Direct Measurement of the In-Plane Thermal Diffusivity of Semitransparent Thin Films by Lock-In Thermography: An Extension of the Slopes Method

We present an extension of the well-known slopes method for characterization of the in-plane thermal diffusivity of semitransparent polymer films. We introduce a theoretical model which considers heat losses due to convection and radiation mechanisms, as well as semitransparency of the material to the exciting laser heat source (visible range) and multiple reflections at the film surfaces. Most importantly, a potential semitransparency of the material in the IR detection range is also considered. We prove by numerical simulations and by an asymptotic expansion of the surface temperature that the slopes method is also valid for any semitransparent film in the thermally thin regime. Measurements of the in-plane thermal diffusivity performed on semitransparent polymer films covering a wide range of absorption coefficients (to the exciting wavelength and in the IR detection range of our IR camera) validate our theoretical findings.


1) Modeling
Consider the geometry shown in Figure 1 of the main text. The power distribution ( ) inside the semitransparent thin film can be calculated using the Beer-Lambert law taking into account multiple reflections of the laser beam at the film surfaces, 1 where is the reflectance and is the absorption coefficient of the film, both taken at the laser wavelength. A fraction of this power is converted into heat ( ) = ( ) and is the heating source of the film. Where is the efficiency of the light to heat conversion. We consider heat S-2 conduction by diffusion in the film. Accordingly, the heat diffusion equation for the presented configuration reads where = ( ⃗, ) is the temperature field inside the sample at the angular frequency = 2 , and is the thermal diffusivity and thermal conductivity of the film, respectively.
Because the film is thermally isotropic, Equation (S2) has cylindrical symmetry. Thus, we can express the temperature field as a Hankel transform: (S4b) The solution of the homogeneous part of Equation (S2) gives the other two coefficients in Equation (S3) by considering the following boundary conditions: where ℎ is the linearized combined coefficient of convective and radiative heat transfer. [2][3] Accordingly, the temperature field given in Equation (S3) can be written as and ℎ ′ = ℎ/ .
The semi-transparency of the film in the infrared (IR) region is also considered. In particular, for the detection range of our IR camera (7.5 -14 m), the signal is collected not only from the sample surface but also from the inside.
Let us consider an effective IR absorption coefficient , which averages the IR absorption over the detection range of our IR camera. This effective coefficient is a good approximation for materials with smoothly varying IR absorption spectra in the corresponding detection range. 4 Accordingly, the signal recorded by the IR camera can be written as, where 0 = {0, } for the IR camera placed in the same side (front-face configuration) or in the backside (rear-face configuration) of the illuminated surface, respectively. The proportionality constant includes the effect of the IR emissivity, sensor area, IR detection range and the derivative of the Planck distribution at room temperature.
Setting 0 = 0 in Equation (S8) allows us to evaluate the resulting integral and obtain an expression for the signal recorded by the IR camera, in the front-face configuration (see Equation (1) in main text). Similarly, to the previous case, we set 0 = and evaluate the integral in Equation (S8). We obtain the following expression for the signal recorded by the IR camera, in the rear-face configuration (see Equation (2) in the main text).
In order to show the equivalence of front-face and rear-face signals for the thermally thin case, we performed simulations using Equations (1) and (2) of the main text and we plot the difference in 'amplitude' and phase. The results are shown in Figures S1a and S1b, respectively.
For these simulations, we have used the same parameters as in Figure 2a of the main text. S-4

2) Experimental section
Photographs of PMMA and LDPE samples for LIT measurements Figure   to resolve accurately with our detector. Therefore, we set the opaqueness threshold to AVis>4 (for the excitation wavelength) and AIR>4 (in the IR range).

S-9
Reference thermal diffusivity measurements of thick PMMA and LDPE samples

1) Preparation of thick reference samples for Xenon flash analysis
1 mm thick PMMA samples were fabricated by compounding and injection molding. PMMA pellets and PR powder were mixed under N2 flow in a twin-screw compounder with a stirring speed of 40 rpm and at a temperature of 240 °C. Then, the compounded material was directly filled into the injection unit. Disks with a diameter of ∼27 mm and a thickness of ∼1 mm were fabricated using an injection force of 6 kN and a tool temperature of 20 °C. In this way, PMMA disks with 0 wt%, 2 wt%, and 6 wt% PR, respectively, were prepared. LDPE samples were taken from the remaining LDPE disks which were already prepared using injection molding (see Experimental section in the main text).

2) Cross-plane thermal diffusivity measurements using Xenon flash analysis (XFA)
Xenon were measured before and after thermal annealing (90 °C, one week).

3) X-ray diffraction (XRD) and small angle X-ray scattering (SAXS)
To demonstrate the amorphous nature of the PMMA samples and the semi-crystallinity of the LDPE samples we performed X-ray diffraction (XRD). The measurements were conducted in Bragg-Brentano geometry on an Empyrean system with a PIXEL solid-state detector (PANalytical, Almelo, Netherlands) using Cu-radiation.

S-10
Small-angle X-ray scattering (SAXS) was performed on the PMMA and LDPE samples using the lab-based system Ganesha Air (SAXSLAB, Denmark), equipped with a rotating anode (copper, MicroMax 007HF, Rigaku Corporation, Japan) and a position sensitive detector (Pilatus 300K, Dectris). The samples (and air as background) were measured as obtained in parallel and perpendicular beam geometry. Different detector positions were used to cover a larger scattering range ( = 4 sin ( 2 2 )).

4) Discussion of polymer microstructure
The As can be seen in Figure S9a and b, the XRD patterns reveal no significant differences in the polymeric structure between the thin and thick PMMA samples prepared for lock-in thermography and Xenon flash analysis, respectively. In both cases, the small Bragg reflections base on the incorporated PR dye and increase slightly with increasing concentration of the dye.
This shows that all samples exhibit a similar polymeric microstructure independent of the PR content, sample thickness, and preparation method (injection molding vs. doctor-blading).
S-12  We found comparable degrees of crystallinity (~ 45%) for all LDPE samples. Consequently, the addition of phenol red does not affect the degree of crystallinity of the LDPE samples.

S-13
Furthermore, also the subsequent hot-pressing of the injection molded LDPE disks does not change the degree of crystallinity.
Usually, one expects for a semi-crystalline film that (i) the amorphous and crystalline regions are randomly distributed over the solid material (isotropy) and that (ii) the small crystallites are polydisperse. Nevertheless, due to external forces like shear stress or elevated temperatures phase segregation between the amorphous phase and the crystallites cannot be ruled out.
Consequently, we performed SAXS experiments in parallel and perpendicular beam geometries to study the in-plane and cross-plane polymeric microstructures, respectively. Figure S10 shows an example of such a study for the LDPE samples with 6 wt% PR. Astonishing, the SAXS patterns taken perpendicular exhibit a broad reflection at ≈ 0.04 Å -1 , which corresponds to a correlation length of about 15 nm and hints to a two-phase LDPE system (segregation of crystallites and amorphous LDPE). This feature is anisotropic spot-like for the thick sample ( Figure S10a) and isotropic for the thin film ( Figure S10b). As a consequence, the crystallites of the thick samples have a stronger tendency to align lamellar-like parallel to the sample surface (i.e., in in-plane direction) compared to the thin samples.
In the mesoscopic scattering signals observed for the thick and thin samples in parallel beam geometry ( Figure S10c and S10d), the oscillation at ≈ 0.04 Å -1 is attenuated, and the patterns appear more isotropic.
Thus, the thick samples (prepared for XFA) and thin films (prepared for LIT) exhibit a different microstructure despite their comparable degree of crystallinity. Both samples show a preferred orientation of the crystallites in the in-plane direction and more random orientation in the crossplane direction. However, the anisotropy is much more pronounced for the thick samples. This difference in the microstructure may be associated with the different fabrication processes and the applied temperature gradients (injection molding vs. injection molding and subsequent hotpressing).
S-14 In Figure S11 the two-dimensional SAXS data of a thick LDPE sample as used for XFA and after additional annealing at 90 °C for one week are compared to illuminate the effect of the temperature/fabrication method on the orientation of the crystallites. All annealed samples show a significantly more preferred orientation of the crystallites compared to the non-annealed state. Note, that the spots appear even triangular for the data of the annealed film in perpendicular geometry and at the same time, the amorphous halo is less pronounced. We conclude that a direction dependent crystal growth is introduced by thermal annealing. The same trend is observed in case of the thin films, but much less pronounced. S-15

5) Comparison of thermal diffusivity data (LIT vs. XFA)
Figure S12 summarizes the thermal diffusivity values obtained for the fabricated PMMA and LDPE samples with 0, 2, and 6 wt% PR, respectively. The in-plane thermal diffusivity of the thin, free-standing samples was determined using lock-in thermography (LIT). The 1 mm thick reference samples were characterized using Xenon flash analysis (XFA). It is to be noted that this measurement technique gives the cross-plane thermal diffusivity.
S-16 As discussed in the main text, we obtain an in-plane thermal diffusivity of ~ 0.12 mm 2 s -1 for all PMMA samples independent of the PR content. The cross-plane values obtained from XFA are ~ 0.11 mm 2 s -1 , and thus close to the data obtained from LIT. This good agreement confirms that the microstructure of the amorphous polymer is similar (see XRD curves in Figure S8, Figure S9a and S9b) leading to isotropic thermal diffusivity values.
However, in the case of LDPE (before annealing), we extracted significantly lower cross-plane thermal diffusivities from XFA (~ 20 %). We relate this discrepancy to the difference in the underlying microstructure as discussed above. Since the degree of crystallinity of all samples is the same, the preferred in-plane orientation leads apparently to a reduced cross-plane thermal diffusivity as measured by XFA.
Thermal annealing at 90 °C for one week led to a more pronounced anisotropy within the thick films. In comparison to the values before thermal annealing, we obtained similar in-plane S-17 thermal diffusivity values from LIT and slightly higher cross-plane thermal diffusivity values from XFA after annealing.
In summary, we found that comparable in-plane and cross-plane thermal diffusivity values are obtained for an amorphous polymer independent of the fabrication method. In case of a semicrystalline polymer, the fabrication method has a strong influence on the polymer microstructure, and thus on the in-plane and cross-plane thermal diffusivity, which renders a quantitative comparison difficult.