somewhat lower than those for polymerized. This may be explained as follows: Recent research in solidstate physics (Grimvall, 1986) suggests that there exists a “minimum heat conductivity” in “strongly disturbed systems”, to which elastomers belong. Such systems are reputed to conduct heat and electricity poorly. It may be assumed that this “minimum heat conductivity” has already been reached for these materials whether they are polymerized or not. Heat of Polymerization. The heat developed during poly-

144 Pamenius & Ohlson/Thermal

properties of impression materials

requests

to:

and Technology

Institute

Box 4064 S-141 04 Huddinge,

Contributions to the error in D are obtained from the slope of the straight line of Fig. 4, which represents dT/dt, and from the graphical solution of Eq. 14b:

and reprint

August 31,199l

M. Pamenius The Karolinska

(22)

16, 1990/Accepted

Sweden

REFERENCES Allen HS, Moore H (1949). A textbook of practical physics. 3rd ed. London: MacMillan & Co., 287-288. Boley BA, Weiner JH (1960). Theory of thermal stresses. New York: John Wiley and Sons, 138-143. Grimvall G (1986). Thermophysical properties of materials. North-Holland, 173-175,229-233. Jorgensen KD (1982). Thermal expansion of addition polymerization (type II) silicone impression materials. Aust Dent J 271377-381.

McCabe JF, WilsonHJ( 1978). Addition curing silicone rubber impression material: an appraisal of their physical properties. Br Dent J 145:17-20. O’BrienWJ,RygeG(1978). Anoutlineofdentalmaterialsand their selection. Philadelphia: W.B. Saunders, 392. Ohlson NG, Pamenius MJ (1991). Dimensional stability and residual strains in elastomers. In: Proceedings of 3rd international conference on residual stresses, Tokushima, 1991. Elsevier (in press). Pamenius M, Ohlson NG (1986). The determination of elastic constants by dynamic experiments. Dent Muter 2:246-250.

Determination of thermal properties of impression materials.

Residual strains are important for dimensional accuracy of impression materials. When these strains are being calculated, knowledge of thermophysical ...
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