《1. Introduction》

1. Introduction

Due to its advantages of high speed, high precision, and low friction, the aerostatic guideway has been widely used in many applications, such as measuring instruments, precision guidance, and chip manufacturing [1,2]. Recently, as a result of developments in overlay accuracy and production efficiency in lithography, the variable-slit system (VS), which is involved in the illumination system, requires improvement in the performance of the positioning accuracy and scanning velocity [3–11]. Installed between a series of optical lenses, the VS has the function of eliminating deformation, and ultimately determines the shape and productiveness of the illumination. Therefore, the VS is considered to serve the engineering objectives of better stability and higher speed. The compact multi-throttle aerostatic guideway is the preferred option in the supporting and guiding structure of a VS. Compared with traditional aerostatic guideways, multi-throttle guideways exhibit the advantages of higher stiffness; however, they are more complicated in structure and have stringent requirements in terms of operating conditions and surface profile error [12–14].

Nakamura and Yoshimoto [15,16] have analytically studied the multi-throttle aerostatic guideway under the hypothesis of laminar and uniform flow. Their results show that the double row orifice, which has broader grooves, improves the tilt stiffness of both the pitch and roll directions. In fact, the analytical method is suitable for the study of single parameters of the microstructure, as it locks other parameters within a narrow value range to maintain the effectiveness of the discharge coefficient. The specific influence of the microstructure on the discharge coefficient has been verified experimentally by Belforte et al. [17]. Under his investigation, the compact multi-throttle aerostatic guideway shows a wide value range in microstructure parameters, which affects the discharge coefficient and, consequently, the loading performance. Thus, the use of this kind of traditional method will lead to overestimation of the loading performance. Because of the gaps in analytical and experimental methods, a performance safety margin of about 50% is stipulated in the design of an aerostatic guideway [18,19]. This percentage is close to the contribution of the microstructure parameters to the rotational stiffness. Therefore, traditional methods are not accurate enough to reflect the microstructure in the rotational stiffness. The finite-element analysis method, which has been gradually popularized and is now recognized, provides a possible way to actually study and design the microstructure of a multi-throttle aerostatic guideway.

Kim et al. [20] studied and put forward a near-wall treatment method on a high-Reynolds-number flow in a narrow and rapid field. Gharbi et al. [21] related the Reynolds number to the thickness of finite volume grids, and established a mesh for highReynolds-number flow. Eleshaky [22] and Zhang et al. [23] used computational fluid dynamics (CFD) and the method of separation of variables (MSV), respectively, to verify that the downstream pressure depression has an effect on the carrying capacity and stability. Gao et al. [24] studied the influence of orifice shape on pressure depression and turbulence intensity, and found that the corner radius performed best for flow smoothness and stability. Thus, the design of the orifice and other throttle structures is determined by characteristic dimensions and target performance [25,26]. Yadav and Sharma [27] used the finite-element method (FEM) to study the effects of tilt angle on the performance of the aerostatic thrust bearing with recesses. At present, the main issue in the research and design of the compact multi-throttle aerostatic guideway is to establish the corresponding relationship between the microstructure parameters and the loading performance. Wen et al. [28,29] put forward a mesh adaptation method to capture and refine target hexahedron grids using the finite volume method (FVM). Although the structure we studied is wider and thinner than a typical structure in this field, it was possible to use the mesh adaptation method to subdivide the mesh in the whole region, especially in the region near the microstructure, based on the *y*^{+} distribution.

The tuning of micron-level changing microstructure parameters, according to the loading performance of the multi-throttle aerostatic guideway including carrying capacity, stiffness, and rotational stiffness, was studied using the mesh adaptation method. The level of impact on the above loading performances determines the design process of microstructure parameters, including the recess diameter, recess depth, groove width, and groove depth. In this paper, a design process is put forward for the microstructure parameters of a multi-throttle aerostatic guideway. The working points of carrying capacity and stiffness are unified under adjustments of the recess diameter and average recess depth. The working points of stiffness and rotational stiffness are unified under two-way adjustment of the gradient recess depth. The proposed design process is applied in the guideway design used in the VS of photolithography.

《2. Model and method》

2. Model and method

《2.1. Model establishment and solving method》

2.1. Model establishment and solving method

The multi-throttle aerostatic guideway model used in this research is based on the VS. The VS is the core component of photolithography. As shown in Fig. 1(a), a VS is installed between the quartz rod and relay lens group in an illumination system, and functions as a significant reshaping diaphragm. As shown in Fig. 1(b), the VS is employed to eliminate deformation and to provide a variable rectangular slit in the reticle stage, according to the changing exposure region. Therefore, the diaphragm scanning accuracy of the VS directly affects the overlay accuracy of a photolithography machine. Fig. 1(c) shows the VS of 90 nm-thread ArF photolithography from the Harbin Institute of Technology (HIT), including the aerostatic guideway that was independently developed. Due to its advantages of higher motion and location accuracy under long-term high frequency and high-speed scanning, this kind of aerostatic guideway is expected to improve the productivity of photolithography. As shown in Fig. 1(d), the motor and diaphragm are assembled on both sides of the aerostatic guideway. Because of the position of the quartz rod and the scanning trajectory of the diaphragm, a cantilever is applied to connect the diaphragm of the *Y*-axis and the aerostatic guide sleeve. Therefore, the rotational stiffness of this guideway directly affects the scanning and positioning accuracy of the diaphragm.

《Fig. 1》

**Fig. 1.** VS of photolithography. (a) Placement of VS; (b) operating principle; (c) VS with aerostatic guideway; (d) cantilever of diaphragm.

Although the current aerostatic guideway has several irreplaceable advantages, continuous improvement is still needed for applied photolithography in the following three aspects: First, at least eight aerostatic guideways are restricted in a narrow space around the quartz rod. Thus, the main issue is to establish effective throttling and to provide the necessary air film stiffness, under the limitations of the throttling structure, orifice number, and distribution position. Second, the rotational stiffness provided by the aerostatic guideway should be adequate to withstand the rotational moment under high acceleration motion, which is a common and unavoidable problem of a cantilever diaphragm structure in an illumination system. Last but not least, the stability of the aerostatic guideway is restricted by the supply pressure and must submit to mutual tradeoff with other loading performances. Increasing the supply pressure will lead to a higher carrying capacity but to easier destabilization. Thus, the supply pressure should not exceed 0.4 MPa. The key to solving these problems is to design the microstructure parameters of the multi-throttle on the guideway working surface.

As shown in Fig. 2, the multi-throttle aerostatic guideway contains recesses and grooves near the orifices. When injected with high-pressure air *P*_{0} (supply pressure), the orifice functions as a Laval nozzle and leads to the first-time throttling. The orifices reduce the pressure to *P*_{d}. Then, the recesses and grooves lead to further throttling and form a narrow air film between the guide rail and the sleeve. To study the influence of these recesses and grooves on guideway performance, the absolutely symmetric structure of the first designed guideway, labeled *G*_{0}, was improved by changing the microstructure. As shown in Table 1, structure *G*_{0} was then experimentally improved into structures of *G*_{1}, *G*_{2}, and *G*_{3} via non-equivalence orifice spacing, a non-equivalence recess diameter, and adding grooves, respectively.

《Fig. 2》

**Fig. 2.** A diagrammatic sketch of the multi-throttle aerostatic guideway surface. *P*_{a}: atmospheric pressure. *d*_{0}: diameter of orifice; *d*_{ri}: diameter of recess; *h*_{g}: depth of groove; *h*_{ri}: depth of recess; *L*_{a}, *L*_{ai}: Length of guide sleeve, spacing between orifices; *L*_{g}: length of groove; *w*_{g}: width of groove.

《Table 1》

**Table 1** Dimensions of the designed aerostatic guideway.

*N*: number of orifices.

The mesh adaptation modeling method [28,29] was employed to calculate the narrow and compact flow field generated by the multi-throttle aerostatic guideway, above. In our study, the depth of the microstructure is 10^{3} smaller than the dimensions of the working surface [30]. Although small, this microstructure depth interacts well with the magnitude of the air film thickness, and then dominates the throttle area. By changing the pressure distribution, the microstructure eventually has an effect on the carrying capacity *W*, stiffness *K _{h}*, and rotational stiffness

*K*. Thus, the computational domain is considered to consist of a region of narrow clearance and the microstructure, which complicate the flow. Because of high speed and wall slip, the mesh near the boundary of the microstructure must be stratified for further turbulent operation. According to the principle that the first interior node of the mesh must be placed at the viscous boundary layer, the actual thickness

_{θ}*y*of the grids were predesigned. The depth of the microstructure dominates the layer number and is incorporated into the fluid calculation as well. First, the dimensionless thickness

*y*

^{+}and the velocity

*u*

^{+}are defined as shown Eq. (1):

where *u* is the true velocity, *μ* is the molecular viscosity, *y* is the true thickness of grids; the wall shear rate, wall shear stress, and coefficient of friction are, respectively: , *C*_{f} = 0.058*Re*^{ -1/5} , *Re* is Reynolds number, and *ρ* is the air density.

Secondly, the actual minimum thickness of the original mesh can be expressed as *y*_{min}, as shown in Eq. (2):

Next, the original mesh is analyzed to calculate the flow characteristics, such as local velocity and pressure. As shown in Eq. (3), the dimensionless velocity *u*^{+} is modified as a mixed function of the linear wall law, which is associated with laminar flow, and of the logarithmic law, which is associated with turbulent flow. New *u*^{+} are transformed by Eq. (1) to u for further compressible flow calculation. The above three steps are repeated to satisfy the conditions of *y*^{+} < 5 at the viscous bottom layer and *y*^{+} < 60 at the logarithmic layer, is the dimensionless velocity by laminar flow, and is calculated by logarithmic flow.

In FVM, three-dimensional (3D) and compressible flow is being solved for turbulence. The indicial notation form in the Cartesian coordinates is given by the following:

where , with *i* = 1, 2, 3 and *j* = 1, 2, 3, represents the properties in three directions, respectively, and is the Kronecker delta symbol, *μ*_{t} is the turbulent eddy viscosity, and *μ* is calculated using Sutherland’s law, *p* is the pressure at clearance, *κ *is thermal conductivity and *C*_{p} is constant pressure specific heat.

The turbulent eddy viscosity is computed according to Eq. (5), and *C*_{μ} is a coefficient in *k*-*ε *model. The transport equations modeled for* k* (turbulent kinetic energy) and *ε* (dissipation rate of *k*) in the realizable *k*-*ε* model are shown in Eq. (6), where *S* is the modulus of the mean rate-of-strain tensor defined as *S* = and is the speed of sound. The constants in these equations have been established to ensure that the model performs well for certain coupled flows [31–33].

At last, the carrying capacity *W* and the tilt moment *M*_{t} of the air film can be found by summing the finite element on the working face *s*_{sur} (surface area of each grid). The stiffness *K _{h}* is the differential of

*W*in the direction of air film thickness , and the rotational stiffness

*K*is the differential of

_{θ}*M*

_{t}on the tilt angle

*θ*.

*L*

_{d}is the arm of force in numerical calculation.

《2.2. Steps of the research and design processes》

2.2. Steps of the research and design processes

Using the aforementioned modeling methods of mesh adaptation, we continued to study the influence of the microstructure parameters. This paper mainly examines the effects of microstructure on the loading performance. In previous studies [22–26], the influences of macrostructure parameters and conditions related to throttling have been widely researched. For example, *W* increases linearly in relation to *P*_{0} and exponentially in relation to decreasing . Nevertheless, these regulations do not function well on the compact aerostatic guideway of the VS, due to the size limitation and the mutual tradeoff between the carrying capacity and air film stability. Therefore, the microstructure of recesses and grooves is expected to improve the loading performance, especially for rotational stiffness, under the condition of a moderate *P*_{0} at 0.4 MPa.

Fig. 3 shows the design processes and the corresponding solutions. First, the macrostructure is defined under the target of pressure homogenization. Then, the air film working point , the recess diameter, the recess depth, and the grooves are researched and designed accordingly. The order is determined by the respective degree of influence of each parameter on the throttling contribution and loading performance. The recesses depth is the main research focus of this study. The ultimate objective is to improve the rotational stiffness and meet the acceleration requirement of the VS.

《Fig. 3》

**Fig. 3. **Processes of research and parameter design.

《3. Analysis and results》

3. Analysis and results

《3.1. Definition of macrostructure parameters and conditions》

3.1. Definition of macrostructure parameters and conditions

With the limits of *P*_{0} = 0.4 MPa and = 9 μm, three structures *G*_{1}, *G*_{2}, and *G*_{3} are modeled by mesh adaptation and solved by FVM. The pressure distribution of half of the working surface is shown in Fig. 4, indicating that these structures meet the principle of pressure homogenization and the requirement of air film stability. To quantitatively study the microstructure parameters that may affect the loading performance, an appropriate macrostructure is needed first to control the variables. As shown in Table 2, CFD results of the carrying capacity, stiffness, and rotational stiffness are compared. *G*_{1} has the simplest microstructures and retains the biggest spacing of growth for *K _{θ}*. Thus,

*G*

_{1}is selected as the infrastructure for further microstructure research and design.

《Fig. 4》

**Fig. 4. **Pressure distribution of structures *G*_{0}, *G*_{1}, *G*_{2}, and *G*_{3}.

《Table 2》

**Table 2 **Guideway loading performances of structures *G*_{0}, *G*_{1}, *G*_{2}, and *G*_{3}.

《3.2. Predesign the working point of the air film thickness》

3.2. Predesign the working point of the air film thickness

According to the adaptive method for calculating the unilateral air film’s performances, the carrying capacity *W* and stiffness *K _{h}* of the guideway are obtained. In fact, the effective working surface of the guideway is a pair of unilateral air films positioned face to face. The average of the thickness of the two air films is the working point, as . The working performances are the differences between the performances of the two unilateral air films, as

*W*=

*W*

_{down}–

*W*

_{up},

*K*=

_{h}*K*

_{h}_{-down}–

*K*

_{h}_{-up}, and

*K*=

_{θ}*K*

_{θ}_{-down}–

*K*

_{θ}_{-up}. Assuming that the other macrostructures and microstructures of the working surface have been determined, the variable factor of the working point should be defined in priority. In addition, the microstructure optimization described in the following section requires a predesigned as well. Therefore, we determine the curves of

*W*and

*K*versus when

_{h}*K*varies from 4 to 14 lm. As shown in Fig. 5, the independent variable De is the working eccentricity, defined as . The minimum of is set to 1 μm to avoid contact friction.

_{h}《Fig. 5》

**Fig. 5. **The influence of on (a) carrying capacity *W* and (b) stiffness *K _{h}* with different working points .

= 0 at the no-load state and increases when loading. As shown in Fig. 5(a), the carrying capacity *W* of the guideway has a significantly positive correlation with . The maximum of is restricted by . Thus, *W* has a positive correlation with as well. Contrasting all the curves at a certain , *W* increases in relation to . Therefore, does this mean that as long as the working point increases, a larger carrying capacity will surely result? The answer is: absolutely not.

According to the = 14 μm curve shown in Fig. 5(a), *W* will no longer increase when the is close to the maximum. Under this condition, the stability of the gas film will be easily destroyed by even a small external disturbance. In fact, such a gas film is not conducive to the stability of the guideway, because the selfadjusting force of the gas film may be insufficient to resist external destabilization. The assumption above can be verified from the *K _{h}* curves in Fig. 5(b).

*K*increases along with when ≤ 6 μm, and

_{h}*K*decreases along with when ≥ 7 μm. According to the = 14 μm curve shown in Fig. 5(b),

_{h}*K*is extremely low when the is close to the maximum. However, in the = 9 μm curve,

_{h}*K*remains stable and higher, regardless of any reasonable in the range. On the other side of the = 9 μm curve in Fig. 5(a),

_{h}*W*is still adequate and acceptable in the variable range of , although the maximum of is limited by . Therefore, = 9 μm (or close to 9 μm, because the influence of the microstructures on the working surface will be taken into consideration in the following study) is the most preferred working point for the guideway in this study, under the mutual tradeoff of

*W*and

*K*.

_{h}《3.3. Influences of recess diameter dri》

3.3. Influences of recess diameter d_{ri}

According to the operating conditions of the guideway, we take the stiffness *K _{h}* and rotational stiffness

*K*as the most important performances for contrast and study. To study the influence of the recess diameter d

_{θ}_{ri}on these performances, we limit the working point to = 9 μm for redesigning. By the adaptive method, we obtain the iso-surfaces of

*K*and

_{h}*K*when d

_{θ}_{ri}varies from 0 to 4 mm with different guide sleeve width

*L*

_{b}, as shown in Fig. 6. As seen from the curves in Fig. 6(a),

*K*increases monotonously with a decrease in the recess diameter, and saturates to a certain small value. As seen from the curves in Fig. 6(b),

_{h}*K*increases monotonously with the recess diameter d

_{θ}_{ri}, and saturates to a certain large value. This monotonous regularity is still applicable when

*L*

_{b}varies from 16 to 32 mm, because

*K*and

_{h}*K*increase along with the effective loading area, which is determined by

_{θ}*L*

_{b}. However, the

*K*and

_{h}*K*present a mutual tradeoff on d

_{θ}_{ri}, no matter how

*L*

_{b}changes.

《Fig. 6》

**Fig. 6.** The influence of d_{ri} on (a) stiffness *K _{h}* and (b) rotational stiffness

*K*with different guide sleeve widths

_{θ}*L*

_{b}.

A single coefficient *k*_{op} is employed to find a uniquely corresponding optimized *d*_{op} (optimization diameter of recess), under the mutual tradeoff of *K _{h}* and

*K*. The pseudo code of the optimization is given in Algorithm 1. First, the

_{θ}*d*that leads to an increase of less than 5% on

*K*and

_{h}*K*is removed for filtration. Secondly, the smaller of the two suspected

_{θ}*d*

_{op}is chosen by the optimization coefficient

*k*

_{op}, shown in Algorithm 1 line 18. Finally, the optimized

*d*

_{op}is employed to find the corresponding

*K*and

_{h}*K*. Taking

_{θ}*L*

_{b}= 20 mm as an example,

*k*

_{op}is assigned to meet the different operational requirements of the VS, shown in Fig. 7. Assume that the VS is operating under extreme conditions and the guideway is working at maximum acceleration. Then, the recess diameter will be optimized to

*d*

_{op}= 2.8 mm when

*k*

_{op}= 0.8 is set to meet the highest acceleration of 80 m·s

^{-2}. According to the motion trajectory planning, we also have two more practical strategies, as follows:

*d*

_{op}= 2.2 mm is carried out from

*k*

_{op}= 0.6, meeting the maximum acceleration duration state when the average acceleration is 61.4 m·s

^{-2}; and

*d*

_{op}= 1.4 mm is carried out from

*k*

_{op}= 0.4, meeting the minimum uniform speed distance state when the average acceleration is 49.1 m·s

^{-2}. As shown in Fig. 7, the results of

*K*and

_{h}*K*are found under these three different operating conditions.

_{θ}*K*increases monotonously with

_{θ}*k*

_{op}, because the greater the acceleration, the greater the required resistance moment and the greater the

*K*. To increase the

_{θ}*K*, the

_{θ}*K*needs to be cut down to compensate. Thus, the optimization method can determine the optimal

_{h}*d*

_{op}and the corresponding

*K*and

_{h}*K*to meet different accelerations.

_{θ}《Fig. 7》

**Fig. 7.** The *d*_{op} results according to *K _{h}* and

*K*by the

_{θ}*k*

_{op}optimization method when

*L*

_{b}= 20 mm.

To further verify the applicability, we optimized the recess diameter *d* again under different widths of the guide sleeve *L*_{b} and rows of orifices *n*. Clearly, a larger *L*_{b} is suitable for a bigger *n*, and results in an influence on the optimized results of *d*. Fig. 8 shows the optimized results of the dimensionless diameter × 100%, which is the average ratio of the effective throttling length along the *L*_{b} direction, and can be used as a direct contrast. It can be seen that all the curves have the maximum point. For a certain *n*, a larger *k*_{op} will lead to a larger, conforming to the abovementioned regulation of the example of *L*_{b} = 20 mm. Furthermore, the maximum of the curves increases monotonously with *k*_{op}, and its growth rate slightly increases with *n*. For a certain *k*_{op}, the maximum of the curves decreases monotonously with *n*, and its reduction rate slightly decreases with *k*_{op}. According to the intersection points of these curves, we can summarize the relationship between *B* and *n*, targeted for a larger . When *k*_{op} = 0.6, *n* = 1 should be better for *L*_{b} < 20, *n* = 2 for 20 < *L*_{b} < 32, and *n* = 3 for *L*_{b} > 32. This relationship always exists when *k*_{op} changes.

The strategy of maximum acceleration duration is the closest to reality, in which *k*_{op} = 0.6. As shown in Fig. 8, the maximum dimensionless of the curves for *n* = 1, 2, and 3 is 20.6%, 18.7%, and 17.1%, respectively, corresponding to a guideway width *L*_{b} of 11, 24, and 39 mm. These are then transformed to a *d* of 2.266, 2.244, and 2.223 mm, respectively. The results show that the optimized recess diameter *d*_{op} = 2.2 mm in Fig. 7 is also applicable for the changing of *L*_{b} and *n*. Therefore, the optimization coefficient *k*_{op} method can be widely applied at different guide sleeve size.

《Fig. 8》

**Fig. 8.** The influence of *L*_{b} on optimized for different *n* and *k*_{op}.

《3.4. Influences of recess depth hri》

3.4. Influences of recess depth *h*_{ri}

To study the influence of the recess depth *h*_{ri}, we once again take *K _{h}* and

*K*as the most important performances for contrast and study. According to the optimization results in Section 3.2, we limit the recess diameter to

_{θ}*d*

_{op}= 2.2 mm. By the adaptive method, we obtain the curves of

*K*and

_{h}*K*versus with for varying recess depth

_{θ}*h*

_{ri}, shown in Fig. 9. The variety regularity of

*K*and

_{h}*K*on the best working points require further research. Finally, we adjust the redesigned working point and confirm it by means of recess depth

_{θ}*h*

_{ri}optimization.

《Fig. 9》

**Fig. 9. **The influence of on (a) stiffness *K _{h}* and (b) rotational stiffness

*K*for different recess depths

_{θ}*h*

_{ri}.

As shown in Fig. 9(a), each curve of*K _{h}* versus has a maximum point for a certain hri. The maximum of

*K*has a significantly positive correlation with the variation of

_{h}*h*

_{ri}. The corresponding working point increases with

*h*

_{ri }as well. According to the current correspondence between and

*h*

_{ri}, the function is fitted as Eq. (11). To meet = 9 μm, or close to this value, as was predesigned in Section 3.1, we limit the

*h*

_{ri}in a scope from 24 to 36 μm for higher

*K*. This is done because a smaller or larger hri will lead to a greater deviation of , resulting in a rapid reduction of

_{h}*K*on the predesigned working point .

_{h}As shown in Fig. 9(b), each curve of *K _{θ}* versus has a maximum point for a certain

*h*

_{ri}. The maximum of

*K*increases insignificantly along with

_{θ}*h*

_{ri}, and gradually increases toward a saturation point. In addition, the corresponding working point remains at 11 μm and hardly changes, as it is not influenced by

*h*

_{ri}. It is clear that

*K*at = 9 μm will be smaller than the maximum at = 11 μm. Thus, it is necessary to adjust to close to 9 μm. Furthermore, the current maximum of

_{θ}*K*still holds the possibility for further improvement.

_{θ}The rotational stiffness *K _{θ}* could be increased without losing stiffness

*K*by means of certain special structures, while adjusting the corresponding to close to 9 μm. The gradient depth of the recesses is intended to realize this outcome, as shown in Fig. 10. Due to the accelerated motion, the tilt angle

_{h}*θ*, although small, will result in a wedge-shaped air film. The film thickness varies according to the location of the recesses, rather than staying at a certain . If the recesses were set at a unified depth, only one of the recesses would provide the maximum stiffness at its local film thickness, as shown in Fig. 10(a). This recess would be located in (or close to) the center of the guideway, and its rotating arm would be at (or close to) zero, which would provide a low rotational stiffness. As for the recesses on the right side, the stiffness would be greatly reduced, because the depth of the recesses would mismatch with the local film thickness, as can be seen from Fig. 9(a). Therefore, it is difficult to increase the rotational stiffness, although the corresponding rotating arm is big enough. This is why it is difficult to improve the rotational stiffness with recesses with a unified depth, no matter how

*h*

_{ri}changes, as can be seen from Fig. 9(b). If the recesses were set to a gradient depth, the performance would be significantly improved, as shown in Fig. 10(b). The depths of

*h*

_{r0,1},

*h*

_{r2}, and

*h*

_{r3}are set to decrease in turn, in order to ensure matching to the local film thickness. Maximum stiffness is expected to be realized by following the matching rules in Fig. 9(a). The center and right recesses would then provide useful rotational stiffness for the guideways. On the other side, the left recesses with a gradient depth prove to be less effective for stiffness but more effective for rotational stiffness, because of the greater depth mismatching with the local film thickness. In brief, recesses with a gradient depth will improve the stiffness on the right side, while reducing the stiffness on the left. Then, taking the rotating arm into consideration, the rotating stiffness of the guideway will be increased. The effect of the gradient depth discussed above is still valid when the tilt occurs on the left side.

《Fig. 10》

**Fig. 10.** A model of a wedge-shaped air film matching with the recesses of (a) unified depth and (b) gradient depth.

To verify the deduction described above, we designed seven kinds of guideways with recesses of gradient depth, and studied their *K _{h}* and

*K*performances. As shown in Table 3, the average depth

_{θ}*h*

_{avg}increases in turn from T

_{1}to T

_{7}, and the depth difference (

*h*

_{r3}–

*h*

_{r0}) increases in a sawtooth manner. It can be seen from the curves of

*K*versus in Fig. 11(a) that the maximum of

_{h}*K*has a significantly positive correlation with the variation of

_{h}*h*

_{avg}. The corresponding working point increases with

*h*

_{avg}as well. This variation tendency conform to the regulation of guideways with unified depth recesses. The guideways of T

_{5}, T

_{6}, and T

_{7}satisfy the scope of 24 μm ≤

*h*

_{avg}≤ 36 μm, while providing acceptable and adequate Kh at the predesigned = 9 μm.

《Table 3》

**Table 3** Parameters of the seven guideways: *T*_{1} to *T*_{7}.

《Fig. 11》

**Fig. 11.** The influence of on (a) stiffness *K _{h}* and (b) rotational stiffness

*K*with gradient recess depth

_{θ}*h*

_{ri}.

As shown in Fig. 11(b), the *K _{θ}* of the gradient depth recesses are significantly higher than those of recesses with a unified depth. More importantly, the corresponding working point of the maximum

*K*is changed. In the comparison of the

_{θ}*K*versus curves, the bigger the difference is in the depths of the recesses, the higher the obtained maximum of

_{θ}*K*will be. The corresponding will also decrease more at the same time. As for the recesses with the same depth difference, such as T

_{θ}_{2}, T

_{3}, and T

_{6}, the maximum of

*K*increases insignificantly along with

_{θ}*h*

_{avg}, conforming to the regulation of the unified depth recesses. The T

_{2}, T

_{3}, T

_{5}, and

_{T6}guideways can significantly reduce the , while providing acceptable and adequate

*K*at the predesigned = 9 μm. The above results verify the deduction that the recesses’ depth should match with the local film thickness for higher Kh. Therefore, a gradient depth of the recesses can be applied to increase the rotational stiffness

_{θ}*K*without losing the stiffness

_{θ}*K*, while adjusting . Taking both stiffness and rotational stiffness into consideration, the T

_{h}_{6}guideway performs better than the others.

《3.5. Influences of grooves》

3.5. Influences of grooves

In a compact multi-throttle aerostatic guideway, grooves are widely applied to balance the air input. The air input is usually determined by upstream structures such as recesses and others. Therefore, it is necessary to optimize the groove parameters contrapuntally. If the previous structure is not limited, the scope of the depth of the grooves *h*_{g} and the width of the grooves *w*_{g} will change greatly with the air input. Therefore, we researched the influence of the grooves after adjustment and recesses optimization. Based on the previous result for T_{6}, we further studied the influence of the *h*_{g} and *w*_{g} on *K _{h}* and

*K*.

_{θ}As shown in Fig. 12(a), *K _{h}* increases monotonously with an increase in

*h*

_{g}, and saturates to a certain value when

*h*

_{g}≥ 80 μm generally. The saturation of

*K*increases when ≤ 8 μm, and decreases when ≥ 8 μm. In addition, the corresponding

_{h}*h*

_{g(max)}decreases with .

*K*is hardly affected by hg when ≥ 12 μm, because the increase of leads to a larger air self-output and reduces the diversion effect of the grooves. Alternatively, the guideway itself could balance the air input, without the help of grooves. Under = 8 μm, the saturation of

_{h}*K*is the highest and the scope of

_{h}*h*

_{g}≥ 50 μm is preferred. In comparison with the predesigned = 9 μm, the maximum of

*K*is a little higher.

_{h}《Fig. 12》

**Fig. 12. **The influence of *h*_{g} on (a) stiffness *K _{h}* and (b) rotational stiffness

*K*for different .

_{θ}As shown in Fig. 12(b), each curve of *K _{θ}* versus

*h*

_{g}has a maximum point for a certain . The maximum of

*K*increases when ≤ 8 μm and decreases when ≥ 8 μm. The corresponding

_{θ}*h*

_{g(max)}increases with , as fitted by

*h*

_{g(max)}= 12 – 40, according to the current correspondence. This is because a lower results in a thinner wedge-shaped air film. Therefore, the change in

*h*

_{g}will be more sensitive to

*K*when

_{θ}*h*

_{g}≤ 70 μm, for a low such as the 6 μm curve. Furthermore, the maximum of

*K*occurs earlier and the corresponding

_{θ}*h*

_{g(max)}is smaller. In addition, a much higher

*h*

_{g}will result in pressure homogenization function of the grooves on a wedge-shaped air film. Therefore, a much higher

*h*

_{g}will lead to similar pressure and stiffness of the right and left sides, resulting in a decrease in

*K*. Under = 8 μm, a higher

_{θ}*K*is obtained in the scope of 45 ≤

_{h}*h*

_{g}≤ 70 μm. In comparison with the predesigned = 9 μm,

*K*is mostly higher for the same

_{h}*h*

_{g}scope. Considering the regulations from Figs. 12(a) and (b), a reasonable scope for

*h*

_{g}should be from 45 to 70 μm, based on the previous result for T

_{6}. This is consistent with the predesigned

*h*

_{g}= 60 μm in Section 3.3.

As shown in Fig. 13(a), each curve of *K _{h}* versus

*w*

_{g}has a maximum point for a certain . The maximum of

*K*increases when ≤ 8 μm and decreases when ≥ 8 μm. The corresponding

_{h}*w*

_{g(max)}increases with , as fitted by

*w*

_{g(max)}= 0.1 + 0.2, according to the current correspondence. This is because a lower , such as = 4 μm, leads to more significant effects of

*w*

_{g}on

*K*. In contrast, a higher , such as = 12 μm, leads to smaller effects of

_{h}*w*

_{g}on

*K*. In addition,

_{h}*K*will decrease when wg increases to the same magnitude as the guideway width

_{h}*L*

_{b}, because a wider

*w*

_{g}will share the effective working surface, decrease the average pressure in the air film, and finally decrease

*K*.

_{h}As shown in Fig. 13(b), each curve of *K _{θ}* versus

*w*

_{g}has a maximum point for a certain . The maximum of

*K*decreases along with the increases of , and the corresponding

_{θ}*w*

_{g(max)}increases with according to the current correspondence. This shows that the curves of

*K*versus

_{θ}*w*

_{g}have similar characteristics to

*K*, because the influences of

_{h}*w*

_{g}on

*K*dominate the variation tendency of

_{h}*K*, especially for the significant influences at lower

_{θ}*w*

_{g}. Considering the regulations from Figs. 13(a) and (b), a reasonable scope for

*w*

_{g}should be from 0.8 to 1 mm, based on the previous result for T

_{6}. This is consistent with the predesigned

*w*

_{g}= 1 mm in Section 3.3.

《Fig. 13》

**Fig. 13. **The influence of *w*_{g} on (a) stiffness *K _{h}* and (b) rotational stiffness

*K*for different .

_{θ}According to the analysis and optimization results shown in Figs. 12 and 13, = 8 μm performs the best in terms of *K _{h}* and

*K*in comparison the others for this specific macrostructure and microstructure. In contrast, the performance is slightly worse at = 9 μm, which coincides with the T

_{θ}_{6}results of

*K*( = 8 μm) >

_{h}*K*( = 9 μm), and

_{θ}*K*( = 8 μm) >

_{θ}*K*( = 9 μm) in Fig. 11. However, adjusting the working point to = 8 μm is unnecessary, because = 9 μm performs better at any eccentricity , except for the maximum = 7 μm shown in Fig. 5(b).

_{θ}《Fig. 14》

**Fig. 14. **C-shaped guide sleeves of the aerostatic guideways for the experiment.

《4. Experiment and verification》

4. Experiment and verification

《4.1. Experimental setup of carrying capacity W and tilt moment Mt》

4.1. Experimental setup of carrying capacity *W* and tilt moment *M*_{t}

As shown in Fig. 14, the guide sleeves G_{1} and T_{6} are made of the aluminum alloy AlZnMgCu_{1.5}, with the surface having undergone anodic oxidation. The guide rail is made of the alloy steel 38CrMoAl, followed by surface nitriding.

The measurement principle of the carrying capacity *W* and tilt moment *M*_{t} are shown in Figs. 15(a) and (b). Unlike in the simulation, the independent variable is the load quality, which is modified at a constant step; this is accomplished by loading discrete weights onto the pallet. The tilt moment is measured by *M*_{t} = *W·L*_{c}. *L*_{c} is the arm of force by experiment. The dependent variable is the relative altitude , which is measured by a coordinate measuring machine (CMM) and processed by the software package QUINDOS 7. As shown in Fig. 15(c), is equal to the surface spacing, evaluated from three or four points on the measured surface. As shown in Fig. 15(d), is equal to the difference of the inclined surface, evaluated from two points at a distance of *L*_{e}.

《Fig. 15》

**Fig. 15. **Measurement principle and experimental setup. (a) Measurement for *W*; (b) measurement for *M*_{t}; (c) setup for *W*; (d) setup for *M*_{t}. CMM: coordinate measuring machine.

《4.2. Verification on stiffness, rotational stiffness, and acceleration performance》

4.2. Verification on stiffness, rotational stiffness, and acceleration performance

The carrying capacity *W* and the tilt moment *M*_{t} can be immediately measured from the setup. Then, the stiffness *K _{h}* and rotational stiffness

*K*are obtained by means of Eqs. (9) and (10). As shown in Fig. 16(a), the

_{θ}*K*curve of T

_{h}_{6}is steadier. As for the eccentricity from the weight of the sleeve and the pallet, cannot be measured from zero. As shown in Fig. 16(b), the rotational stiffness of T

_{6}is improved by the design of the microstructure parameters. The result shows that

*K*= 2.14 × 10

_{θ}^{4}Nm·rad

^{-1}; this value has increased by 69.8% and aligns with the CFD result.

《Fig. 16》

**Fig. 16.** CFD and experimental (Exp.) results of (a) stiffness *K _{h}* and (b) rotational stiffness

*K*of G

_{θ}_{1}and T

_{6}.

In the scanning test of the applied ArF photolithography, the aerostatic guideway was driven and monitored by an Elmo recorder after the design of the microstructure parameters. As shown in Fig. 17, the average scanning acceleration reached 67.5 m·s^{-2 }, which meets the design specification of 61.4 m·s^{-2} mentioned in Section 3.3.

《Fig. 17》

**Fig. 17.** Velocity monitored by an Elmo recorder in the scanning test of the applied ArF photolithography.

These results show that the design and tuning of microstructure parameters can improve the rotational stiffness of the guideway under the condition of a medium pressure supply, with no loss of stiffness. This practical application shows that the multi-throttle aerostatic guideway designed by the proposed method satisfies the requirement of high acceleration scanning motion in photolithography.

《5. Conclusions》

5. Conclusions

The following conclusions can be made:

(1) The relevance between the working point and the microstructure parameters was established using the mesh adaptation method. Furthermore, the effect of the microstructure parameters at the micron level on the loading performance of the multi-throttle aerostatic guideway was revealed. This study shows that the diameter and depth of the recess have a relatively significant influence on the rotational stiffness. In particular, the method of designing a gradient recess depth in order to tune the working point was discovered.

(2) A type of design process for the microstructure parameters of a multi-throttle aerostatic guideway was put forward, with the aim of improving the rotational stiffness. The working points of carrying capacity and stiffness were unified by tuning the recess diameter and average recess depth. The working points of stiffness and rotational stiffness were unified by two-way adjusting the gradient recess depth. To a certain extent, the design process can lift the restriction of the supply pressure and solve the mutual tradeoff among loading performances. Thus, the rotational stiffness can be effectively promoted under medium pressure by the design of the microstructure parameters.

(3) The experimental results showed an increase in rotational stiffness of 69.8% as a result of the microstructure parameters design. The alignment of the CFD and experimental results verified the effectiveness of the design process. A compact multi-throttle aerostatic guideway was applied in the VS of ArF photolithography, based on the presented microstructure parameters design process. In the scanning test, the average scanning acceleration reached 67.5 m·s^{-2} , which meets the design specification. Furthermore, this design process for the microstructure of a multi-throttle aerostatic guideway is expected to improve the scanning performance of other components in lithography, laying the foundation for crossgenerational developments in accuracy and productivity.

《Acknowledgements》

Acknowledgements

This work was funded by the National Natural Science Foundation of China (51675136), the National Science and Technology Major Project (2017ZX02101006-005), and the Heilongjiang Natural Science Foundation (E2017032).

《Compliance with ethics guidelines》

Compliance with ethics guidelines

Zhongpu Wen, Jianwei Wu, Kunpeng Xing, Yin Zhang, Jiean Li, and Jiubin Tan declare that they have no conflict of interest.