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Stress in the layer is an essential parameter to avoid cracking or distortion of wavefront. We present a bench for stress measurement, lower than 10 MPa, from birefringence measurement on organic/inorganic coatings manufactured by sol-gel technology. We use the induced birefringence by Brazilian tests to standard our measurement. The results obtained with our layer are compared to Stoney measurements.
© 2016 Optical Society of America
1. Introduction
For years, CEA has investigated the laser damage of optical components for high power laser and its influence on laser beam propagation. The studies are about the materials (silica, KDP, DKDP) or thin layers. The laser damage of optical component is studied with different benches as Lutin [1] and Socrate [2]. However, to understand the phenomena that produce laser damage, we need to determine the mechanical parameters of optical coatings or optical material in order to refine the physical models [3].
Currently, we are studying optical layers having elastic properties required to absorb mechanical shock that causes the laser damage. One of the used characterization techniques is the generation of laser ultrasound. This technique has already helped us to classify the mechanical properties of sol-gel materials according to the implemented synthesis [4]. Today, we introduce a birefringence bench that was inspired by a previous bench used for the study of (D) KDP [5]. The silica substrate will help us to determine stresses because its index changes when a thin layer is deposited.
In this paper, we describe the experimental means; the simulations of the induced mechanical stresses inside a silica disc hold in a V support and the optical simulation. This simulation will be compared to experimental results through a link that has been determined by prior internal standardization. The studied materials are thin layers of ormosil (organically modified silicates) made by sol-gel process. This hybrid material is a mixture of silica which gives rigidity and polydimethylsiloxane (PDMS), an organometallic polymer, which provides the elastic character to the final material. Bulk ormosils were studied by MJ. Mackenzie [6]. They have a wide range of variation of Young's modulus, therefore of stresses according to the PDMS rate. The stress of these stacks can be low even lower than 100 MPa.
2. Stress in thin layers
In the case of thin films on a substrate, stresses are related to the growth of the layer on the substrate and its thickness or are thermally induced.
2.1 Stress characterization method for thin layers
The techniques used to measure stresses of films are: X-ray diffraction [7], the indentation measurements [8], curvature measurements using the Stoney's formula [9, 10], the photoelasticity [11], and microelectromechanical measurements [12].
2.2 Bend radius measurement
The curvature radius measurements are familiar. They are carried out by profilometry on silicon substrates or by interferometry on glass substrates. The profile of a surface is measured before and after coating to determine the sag (maximum deformation) Δf. We use Campbell formula [13] (Eq. (1)) to find the stress σLayer in the stack layer/substrate.
where es and eLayer are the thicknesses of the substrate and the layer, R is the radius of curvature of the distortion; M is the biaxial modulus that takes into account the Young's modulus and the Poisson's ratio of the substrate. These stresses induce distortion which is either convex when the layer is in compression, either concave when the layer is in tension (see Fig. 1). The distortion is measured on the bare face (uncoated face) to avoid taking into account the thickness heterogeneities coming from the deposit method [14].These measurements are carried out by a profilometer Dektak 8 of Veeco with coatings made on silicon wafer of 500 µm thickness and 100 mm diameter using the Stoney's method [15].
3. Experimental bench for measuring the birefringence
3.1 Experimental bench description
The experimental set up is shown in Fig. 2. A polarized HeNe laser beam is directed with two mirrors (1, 2) and focused by a lens (1) of 800 mm focal length on the sample to be measured which is placed between two Thompson glans serving as polarizer (2) and as analyzer. The beam exiting from the analyzer is focused on a photodiode by the lens (2) of 200 mm focal length. An interference filter in front of the photodiode blocks all unwanted ambient lighting. The output signal is measured by a voltmeter HP34401A,
The sample is mounted in a V-holder on the x&y motorized translation stages to perform mappings (with a resolution of 1 micron) with travels going up to 100 mm. The analyzer is put on a motorized rotation stage for adjusting its position in relation to the polarizer. The motorized linear stages are piloted thanks to an ESP 300. The first polarizer is a dichroic film polarizer that attenuates the HeNe beam to avoid detectors saturation. A silicon photodiode placed on the reflection of polarizer 2 controls temporal fluctuations of the laser beam. This detector being connected to a synchronous detection SR830 requires a chopper. The V-holder is made of ertalon. Its tightening is controlled by an application force sensor whose its output is displayed on an oscilloscope TDS100. All equipments are driven by a computer via IEEE 488.
3.2 Signals stability
The stabilities of the laser and the force sensor are checked by mappings (Fig. 3(a) and Fig. 3(b)) representing their variations over time (time put to make these mappings). We estimate from these data that fluctuations expressed in terms of peak to valley (PV in %) and standard deviation (σ in %) of the laser source and the force sensor are low and can be neglected thereafter because σ << 1%.
Fig. 3 (a). Force sensor fluctuation over time in mV. (b). Force sensor fluctuation over time in mV.
3.3 Malus's law
To determine the correct analyzer position θ0 in relation to the polarizer, we draw the variation of normalized intensity INormalized depending on the analyzer angle θ (Fig. 4(a)) and we fit with the Malus's law: INormalized = cos2(θ−θ0). Experimentally, the analyzer and the polarizer are parallel to θ = 82° and crossed for θ = 172°. For some experiments, we set the analyzer to 165 °.
Fig. 4 (a) Experimental measurement and corresponding theoretical fit. (b). Experimental determination of the size ωx of our laser.
3.4 Laser beam size
We use the knife method to know the size of the laser beam on the sample. For a gaussian beam, the values for x1 = -ωx and x2 = ωx are defined between 0.92135 and 0.0786496 respectively whatever ωx (or ωy) with IDetector(x0) corresponding to Eq. (2).
IDetector(x0) represents the flux received by the detector when the knife is in x0 .The experimental curve, Fig. 4(b), gives 2ωx = 451 µm, for y we found 2ωy = 453 µm.
4. Link between stress and birefringence
4.1 Mechanical simulation
The holder being a V setting (Fig. 5), we apply a force F determined by a force sensor at the top of this holder. This force is split into two equal forces that apply to 2 upper contact points of the set holder/sample. The sample is the seat of a superposition of two perpendicular Brazilian tests, which have the advantage of having literal solutions. For the case of a thin disc (t = 5 mm << R = 25 mm) subjected to a vertical linear force F of 20 N/mm, the stresses σx, σy and the shear τxy can be expressed in the 3 following equations [16]:
For the same disk subjected to a vertical linear force of 20 N/mm and a horizontal linear force of 20 N/m, the total stress field is given by:
These quantities are displayed (in MPa) by the respective mappings in Fig. 6(a), Fig. 6(b) and Fig. 6(c):
According to C. Klein [17, 18] and to study the influence of a deposit on a substrate, we express the stress σx, σy, and the shear τxy in cylindrical geometry σr, σθ, τrθ [19] (Eq. (3) to Eq. (5)) that are represented in Fig. 7(a), Fig. 7(b) and Fig. 7(c) respectively (expressed in MPa).
4.2 Optical simulation
An isotropic material has a spherical index ellipsoid that is deformed when it is subjected to loading. That causes a non-uniform birefringence state in the material. We shall limit demonstration to symmetry planes given by the direction of the linear force F1 and F2 (see Fig. 5).
The index variation Δnij is linked to the initial index by following relation [20]
with qikjl the piezo-optical tensor, σkl the stress tensor, pijmn the photoelastic tensor and emn the strain tensor.For a silica disk subjects to a stress due to isotropic deposit of a homogeneous layer, the variations of optical index δn1 and δn2 according to the both main directions (1) and (2) of stress tensor plane (u1 and u2) are given by [17] where q// and q┴ are the parallel and perpendicular elasto-optical constants and n is the initial optical index [18]:
Passing through the disc, the phase along the propagation direction (z) of the wave is given by ϕ(Μ) = k.(nM - 1).t where k = 2.π / λ is the wavenumber, nM is the refractive index of the material at the point M and t is the thickness of material. Under the action of a radial stress at each point M of an isotropic material, a birefringence on neutral axes, ur and uθ polar axes, in this point is induced. This birefringence leads to a variation of the phase ϕ(Μ) such as
where δt is the thickness variation due to the stress either:
Silica disk has a thickness t much smaller than its diameter ϕ, we are in geometry called “window” where the stress along z is null (σz = 0). The wave is transverse so δn depends on δn1 and δn2. The phase shifts at M are:
According to the mechanical simulation made and taking as values for the silica [21]: 73 GPa for E the Young's modulus, 0.16 for ν the Poisson's ratio, 0.633 µm for the wavelength of the HeNe laser, 5 mm for t the disk thickness, 1.45 for n the refractive index, 2.8 10−6 for q// and 0.58 10−6 for q┴ [16], we get phase mappings for δϕ1 (Fig. 8(a)) and for δϕ2 (Fig. 8(b)). Then we are reduced to an ellipsometry problem. The vertically polarized incident wave is projected on the neutral axes u1 and u2. Passing through the plate, each component of the amplitude is shifted by δϕ1 (M) to the projected component on u1-axis and δ ϕ2 (M) to the projected component on u2-axis. These two amplitudes are then projected on the analyzer-axis to be added. The collected intensity is the module of these summed amplitudes.
For any αP polarizer angles and any αA analyzer angles, the transmitted intensity I through the analyzer, at a point M located in an αE direction in the rectangular coordinate system Oxy is given by Eq. (20):
This theoretical intensity is shown in Fig. 9(a) and Fig. 10(a) and compared to experimental results (Fig. 9(b), Fig. 10(b)).
This theoretical intensity is shown in Fig. 9(a) and Fig. 9(b). As expected, the highest signal intensities are spatially correlated with the stress location. Figure 10(a) enables observing the rapid increase of the stress (and thus of the intensity) at the four sites located under the contacts. The maximum values are ranging between 0.84 and 1, the differences coming from the mesh used to calculate the theoretical solution made with 0.25X0.25 mm2 resolution. A higher magnification of one of the zones will be proposed in sub-section 4.3. Figure 9(b) shows the laser intensity obtained during the experiment made with 0.5X0.5 mm2 resolution.
Looking the patterns along the two diagonals of the square zone, the same contours are observed between theoretical and experimental results confirming the symmetries observed in Fig. 9(a). The small patterns orientated along the circle in Fig. 9(b) probably come (1) from a no perfect contact (friction) between the sample and the holder or (2) from a no radial force at the contact. Figure 10(b) shows three peaks with maximum value ranging between 0.88 and 1, in agreement with theoretical results. The maximum at the last peak is equal to 0.53. This difference is due to the spatial discretization of the sensor and maybe due to a local defect on the analyzed surface because the surface of this peak seems comparable to the opposite peak surface. These observations show a perfect agreement between both theoretical and experimental analyses.
4.3 First attempts
The four contacts being linear and similar, we choose a sector among four to perform the calibration by several birefringence mappings with a load of 50 N on an area of 3x3 mm2 (see Fig. 11(a)) and on smaller zone of 0.4X0.4 mm2 to achieve more accurate data (see Fig. 11(b)). Figure 12 shows the mapping obtained with lower load (0.6 N). As the sol-gel layers are measured between a polarizer and analyzer having an angle of 82° (see Chap. 6), we must calibrated our experience with an analyzer angle of 165°.
Fig. 11 (a). Birefringence mapping made on 3X3 mm2 in 100 µm step expressed in V. (b). Birefringence mapping made on 0.4X0.4 mm2 in 20 µm step expressed in V.
Before calibrating, we will discuss about the size of the contact area in order to determine the applied stress. Indeed, according to the load, the contact surface will gradually increase but its knowledge is required. By using our holder made of Teflon, a Hertz plane / cylinder contact is assumed to have 4 lines of contact. We remind that the mechanical characteristics of Teflon (ETeflon = 800 MPa and νTeflon = 0.46 [22]) lead to consider Teflon as a soft material compared to silica.
4.4 The Hertzian contact stress
The mathematical theory of contact problems in 3D was developed by Hertz in 1881 [23]. Hertz shown that the contact surface is an ellipse where the parameters a and b can be calculated from the geometrical parameters of the bodies in contact (Fig. 13). Then the pressure distribution q[x, y] on the contact area is given by [24, 25]:
withwhere q0 is the pressure at contact center, a and b are the semi axes of the ellipse q [x, y] and P is the total force.Under this pressure at point (0, y), there is a deformation w1 in point P1 and a deformation w2 in point P2. If z1 and z2 are the initial drop (without contact) between P1, P2 and the tangent planes to the tops of quadratic forms zi:
The total deformation δ is equal:
We have (C1 + C2) = 0 since the deformation δ is independent of the orientation of the y-axis so:
According Prescott [25], the deformations w1 and w2 under the normal force action are:
w1 = λ1 Φ [x, y] and w2 = λ2 Φ [x, y] with:
andwhere Ei is the Young's modulus and νi is the Poisson's ratio.
Φ [x, y] represents the potential of a point of the surface subjected to a density P' and r is the distance between two points M and M' of the surface having the coordinates (x, y) and (x ', y ') .This gives:
In our case we have a linear contact at 4 contact areas between the sample and our holder. The linear pressure [24, 25] is:
with the maximum contact pressure q0:
q0 is located in the center at y = 0.
The problem is 2D in the case of a linear contact, so it is independent of x thus A = 0 and if z = 0. Finally, we have:
Where s is solution of equation:
The knowledge of B enables to calculate the half width of contact b:
As λ1 = λSilica = 4.23.10−12 Pa−1, λ2 = λTeflon = 3.14.10−10 Pa−1, with R = 25 mm and t = 5 mm, we can draw b according to P (Fig. 14).
Knowing b, we can calculate the maximum pressure of contact q0 = 2P / (π t.b) in MPa. q0 is given in Fig. 15(a) and the stress q(y) also denoted σ is drawn in Fig. 15(b) as a function of P and y.
Fig. 15 (a). q0 (in MPa) as a function of a load P(in N) for t = 5 mm and R = 25 mm. (b). σ(MPa) as a function of a load P(N) and y(in mm) for t = 5 mm and R = 25 mm.
The real stress is measured using a helium-neon laser beam having a waist of 351 microns for comparison with σ. When evaluating the stress at the center, we can see, Fig. 16(a) and Fig. 16(b), the difference between the real stress and the measured stress which is significantly weaker when P is weak.
Fig. 16 (a). The punctual stress (in MPa) and the measured integrated stress according to P(in N). (b). Zoom between the punctual stress (in MPa) and the measured integrated stress according to P(in N).
4.5 Standardization
On the chosen sector, we have carried out a series of mapping of 3X3 mm2 then of 0.4X0.4 mm2 with a load going to 0 up to 50 N with an oriented analyzer at 165 °. The load is given by a screw which presses on a load sensor in contact with the holder. This sensor is connected to an oscilloscope TDS 100 which will indicate its possible drifts. The uphill and downhill relationships load/voltage are given in Fig. 17. This Figure shows of hysteresis neither for sensor nor for silica (elastic range).
We have progressively varied the load step by step. For each load both mappings, we have been carried out, and treated to match the maximal value with applied load (see Fig. 11(a). and Fig. 11(b)). The maximal value (measured voltage) corresponds to the output laser intensity i.e. the phase shift (Malus's law). The phase shift gives Δn because t the sample thickness and λ the wavelength of laser are known. To know σ, we use the curve drawn in Fig. 16(a). In this way, σ, Δn and the measured voltage are linked. Figure 18(a) shows the stress as function of the extremums of the measured voltages. Figure 18(b) gives the stress as function of the Δn.
5. Investigated materials
5.1 Sol-gel process
The materials are deposited onto a silica substrate using spin coating. This deposition technique requires the preparation of a sol resulting from organometallic precursor hydrolysis step under acidic or alkaline catalysis conditions. A step of thermal treatment, chemical or ultra-violet curing used to strengthen the mechanical properties of the layer and confer the final properties by forming the metal oxide condensation.
5.2 Ormosil coatings
Ormosils materials (Organically modified silicates) are prepared from a classical sol-gel synthesis of polymeric silica coupled with the addition of elastomers, such as polydimethylsiloxane (PDMS). PDMS reacted in acidic catalysis with the silica precursor, tetraethylorthosilicate (TEOS). The obtained material has good elastic properties compared to the pure silica. The mechanical properties can be adjusted regarding the chemical formulation. The deposited thickness varies from 0.5 to 2 µm, and all the samples underwent the same heat treatment at 120 ° C for 90 mn.
6. Measured sol-gel layers
Several sol-gel layers made of a mixture of polymeric silica with PDMS have been prepared on silica substrates with a variation of PDMS rate, of thickness and of catalyst. There are summarized in Table 1.
Table 1. Measured sol-gel samples
Layers of the same type were manufactured on a silicon substrate in order to measure stress with a Dektak profilometer using the Stoney's method [9, 10, 15]. The thicknesses are determined by UV-visible spectrophotometry [17] from a transmission spectrum (see Fig. 19). Their homogeneity in thickness is estimated using a reflection mapping obtained by a reflectometer measurement [14, 28, 29].
Figure 20 is an example of reflectivity mapping at 780 nm obtained evidencing the well-known radiality of sol-gel layers made by spin-coating [30]. This mapping is carried with 0.5 mm step along x-axis and y-axis. Nevertheless, the standard deviation of the variation of average thickness is equal to 3.7 nm, which is insignificant and can be neglected. The thickness variations also cause a change in transmission, which is less than 1% [14].
All these samples have been measured by photoelasticity with our bench. Beforehand, we measure the uncoated silica substrates that may have residual stress caused by the cleaning acid attack used to remove the sol-gel previous layer. At the point M of the sample, the forces in the thin layer are balanced by the reaction of the substrate [19] that being:
As tL << tS, we only measure by the photoelasticity that the stress σS in the silica, which enables to determine the stress in the layers σL thanks to Eq. (38). The aspect ratio tL / tS is a drawback of our method. Each substrate and each layer are mapped on an area of 80x80 mm2 with 1 mm step in x and y between a polarizer and analyzer having an angle of 82 ° between them. The mappings are then processed to extract the 4 points Mi located at 20 mm from the sample center and on the bisectors of angles between the polarizer and the analyzer. We convert the intensities of these points into variation of ΔnS index by the Eq. (39):
where: ΔV is the voltage difference between a point Mi and the central point M0, λ is the laser wavelength that is 0.633 µm, tS is the sample thickness, I0 is the maximum voltage given by the photodiode when the latter is illuminated by the laser when the polarizer and analyzer are parallel, C is a coefficient corresponding to 0.5 sin 2 (αA-αE). sin 2 (αE - αP), in our case it is equal to 0.481. The ΔnS found are translated into stress σS in the silica and therefore in stress σC in the layer. The results of the parametric study are:
- According to the thickness, for the same catalysis and the same PDMS rate: the layer stress increases wit layer thickness, (see Fig. 21),
- According to PDMS rate, for HCl catalysis: the layer stress decreases with PDMS rate (see Fig. 22(a)).
Fig. 22 (a). Stress according to PDMS rate with HCl catalysis. (b). Stress according to PDMS rate with TFS catalysis.
- According to PDMS rate, for TFS catalysis: the layer stress decreases with PDMS rate (see Fig. 22(b))
The stress comparison between TFS and HCl catalysis suggests that the TFS catalysis induces higher stress in the ormosil layer than in the layers made with HCl catalysis.
Despite the weakness of birefringence signals due to small stress values in our sol-gel layers of ormosils, we can note there is a correlation between our measurement and the ones made with the profilometer. Film stress tends to decrease with the PDMS rate and therefore the Young's modulus Et decreases with the PMDS rate, same as observed in the bulk material [6].
7. Conclusion
This characterization study of mechanical stress in ormosil layers using birefringence measurements has permitted to follow the mechanical behavior of these materials deposited as thin layer. The bench has been calibrated using a biaxial Brazilian test and a silica sample. The contact areas have been investigated with the Herzian contact theory taking into account the laser beam profile. This method has been compared to a more conventional method to measure the stress by measuring the sag. Both methods have given similar results
The developed materials, prepared from a mixture of polymeric silica with PDMS using sol-gel process, show low elasticity moduli and low stress. The elasticity moduli are of the order of several MPa. In both cases, the elastomer rate is the parameter controlling the stress intensity. The increase of PDMS content in sol-gel layer reduces the coating stress. The trend is the same as for bulk materials. These are directly connected to the mechanical properties (Young's modulus) of the coating.
Optimized adaptation of the initial samples and measurement conditions yield more tangible results for this low level of stress and will strengthen these preliminary results.
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