**4.1 Thermal diffusivity of Sorel cement without PVAc**

**Figure 6** presents the experimental curves of normalized amplitude (a) and phase (b) of the photothermal signal and the corresponding theoretical ones for three values of the thermal diffusivity 0.2, 0.4 and 5.0 m<sup>2</sup> /s versus square root modulation frequency for the reference sample (MOC without PVAc). Furthermore, **Figure 6** also presents the experimental normalized amplitude and phase of the photothermal signal and the corresponding theoretical ones for three values of the thermal conductivity 0.01, 0.3 and 0.7 w/mk versus the square root modulation frequency for magnesium oxychloride cement.

As noted in the previous section, the PTD signal is insensitive to thermal conductivity. We can only determine the values of the thermal diffusivity of the magnesium oxychloride cement. The theoretical curve which coincides best with the experimental curve is obtained for 0.4 10<sup>7</sup> m<sup>2</sup> s 1 .

### **4.2 Thermal diffusivity of Sorel cement with PVAc**

We will proceed the same way for the rest of the magnesium oxychloride cement samples to which PVAc has been added with different percentages using the same method. **Figure 7** shows the normalized amplitude and phase of experimental

#### **Figure 6.**

where Ac = αcI0/2kc are constant numbers and α<sup>c</sup> is the optical absorption

*Theoretical variation of the normalized amplitude (a) and phase (b) of the photothermal signal with the square root of modulation frequency for three values of thermal diffusivity and thermal conductivity.*

By applying the continuity conditions of temperature and heat flow at the different interfaces allows us to determine the expression of the periodic tempera-

ð Þ <sup>1</sup> � *<sup>c</sup> <sup>e</sup>σclc* <sup>þ</sup> <sup>1</sup> � *<sup>g</sup>*

**Figure 5** shows the variations of normalized amplitude and phase for three values of thermal conductivity with equal thermal diffusivity D = 0.4 10�<sup>7</sup> m<sup>2</sup> s

and well-defined values of the properties of the ink layer [11]. In addition, **Figure 5** also shows the variations normalized amplitude and phase for different values of

. We notice that the PTD signal is sensitive to both the conductivity and the thermal diffusivity of sample (substrate). Since the value of thermal diffusivity is determined in the previous case (sample without layer), the value of thermal

**4. Determination of the thermal diffusivity of the samples (without**

sivity is measured by the coincidence between the experimental curves of

In this section, we will study the thermal diffusivity of samples (MOC without and with *PVAc*) using the Photothermal Deflection technique PTD. Thermal diffu-

h i

½ð Þ <sup>1</sup> � *rc* ð Þ <sup>1</sup> � *<sup>c</sup> <sup>e</sup>σclc* <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *rc* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>c</sup> <sup>e</sup>*�*σ<sup>c</sup> lc*

*c* � �

> *c* � �

), rc = αc/σc,b=kbσb/ksσs,c=kcσc/ksσs, and g = kfσf/ks.

½ð Þ <sup>1</sup> � *rc* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>c</sup> <sup>e</sup>σclc* <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *rc* ð Þ <sup>1</sup> � *<sup>c</sup> <sup>e</sup>*�*σ<sup>c</sup> lc*

h i

*c* � �

�

ð Þ <sup>1</sup> � *<sup>c</sup> <sup>e</sup>*�*σclc*

(7)

�1

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>c</sup> <sup>e</sup>σclc* <sup>þ</sup> <sup>1</sup> � *<sup>g</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>c</sup> <sup>e</sup>*�*σclc*

coefficient of the black layer.

**Figure 5.**

ture T0 at the sample surface:

*<sup>T</sup>*<sup>0</sup> <sup>¼</sup> *Ec*½ð Þ <sup>1</sup> � *<sup>b</sup> <sup>e</sup>*�*σsls*

where Ec = Ac/(α<sup>c</sup>

**black layer)**

**128**

�2 1ð Þ <sup>þ</sup> *c rc <sup>e</sup>*�*α<sup>c</sup> lc* � � ð Þ <sup>1</sup> <sup>þ</sup> *<sup>b</sup> <sup>e</sup>σsls*

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�ð Þ <sup>1</sup> � *<sup>b</sup> <sup>e</sup>*�*σsls* <sup>1</sup> <sup>þ</sup> *<sup>g</sup>*

<sup>2</sup> � <sup>σ</sup><sup>c</sup> 2

thermal diffusivity with k = 1 Wm�<sup>1</sup> K�<sup>1</sup>

conductivity can be determined with great precision.

photothermal signal to the corresponding theoretical ones.

*3.2.2.2 Illustration of theoretical model*

�2 1ð Þ � *c rc <sup>e</sup>*�*α<sup>c</sup> lc* �� *<sup>=</sup>*½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>b</sup> <sup>e</sup>σsls* <sup>1</sup> <sup>þ</sup> *<sup>g</sup>*

*c* � �

*Experimental and theoretical variation of the normalized amplitude (a) and phase (b) of the photothermal signal with the square root of modulation frequency for three values of thermal diffusivity and thermal conductivity.*

**Figure 7.**

*The normalized amplitude (a) and phase (b) of experimental photothermal signal versus square root of modulation frequency of the magnesium oxychloride cement with PVAc fitted with theoretical curves (line).*

photothermal signal versus square root of modulation frequency of the magnesium oxychloride cement with PVAc fitted with theoretical curves. The theoretical curves that best coincide with the experimental curves allow deducing the thermal diffusivity of the samples (**Figure 7**). The difference between theses curves is due to their different thermal diffusivity. We note that the addition of PVAc significantly influences on the diffusivity values. Indeed, the thermal diffusivity decreases with the percentage of PVAc and reaches their minimum values at 10% and begins to increase after this value. The reduction of thermal diffusivity of cement is due to the insulating effect of polyvinyl acetate particles. The thermal diffusivity of PVAc is measured by the same technique (PTD) [12].
