1 Introduction

Monocrystalline quartz is a multifunctional material with special optical, piezoelectric, and thermophysical properties (Baek et al. 2018; Trukhin 2019; Yang et al. 2018; Molaei and Siavoshi 2020). It is used for elements of precision optics, resonators, accelerometers, gyroscopes, as substrates for microelectronics (Danel and Delapierre 1991; Lee et al. 2018; Yu et al. 2018; Beitia et al. 2015; Matko and Milanovich 2019; Bai et al. 2020). One of the most effective sensors based on monocrystalline quartz is quartz crystal micro- and nanobalance (QCM and QCN) (Hampitak et al. 2020; Rudolph et al. 2021). The quality and level of roughness can degrade the properties of sensitive elements based on quartz crystals. For example, a decrease in roughness from 600 to 2 nm leads to a proportional decrease in the frequency shift and QCM scattering (Cho et al. 2007). The sensitivity of the recognition layers of biosensors is largely determined by their thickness and surface roughness of a quartz crystal (Rianjanu et al. 2019; Leppin et al. 2020; Dong et al. 2019). Such a sensor element requires high-precision processing to a roughness comparable to the parameters of the crystal lattice. Roughness affects not only the recognizing properties of the deposited layers—it is interrelated with the characteristics of crystal vibrations in some surface modes (Hao et al. 2018). To obtain surfaces with a certain roughness, various techniques are used: chemical–mechanical (Xia et al. 2020), laser (Hildebrand et al. 2011; Weingarten et al. 2017), magnetorheological (Cao et al. 2019; Bedi and Singh 2016; Gorodkin and Novikova 2012), chemical (Bo et al. 2014), ion-beam (Zhou et al. 2019), ultrasonic abrasive (Guzzo et al. 2003), combined methods (Gupta et al. 2020; Zhong 2008) and others.

To control the surface roughness of optical and piezoelectric materials, in particular quartz crystals, non-contact optical profilometry and atomic force microscopy (AFM) are common methods (Cho et al. 2007; Dokou et al. 2002; Lapitskaya et al. 2020; Kuznetsova et al. 2020). Optical non-contact profilometry usage provides fast scanning of relatively big areas of interest (Sadyrin et al 2020b; Burlakova et al. 2019), but it’s can give incorrect roughness values due to the influence of the optical properties of the material and the phenomena of reflection and absorption of light. AFM, due to the high accuracy of direct measurement of surface topography, makes it possible to determine roughness with angstrom accuracy (Misumi et al. 2019; Anishchik et al. 2005; Warcholinski et al. 2019).

Any polishing leads to changes in the properties of the surface layers of the material (Kanaev 2019) and in sensitive MEMS technologies the degree of this change must be taken into account. Mechanical polishing can lead to weakening / hardening of the surface layer of the material and formation of the smear layer (Sadyrin et al. 2020a). Laser polishing, due to high temperatures, can lead to phase transitions in the material (Ma et al. 2017). Polishing using focused ion beam is time and resource consuming technique, moreover it poses serious limitations on the sample size (Malshe et al. 1999; Vasiliev et al. 2018). Ultrasonic abrasive polishing is often conducted by skillful handwork, thus the process becomes time-consuming and not well accepted concerning the reliability of the polished surface (Hocheng and Kuo 2002). Among all types of polishing, the most promising is magnetorheological one, which provides the smallest size and number of micro-roughnesses among other methods (Cao et al. 2019; Bedi and Singh 2016; Gorodkin and Novikova 2012; Kim et al. 2009).

Mechanical properties of quartz play a significant role in the QCN design, especially taking into account the decrease in their size and thinning of the sensitive elements (Singh 2016; Liang et al. 2015). The mechanical properties affect the electromechanical coupling coefficient, which is important for the recognition functions of quartz elements (Lamberson and Ramesh 2017). The good resistance of quartz structures to vibrations, shocks and other influences directly depends on its crack resistance. Crack resistance of the material is characterized by the critical stress intensity factor KIC (Lapitskaya et al. 2019). The indentation method is widely used to determine KIC in elements of microtechnology and silicon wafers (Ferguson et al. 1987; Guzzo 2001). The accuracy of KIC determination depends on the accuracy of visualization of the formed cracks.

The aim of the present research was to carry out comparative studies of surface morphology, roughness, mechanical properties, and crack resistance of AT-cut quartz crystals after chemical–mechanical and magnetorheological polishing. AFM was used to increase the accuracy of determining the surface morphology and crack resistance by indentation method for visualizing the surface topography in the initial state and with an imprint and cracks.

2 Materials and methods

2.1 Sample preparation

As an object of research, we took quartz elements—two AT-cut (or yxl/+ 35° cut) plates of crystalline quartz with a diameter of 12 mm and a thickness of 3 mm, which are mass-produced and are widely used in the manufacture of precision resonators and complex filters for devices for stabilization and frequency selection. Initially, both plates were processed by chemical–mechanical polishing (CMP), then one of them was additionally subjected to magnetorheological polishing (MRF). The CMP was produced with a flat polisher using a slurry containing free abrasive cerium oxide. The MRF was carried out on a 5-axis CNC machine (ITMO, Minsk, Belarus) with a small-sized magnetically controlled tool using a water-based polishing liquid with the addition of a nano-diamond abrasive. The amount of material removed from the quartz surface during MRF processing was at least 2 μm to ensure the removal of the damaged layer formed after CMP and the performance of previous operations.

2.2 Experimental procedures

Crack resistance was determined by the indentation method. Three imprints were made on each of the plates at different loads using a PMT-3 microhardness tester (LOMO, St. Petersburg, Russia). A Vickers tip was used as an indenter. The load on the indenter varied from 0.01 to 2.5 N. In our work, we used AT-cut (or cut yxl/+ 35°) monocrystalline quartz. During the manufacture of monocrystalline quartz plates, the orientation on them is not indicated in any way. Therefore, in this work, we evaluated the influence of anisotropy on the value KIC by rotating the crystal during indentation. The effect of the angle of rotation of the quartz plate relative to the indenter (initial 0°, 45°, and 90°) on the shape of the imprint, the type and length of cracks, and the magnitude of crack resistance were obtained. Different orientations of the specimen were used in order to exclude the influence of the orientation of the quartz plate on the values of crack resistance. When processing the data for calculating KIC the direction of the largest total crack length was chosen. The comparison was carried out at a load of 1.5 N.

A Dimension FastScan AFM (Bruker, Santa Barbara, State of California (CA), USA) in PeakForce quantitative nanoscale mechanical mapping mode with a standard silicon cantilever type NSC-11 (Mikromasch, Tallinn, Estonia) with a radius of curvature of the probe tip of 10 nm and a console rigidity of 4.89 N/m was used to visualize imprints and cracks, to quantify adhesion forces and surface roughness (Khort et al. 2020; Lapitskaya et al. 2018; Kuznetsova et al. 2018; Nikolaev et al. 2020). Determination of the surface roughness of the samples was carried out in the semi-contact mode according to ISO 19606:2017. Scanning fields were as follows: 91.8 × 91.8, 10 × 10 and 1 × 1 µm2. The measurements were carried out in 3 different areas of the sample at a distance of about 2 mm from each other. Resolution of the scanning frame was 512 × 512 points. Measurement errors for the axes were as follows: XY: ± 0.2 nm, for Z: ± 0.03 nm. The adhesion force was determined by the maximum force of separation of the cantilever tip from the surface of the samples from the force curves of the “approach-withdrawal”.

Microhardness H, Young’s modulus E, and deformation η (plastic and elastic components of deformation) of quartz were determined using nanoindentation (NI) technique on a Hysitron 750 Ubi NI device (Bruker, Minneapolis, MN, USA) equipped with a Berkovich diamond indenter with a radius of curvature of 150 nm for continuous recording of deformation curves. As a result, data on the applied load and the corresponding indentation depth was obtained as a function of the dependence F = f(hBer). The load was 10 mN. Additionally, studies of microhardness and modulus of elasticity by depth were carried out at a load ranging from 0.2 to 5.0 mN. Elastic recovery ηelast and plastic deformation ηplast were calculated from the shape of the deformation curves F = f(hBer) (Jha et al. 2012). The mechanical work Wtolal, performed during indentation was only partially spent on plastic deformation Wplast (Jha et al. 2012). When the applied load was removed, part of the work (the work of elastic deformation Welast) was released. Formula (1) contains information characterizing the plastic properties of the test specimen:

$${\eta }_{elast}=\frac{{W}_{elast}}{{W}_{total}},$$
(1)

where Wtolal = Welast + Wplast.

The plastic component was calculated as

$${\eta }_{plast}=\left(1-{\eta }_{elast}\right)\cdot 100\mathrm{\%}$$
(2)

To determine KIC several formulas were used to check the correctness of the obtained values and the influence of the parameters included in the formulas on the values KIC (Sergejev 2006; Lawn and Fuller 1975; Evans and Wilshaw 1976; Niihara et al. 1982; Niihara 1983):

$${K}_{IC}=0.0515\frac{P}{{c}_{1}^{\frac{3}{2}}},$$
(3)
$${K}_{IC}=0.079\frac{P}{{a}^{\frac{3}{2}}}\mathrm{log}\left(4.5\frac{a}{{c}_{1}}\right)$$
(4)
$${K}_{IC}=0.035{\left(\frac{{c}_{1}}{a}\right)}^{-\frac{1}{2}}\cdot {\left(\frac{{H}_{V}}{E\Phi}\right)}^{-\frac{2}{5}}\cdot \left(\frac{{H}_{V}{a}^{\frac{1}{2}}}{\Phi}\right)$$
(5)
$${K}_{IC}=0.048{\left(\frac{{c}_{1}}{a}\right)}^{-\frac{1}{2}}\cdot {\left(\frac{{H}_{V}}{E\Phi}\right)}^{-\frac{2}{5}}\cdot \left(\frac{{H}_{V}{a}^{\frac{1}{2}}}{\Phi}\right)$$
(6)
$${K}_{IC}=0.0726\frac{P}{{c}_{2}^{\frac{3}{2}}}$$
(7)
$${K}_{IC}=0.129{\left(\frac{{c}_{2}}{a}\right)}^{-\frac{3}{2}}\cdot {\left(\frac{{H}_{V}}{E\Phi}\right)}^{-\frac{2}{5}}\cdot \left(\frac{{H}_{V}{a}^{\frac{1}{2}}}{\Phi}\right)$$
(8)
$${K}_{IC}=0.014{\left(\frac{E}{{H}_{V}}\right)}^{\frac{1}{2}}\cdot \left(\frac{P}{{c}_{2}^{\frac{3}{2}}}\right)$$
(9)

where P is the load on the indenter, N; \({c}_{1}\) is the length of the crack near the imprint, m; \({c}_{2}\) is the length of the crack from the center of the imprint, m; a is the length of the half-diagonal of the imprint, m; HV is Vickers hardness, GPa; E is the modulus of elasticity, GPa; Ф is a constant, an indicator of the bond reaction in the crystal lattice, Ф ≈ 3.

Formulas (3)–(6) include the length of the crack near the indentation \({c}_{1}\), and in (7)–(9), the length of the crack from the center of the indentation \({c}_{2}\). For selection of the appropriate formula, the ratio \({c}_{2}\)/a should be determined. If \({c}_{2}\)/a < 2.0, then Palmquist cracks are formed in the sample and the calculation is carried out according to the formula (3)–(6) (Sergejev 2006; Lawn and Fuller 1975; Evans and Wilshaw 1976; Niihara et al. 1982); if \({c}_{2}\)/a > 2.0, then median cracks are formed in the sample, and the calculation is carried out by the formulas (7)–(9) (Lawn and Fuller 1975; Niihara et al. 1982; Ebrahimi and Kalwani 1999; Lawn et al. 1980).

In the formulas (3) and (7) only the length of the crack and the applied load are taken into account and there is no dependence on the mechanical properties of the material, in (2) the length of the subdiagonal of the imprint is added to the length of the crack and the load, in (5), (6) and (8)—all indentation parameters, crack length and mechanical properties of materials are used and there is no direct dependence on the applied load. In most formulas used earlier in the literature, loads of more than 1 N were used. In those cases when lower loads (0.25–0.75 N) were used, the values of a, \({c}_{1}\) and \({c}_{2}\) were determined inaccurately and because of this, at low loads, the difference in a,\({c}_{1}\), and \({c}_{2}\) between quartz slices of different orientations was not revealed (Guzzo 2001). The task was to select a calculation method that is most adequate for low loads (from 0.01 to 0.5 N) to obtain the correct value of KIC.

The HV values obtained by the NI method were used in the calculations. In this case, H was taken from the Berkovich imprint, obtained at small (1–5 mN) loads and guaranteed to be free of cracks. This approach seems to be more correct, since microhardness imprints often contain cracks that are not visible to optics.

3 Results and discussion

AFM images of the surface (fields of 10 × 10 and 1 × 1 μm2) of quartz wafers after CMP and MRF are shown in Fig. 1. On a field of 10 × 10 μm2, the surface after CMP looks more homogeneous, without a strongly pronounced direction (Fig. 1a). Quartz after MRF demonstrates elongated depressions oriented in one direction on the surface, there is a pronounced directionality at an angle of about 20° to the Y axis (Fig. 1b). Comparison of the surface profiles shows that the relief drop after CMP was about 5 times higher than that of quartz after MRF. After CMP, particles with a size of 100–150 nm are evenly distributed on the quartz surface (Fig. 1c).

Fig. 1
figure 1

AFM images of the surface morphology of quartz plates after chemical–mechanical polishing (a, c) and magnetorheological polishing (b, d) in fields of 10 × 10 (a, b) and 1 × 1 (c, d) μm2

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Their formation on the surface of monocrystalline quartz is associated with the separation of small volumes of material during processing, their repeated mixing with the participation of polishing liquids. Surface morphology significantly depends on the type of polishing and the used abrasive and liquid during polishing (Bo et al. 2014; Dokou et al. 2002). Presence of the block (grain) structure on the surface after CMP shows that a modified layer was formed. Also, on the surface, like in quartz after CMP, particles with a size of 20–30 nm are distributed. The surface morphology (field 10 × 10 μm2) of the quartz after MRF is similar to the morphology obtained in Zhao et al. (2018). The fine structure of quartz after MRF, revealed at a field of 1 × 1 μm2, shows directional structure elements (Fig. 1d). Lines are visible on the surface, oriented in the same way as depressions, at an angle of about 20° to the Y axis with a period of 20 nm (Fig. 1d, arrows). The inset to Fig. 1d shows an ordered surface structure corresponding to the crystal lattice. Material carryover during MRF does not occur randomly, but according to the directions of the crystal lattice. The direction at an angle of about 20° to the Y axis in the images of the surface with large depressions obviously correspond to the direction in the quartz lattice with a longer bond length and, accordingly, weaker. In this direction, the material was removed first. The inset shows this direction and perpendicular to it. All blocks are 20–30 nm in size and oriented in a given order by the lattice. It can be safely said that the modified layer on the quartz surface after MRF is absent. The lines of preferential grooves on the quartz surface will serve as a guide for its orientation under the indenter when making imprints to determine KIC.

The roughness of the surface of a quartz plate after chemical–mechanical polishing is significantly higher than after magnetorheological polishing. Ra, Rq, and Rz for quartz after MRF were 0.4, 0.6, and 0.8, respectively. The roughness values of the quartz wafer after MRF are in good agreement with (Zhao et al. 2018). On a quartz sample after CMP, the surface layer affects the roughness value: Ra, Rq, and Rz are 4.0, 10.1, and 12.1 nm, respectively, and these values correlate with the results (Yuan et al. 2003). When examining the surface morphology of the quartz plate after CMP, the sites of rupture of the altered layer were identified (Fig. 2). The layer thickness was 70–80 nm (Fig. 2c, red profile). In places of rupture, a completely different surface structure was revealed.

Fig. 2
figure 2

The surface of a quartz plate after chemical–mechanical polishing with areas of damage to the upper layer: a, c morphology obtained on AFM and surface profile, respectively; b morphology obtained on NI

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The roughness of a surface with open areas (without a layer) is 3–4 times less than a surface with a layer. In the mode of adhesion forces (Fig. 2a, adhesion) it was found that the quartz surface without a layer demonstrated higher values of the adhesion force compared to the surface with a layer (Fig. 2c, blue profile). The adhesion force on the surface of quartz with a layer was 13.07 nN, and without a layer—21.97 nN. On the surface of quartz without a modified layer, directional lines of grooves were also revealed, while they are deeper than in quartz after MRF.

The NI method was used to determine the physical and mechanical properties of quartz plates (Table 1). It has been established that the presence of a modified layer has a significant effect on the value of mechanical properties. NI imprints on the surface after CMP with a modified layer were twice as large as without it (Fig. 2b). The Young’s modulus E and microhardness H obtained on a quartz surface without a layer were 101 ± 1 and GPa and 13.7 ± 0.1 GPa while those with a layer were 88 ± 7 GPa and 8.7 ± 2.4 GPa, respectively. Quartz after CMP can be considered as a sample with a “soft” coating of 70–80 nm thickness. Therefore, the quartz substrate makes a significant contribution to these values. Judging by the large size of the imprints, E and H of the modified layer itself is much lower. The modulus of elasticity and microhardness of quartz after MRF are virtually the same (Table 1) as for quartz after CMP without a layer and amounted to E = 99 ± 1 GPa and H = 13.5 ± 0.1 GPa. The values for H and E are close to the ones obtained by the NI method in (Whitney et al. 2007; Yu et al. 2012).

Table 1 Roughness, physical and mechanical properties of quartz plates after various polishing techniques
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According to the dependence of the modulus of elasticity and microhardness on the maximum indentation depth, it was found that under a load of 100 μN, the indentation depth for quartz after CMP was 60 nm, and for quartz after MRF—10 nm (Fig. 3a, b). With a maximum possible instrument load of 10 mN, the indentation depth for quartz after CMP is 200 nm, and for quartz after MRF it is 148 nm. Thus, the presence of a “soft” layer on the surface of the sample after CMP leads to higher values of the indentation depth. After CMP, the observed values of microhardness on the sample increase with increasing load and, accordingly, the penetration depth.

Fig. 3
figure 3

Dependences of the elastic modulus and microhardness of silicon wafers on the indentation depth after CMP (a) and MRF (b) and indentation curves (c,d)

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The indentation curves characterize the level of deformation (plastic and elastic) at the indentation site (Fig. 3c, d). For “soft” materials, the area under the curve is much larger and, accordingly, the plastic component of deformation is larger (Oliver and Pharr 1992). For “hard” materials, the elastic component of deformation is larger and the area under the curve is smaller (Oliver and Pharr 1992). The difference in modulus of elasticity at 11 GPa and microhardness at 5 GPa results in a difference in the shape of the curves for quartz after MRF and after CMP (Fig. 3c, d). The curves for quartz after MRF and for quartz after CMP without a layer coincide (Fig. 3d). For quartz after MRF, the elastic component of deformation is 75.3%, and the plastic component is 24.7%. Quartz after CMP without a layer has virtually the same values. In quartz with a layer, the plastic component increases to 30.7%, while the elastic decreases to 69.3%. The increase in the plastic component in quartz with a changed layer is explained by the shape of the curve, with a long initial section close to the X axis and a significant deflection of the curve.

The shape of Vickers imprints on the surface of quartz after CMP and MRF polishing was practically the same when visualized by optical microscopy. The contour of the imprints was not precisely defined, it is only possible to approximately determine the size of the diagonal. AFM imaging revealed a significant difference both in the shape of the contour and in the morphology of the Vickers indentation faces obtained on quartz (Figs. 4, 5). The shape of the imprints on quartz after CMP demonstrated symmetry, with smooth contour lines (Fig. 4a, b). A small bead is located along the indentation contour from the torn altered layer, the slopes of the imprint were covered with a deformed layer and the slip lines typical of a crystalline material on them were not presented. The tops of the imprint were rounded. Fragments of cracks were visible in the fractures of the modified layer, which at a load of 0.01 N were not even visible on the surface behind the indentation contour. With an increase in the load, the cracks break the layer, and partially peels off the surface (Fig. 4d). On the figures with absent surface layer (Fig. 4d), slip lines and cracks are visible. Thus, it can be assumed that the material is cracking, but under the layer it is not possible to reveal this from the surface.

Fig. 4
figure 4

AFM images of indentations at a load of 0.01 N (a, c) and 1.5 N (b, d) on quartz after chemical–mechanical polishing

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Fig. 5
figure 5

AFM images of indentations under a load of 0.01 N (a, c) and 1.5 N (b, d) on quartz after magnetorheological polishing

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Imprints on quartz after MRF possess a structure typical for “hard” materials (Fig. 5). The shape of the imprint is incorrect, which corresponds to the fact that the deformation pattern in an imprint on a crystal depends on the orientation of the crystal lattice with respect to the indenter (Eidel 2011). The largest crack in all imprints has a direction of 20° along the Y axis. According to (Pizzagalli et al. 2013), the crack propagates primarily in the direction of the least strength of the crystal lattice. In this case, this direction coincides with the direction of the predominant entrainment of material in Fig. 1b, d. Chips of the material follow the contour of the imprint. Among the indentation contour at 0.01 N, only one edge was straight (Fig. 5c). There were no straight edges among the imprint contour at 1.5 N (Fig. 5d). On the imprint faces there were “steps” typical of crystal slip lines (Fig. 5c, d) (Januchta and Smedskjaer 2019). In the fields of smaller size, it can be seen that at low loads, the width of the steps was greater, about 200–400 nm (Fig. 5c), and with increasing load, the width decreased to 100–200 nm (Fig. 5c, d).

Since the quartz plates (disks) have a symmetrical shape without marks characterizing the orientation, the effect of the angle of rotation (0°, 45°, and 90°) of the quartz plate on the shape of the imprint, the type and length of cracks, and the magnitude of crack resistance was obtained. Rotation by 45° and 90° was performed relative to the original position, which corresponded to the imprints described above (Fig. 4c and 5c). The shape of the imprint on quartz after MRF changed significantly with the rotation angle (Fig. 6a). The most regular equilateral shape of the imprint was obtained at an angle of rotation of 45° and it was similar to the shape of the imprints obtained in (Whitney et al. 2007).

Fig. 6
figure 6

Shape of Vickers indentations from the angle of rotation of quartz plates after MRF (a) and CMP (b)

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The shape of the imprint at an angle of rotation of 90° was, as in the initial one, of the curved form. In quartz after CMP under a soft modified layer, the shape of the imprints at all angles of rotation was correct and the appearance of the prints at each angle was the same (Fig. 6b). The length of cracks, depending on the angle of rotation, for quartz after MRF changed and a for quartz after CMP remained close (Table 2). To determine KIC, the most suitable orientation for quartz after MRF was the initial one, since it corresponded to the longest cracks. The length of the half-diagonal of the imprint a, the lengths of cracks near \({c}_{1}\) and from the center of the \({c}_{2}\) of the indent, as well as the ratio of the length of the crack from the center of the indent to the length of the half-diagonal \({c}_{2}\)/a are given in Table 3. The presence of a layer on the quartz surface after CMP interferes with the reliable identification of the lengths of cracks on the surface of the samples. The difference between the length of cracks near the indent \({c}_{1}\) after CMP and MRF reaches 30%, between the length of cracks from the center of the indent up to 10%. In all the cases, the crack of shorter length was observed for quartz after CMP. This can affect KIC values for completely identical materials and lead to significant differences in values.

Table 2 The half-diagonal length of the indentations a, the crack lengths \({c}_{1}\) and \({c}_{2}\), and the \({c}_{2}\) / a ratio for monocrystalline quartz plates after CMP and MRF, depending on the rotation angle
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Table 3 Diagonal length of imprints a, crack length \({c}_{1}\) and \({c}_{2}\) and the relation \({c}_{2}\)/a
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The table shows that the ratios \({c}_{2}\)/a, which is the criterion for choosing the formulas for calculating KIC for quartz, differ. For quartz after CMP for loads up to 1.0 N, the calculation should be performed using the formulas (3)–(6), since the value of \({c}_{2}\)/a < 2.0, and for loads above 1.0 N the value of \({c}_{2}\)/a > 2,0 and the calculation was carried out according to the formulas (7)–(9). For quartz after MRF at all loads \({c}_{2}\)/a > 2.0 and the calculation was carried out according to the formulas (7)–(9).

The KIC values at low loads from 0.01 to 0.5 N determined by the formulas (3), (4), (7) and (9) proportionally depend on the applied load P (Fig. 7a, c) and with decreasing load the KIC values decrease as well. At loads of 0.5 N and more, the KIC values for quartz in both types of polishing are more stable and slightly change relative to the average value (Fig. 7b, d). The KIC values obtained by formulas (5), (6) and (8), which do not include the load in the calculation, and in the region of low loads slightly deviate from the average value.

Fig. 7
figure 7

Dependence of KIC on the load from 0.01 to 0.5 N (a, c) and from 0.5 to 2.5 N (b, d) for quartz after chemical–mechanical polishing (a, b) and magnetorheological polishing (c, d)

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A significant decrease in KIC values in the load range of 0.01–0.2 N did not allow to use the formulas that include load. Formulas that do not include load give consistent results over the entire load range, high to small, and average values compared to the range obtained from the no load formulas. When comparing the KIC values of quartz, one should pay attention to the formula by which it was calculated, since with the same crack length and half-diagonal of the imprint, the values may differ by 4 times. At low loads less than 0.5 N, one should pay attention to the type of polishing with which its surface was treated. At loads over 1.0 N, the type of polishing does not significantly affect the KIC value.

According to the average KIC values determined in the area of loads above 0.5 N (Fig. 7b, d), it was obtained that the values for quartz after CMP are higher than for quartz after MRF (Fig. 8). However, this is only due to insufficient detection of the end of the crack as a result of the presence of a softer layer on quartz after CMP. The values obtained by the formulas (3), (4), and (9), which include only the crack length, half-diagonal length of the imprint and load, demonstrate a wide scatter. The values of KIC, determined by the formulas (5), (6) and (8), in comparison with other formulas, are close to each other. The results of formula (7) at loads less than 0.5 N significantly depend on the load, as well as formulas (3), (4), and (9), and with decreasing load, the values of KIC also decrease. At the loads above 0.5 N, the results of the formulas are close in values to those obtained by formulas (5), (6) and (8). Thus, we can propose that the formula (7) can be applied at loads of 0.5 N and higher. The values obtained by the formulas (5), (6) and (8) can be considered the most reliable ones, since they are very close to the values obtained in (Whitney et al. 2007), and are close to each other. Moreover, the applicability of these formulas depends on the value of \({c}_{2}\)/a. If Palmquist cracks are formed (\({c}_{2}\)/a < 2.0), then (5) and (6) are applied, and if median cracks (\({c}_{2}\)/a > 2.0), then the formula (8) is recommended.

Fig. 8
figure 8

Average KIC values determined by formulas (3)–(9)

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The calculation of KIC for quartz samples after two different treatments at different angles of rotation of the samples relative to the indenter was carried out according to the formulas (5), (6) and (8). It was found that since the angle of rotation affects the length of cracks at the indentation, it also affects the value of KIC. At an angle of 45°, KIC increases in quartz after MRF, and decreases in quartz after CMP (Fig. 9a). At an angle of 90°, the opposite is true (Fig. 9b). The KIC values, depending on the angle, can vary by 20–30%, both upward and downward. Hence, we can conclude that when determining KIC in a single-crystal material, one should take into account the orientation of the sample relative to the indenter, otherwise the error may amount to 20–30%.

Fig. 9
figure 9

Average values of KIC determined by formulas (5), (6) and (8) of quartz plates after MRF (a) and CMP (b)

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4 Conclusions

The surface of monocrystalline quartz after two types of polishing techniques (chemical–mechanical and magnetorheological) was investigated in this work. The effect of treatment on surface morphology, roughness, elastic modulus, microhardness and crack resistance was evaluated. The criterion for crack resistance was the critical stress intensity factor KIC. Various methods for calculating KIC were compared in the load range from 0.01 to 2.5 N. The main conclusions are as follows:

  • using AFM and NI methods, we have comprehensively described the surface morphology, physical and mechanical properties of the quartz surface after chemical–mechanical and magnetorheological polishing;

  • on the surface of quartz after chemical–mechanical polishing, there is an altered layer with a thickness of 70 nm, which reduces the E and H values of the surface, hides cracks and makes it impossible to determine the true KIC at loads below 1 N;

  • magnetorheological polishing achieves surface roughness Ra = 0.4 nm without the presence of extraneous layers;

  • it is recommended to measure KIC over several indentations, using different angles of rotation of quartz samples relative to the indenter for a more accurate determination of the fracture toughness value;

  • KIC was determined to be 0.88–1.20 MPa m1/2 when the specimen was oriented relative to the indenter, giving the maximum crack length, and 1.27–1.74 MPa m1/2 when the specimen was oriented relative to the indenter, giving the minimum crack length. Differences in values associated with the orientation of the indenter relative to the quartz sample can reach 20–30%;

  • when comparing KIC with the literature data, one should take into account the formula by which the value and technology of quartz surface treatment were calculated. It is shown that for calculating KIC one should also take into account the physical and mechanical characteristics of the material, the geometric parameters of the imprint.