A new model for surface roughness evolution in the Chemical Mechanical Polishing (CMP) process G. Savio, R. Meneghello, G. Concheri DAUR - Laboratory of Design Methods and Tools in Industrial Engineering - University of Padua - Italy Abstract Surface polishing is a typical example of a machining process based on mixed chemical-mechanical phenomena, as pointed out in the recent literature on the polishing process (CPM - Chemical Mechanical Polishing). In this work, a model is proposed for the assessment of surface roughness evolution in the polishing process of glass moulds, used in the manufacturing of ophthalmic lenses, in order to identify the influence of the operating parameters on the material removal rate (MRR). In this model the evolution of surface roughness during the polishing process is based on Reye hypothesis. According to such hypothesis, the removed material in a specific time interval is proportional to the friction work: the removed material per unit area can be computed by aduately integrating the bearing ratio curve (Abbott-Firestone) of the surface; the friction work per unit area is proportional, according to the dynamic friction coefficient, to the integral of the product of pressure and velocity in the time interval. A similar result can be also obtained adopting other wear models, e.g. the Preston or Archard approaches. The model validation was performed on ground glass flat samples polished with increasing values of MRR. Pressure and velocity distributions on the sample surface were established according to the polishing machine operating parameters by means of the Hertz theory; the surface roughness of the sample was mapped using an atomic force microscope (AFM). The developed model shows a satisfactory estimate of surface roughness evolution during the polishing process and confirms the experimental results found in literature.
NOMENCLATURE CMP = Chemical Mechanical Process MRR = Material Removal Rate V = volume of removed material A = area t = polishing time h = volume of removed material per unit area z = ordinate of profile/surface coordinate system const = a generic constant in a model p = pressure v = relative sliding velocity between tool and workpiece w= angular velocity of wheel spindle (tool) v tr = tangential component of the angular velocity of the tool v av = feed rate = radial tool deformation p a = feed step E = modulus of elasticity of the workpiece E = modulus of elasticity of the tool E = uivalent modulus of elasticity E = ( / E + E ) R = curvature radius of workpiece surface R = curvature radius of tool surface R = / R + R / R = uivalent curvature radius ( ) R ac = contact area radius R ac = R / /t k = Reye s constant k p = Preston s constant k w = wear coefficient k s = proposed model constant µ = friction coefficient z = height of the surface from the mean plane Sa = arithmetic mean deviation of the surface Sa = MN Sq = root mean square deviation of the surface Svv = highest depth Svv = MIN z( x i, y )) ( j Sq = N M j= i= MN N z( x i, y j ) M j= i= z ( x i, y j ) St = highest peak from the highest depth St = MAX ( z x, y )) + MIN( z( x, y )) Introduction ( i j i j The glass polishing process is a typical polishing wear phenomenon in which the polishing pad (a polyurethane layer superimposed to a rubber bulk) and the abrasive slurry (a suspension of cerium oxide in water) perform the removing action on the glass workpiece. The involved mechanisms are both mechanical
and chemical (CMP). The mechanical action is based on the physical interaction between the abrasive particles and the glass surface; the chemical mechanism concerns the chemical reactions between abrasive slurry and glass. The study of factors influencing the CMP led to the proposal of various hypotheses and models: however the basic material removal mechanisms in CMP are not yet well understood []. The first fundamental study on glass polishing was made by Preston []. Brown et al. [3] proposed a model where the Hertz theory was applied to the interaction between the abrasive particles in suspension, assumed as spherical, and the polished workpiece surface. Cook [4] suggested a detailed mechanical-chemical model for the polishing process, taking into consideration the following phenomena: (a) the rate of molecular dissolution and water diffusion into the glass surface, (b) the subsuent glass dissolution under the load imposed by the polishing particles, (c) the adsorption rate of dissolution products onto the surface of the polishing grain, (d) the rate of silica re-deposition onto the glass surface, and (e) the aqueous corrosion rate between particles. Yu et al. [5] proposed a CMP model that includes the effects of the polishing pad roughness and the dynamic interaction between pad and wafer surface. Runnels et al. [] employed a feature-scale erosion model to estimate the stresses induced by the flowing slurry and the polishing rate on the feature surfaces. Venkatesh et al. [6] [7] highlighted the influence of temperature and the effects, both separate and combined, of mechanical and chemical actions. C. Wang et al. [8] proposed a model that more closely describes the relationships between polishing parameters and MRR, based on the hypothesis that the probability of contact between polishing pad and the polished surface be proportional to the slurry fluid film thickness. L. Wang et al. [9] pointed out the influence of the cerium oxide concentration on the removed material volume. In previous studies, the geometric characterization of the surface of glass ground surfaces [] was given and the evolution of the surface texture at different steps of the polishing process was compared with the increase of geometry deformation of surface in terms of curvature [], even if no models were proposed to describe such phenomena. In this paper, a model based on the analysis of the Abbott-Firestone curve is proposed in order to estimate the evolution of surface texture during the polishing process. In order to apply the proposed model the local distributions of pressure and velocity between the tool and the workpiece were evaluated by the Hertz theory []. The model was validated in the polishing process of glass moulds for ophthalmic lenses. Background. Surface roughness characterization In the present work the characterization of surface roughness conforms to ISO 487 standard [3] and Stout 3D areal parameters [4]; in addition, two 3D parameters are defined (see Nomenclature).
The bearing ratio curve of a profile (Abbott-Firestone curve) is defined as the curve representing the material ratio of the profile as a function of level (figure ); this curve can be interpreted as the cumulative probability function of the ordinate values z of the profile. The Abbott-Firestone curve and the roughness parameters defined by the ISO 487 standard can be extended from D to the 3D analysis of the surface texture. In the following, the considered surface roughness parameters are Sa, Sq, Svv and St, extracted from primary surface being filtered with a second order polynomial fitting of the surface. Alternatively such parameters can be estimated by using the Abbott-Firestone curve. In 3D surface texture analysis, the material volume per unit of area, relevant to a specific height z*, is given by the area comprised between the Abbott-Firestone curve and the horizontal line through z* (hatched area in figure ). z [µ m] z* - - -3..4.6.8 Material ratio Figure : Abbott-Firestone curve and volume material per unit area relevant to height z*.. Reye, Preston and Archard hypotheses As previously discussed, in the literature many hypotheses are presented, stating the relationship between the volume of removed material and physical parameters, such as velocity and pressure, involved in polishing and wear processes. The analytical definition of the removed material volume brings together the proposed models (Reye, Preston and Archard): dh = const p v dt () The energetic approach by Reye (86) [5], relevant to the study of wear, assumes that the volume of material removed by wear per unit time is proportional to the work carried out by the friction force.
Preston (97) [], on the basis of experimental data, shows that the rate of glass polishing is proportional to the rate at which the work of friction is done on each unit area of the glass surface. The wear hypothesis attributed to Archard (953) [6-7] is based on the assumption that the ratio, named wear rate, between the volume of removed material per unit time and the sliding length is proportional to the ratio between normal load and surface hardness. According to these hypotheses, the constant in () assumes respectively the values µ/t k, k p and k w /H. In any case, from the three hypotheses described it can be derived that in a wear or polishing process, the volume of removed material is proportional to the product of pressure, velocity and time. 3 Polishing operation: kinematics and pressure distribution The polishing machine used in glass mould manufacturing is uipped with a workpiece spindle and a wheel spindle. The first spindle moves along x and z axes; the second one rotates at angular velocity w and translates along axis y. Moving over the mould surface, the tool follows a meandering path: firstly it travels along the y axis at a constant feed rate v av ; when the tool completes its run in y direction, the lens is moved of a constant feed step p a in x direction. When the tool is pushed against the mould surface the radial deformation is induced, producing a pressure distribution on the contact surface that can be estimated according to the Hertz theory [], disregarding the friction contribution. The assumption of a frictionless process is based on the presence of the cerium oxide (CeO ) suspension in water which acts as a lubricant during machining. According to these hypotheses, the resulting pressure distribution is ellipsoidal-like and may be expressed as: 8 E p( x, y ) = Rac ( x + y ) () 9 π R Y-axis wheel spindle spherical polishing tool glass mould workpiece spindle X-axis Z-axis Figure : Functional scheme of the polishing machine
Given the pressure distribution law, the feed rate (y-direction), the feed step (xdirection), the angular velocity of the tool and the tool deformation, it is possible to determine the pressure plot on a specific point of the glass mould surface as a function of time, as depicted in figure 3. Pressure [Mpa].8.6.4. 5 6 7 8 9 3 4 Time [s] Figure 3: Pressure distribution in a point of glass mould surface. 4 The surface roughness evolution model 4. Relationship between material removal rate and roughness From uation (), and considering that the removed material volume per unit area can be expressed as dv/a, the following uation holds: dv A = const p v dt (3) Introducing in uation (3) the pressure distribution and speed, as presented in the previous paragraph, and integrating it, the (3) becomes: V A. 5 6 vtr = ks E R (4) 7 vav pa From uation (4), once experimentally determined the k s constant, it is possible to estimate the volume of removed material per unit area, relevant to specific values of the polishing process parameters. The volume of removed material per unit area can be related to the Abbott- Firestone curve assuming that the semi-polished surface, approximates the original ground surface from which the volume of material above a given height z has been removed (see fig. 4). Correspondingly, it can be assumed that the volume of removed material per unit area is uivalent to the area included between the Abbott-Firestone curve of the ground surface and the horizontal straight line at height z (hatched area in figure ). By eliminating the hatched area in figure, a new Abbott-Firestone curve is obtained which approximates the actual Abbott-Firestone curve of the semi-polished surface and that can be used to extract the 3D surface roughness parameters.
Figure 4: AFM plots of a ground and a semi-polished surface of glass moulds Finally, it is possible to determine a one-to-one relationship between the removed material volume per unit area and the value of the 3D roughness parameters for a given step of the polishing process, relevant to a defined set of operating parameters. 4. The uivalent elastic modulus E In order to assess the elastic modulus of the tool, a compression test was executed. By applying the Hertz theory [], the elastic modulus is expressed as a function of the applied load, the tool deformation,.5, and the uivalent curvature radius of the workpiece-tool system, R.5 : 3 F 3 m E = = (5).5.5.5 4 R 4 R where m represents the angular coefficient of the linear fitting relevant to the experimental curve F(.5 ). The testing apparatus was set with the purpose of obtaining an infinitely rigid, plane measuring support for the tool: therefore E =E and R =R. 5 The experimental campaign The experimental verification of the model was done by polishing a set of glass (Schott K5) ground plane samples []. The polishing parameters are shown in table. Table: Machining parameters p a [mm] v tr [m/s] v av [m/s].3 3.66.4.5.3 3.66.8.5.3 3.68.8.8.3 3.66.3.5.3 3.66.35.5.3 3.68.35.8.3 3.68.67.8.5 3.66..5 [mm]
Each sample was measured before and after polishing, to estimate 3D surface roughness parameters and the Abbott-Firestone curve, by mean of an AFM Explorer Topometrix, over a µm µm area. 6 Results 6. Estimate of E The elastic modulus of the tool was experimentally assessed as described in paragraph 4.. In figure 5, the experimental curve F(.5 ) is given: it is evident the expected linear trend which confirms the linear elastic behaviour of the composite tool in the range of permissible deformations during polishing. The value of the angular coefficient of the linear fitting is m=94.7 N/mm.5. From uation (5), with a tool curvature radius R = R = 34.9 mm, an elastic modulus E = E = MPa is calculated. F [N] 5 5.5.5 [mm.5 ] Figure 5. Results of the compression test (F vs.5 ). 6. Estimate of k s and validation of the model From the 3D measurement of the surface roughness of each ground sample, both the initial Abbott-Firestone curve and the 3D roughness parameters (Sa, Sq, Svv, St) are acquired; moreover, by analysing the Abbott-Firestone curve, as described in paragraph 4., the minimum, mean and maximum curves of the roughness parameters as a function of the volume of removed material per unit area are theoretically derived (figure 6). From the measurements of the semipolished samples the actual values of Sa, Sq, Svv and St are determinated. By adopting these values of Sa, Sq, Svv and St, the relevant actual volume of removed material is assessed. In such a way a relationship between the machining parameters and the actual volume of removed material is established. From uation (4) the estimate of a mean value of k s of.47-3 Pa -, in accordance to the abovementioned references in the literature [6], can be derived (figure 7). In figure 6, the comparison between the mean values of the effective roughness parameters and the theoretical values predicted by the model as a function of the volume of removed material per unit of area is presented.
.8.6 µm.4 Sa theoretical Sa measured 5 4 µm 3 Svv theoretical Svv measured. 3 4 5 V/A [µm 3 /µm ] Figure 6. Experimental versus predicted values of roughness parameters as a function of removed material per unit of area. 5 3 4 5 V/A [µm 3 /µm ] V/A [µm3/µm] 4 3 Sa Sq Svv St.E+ 5.E+.E+3.5E+3 av. 5 6 v tr E R 7 v p a [Pa µm] Figure 7. Determination of the constant k s. 7 Conclusions In this paper a model is presented integrating two different components: the first one, based on the Reye theory, defines the volume of removed material during the ground glass polishing process as a function of the machining parameters; the second one, starting from the Abbott-Firestone curves, defines the evolution of surface texture as a function of the removed material volume. Integrating the two components, a one-to-one correspondence between polishing parameters and surface texture parameters can be established. The integrated model is able to predict the surface texture as a function of the machining parameters within the statistical uncertainty typical in roughness measurements and gives an estimate of the Preston coefficient consistent with the values of previous works. Acknowledgements The authors are grateful to IODA srl for the technical and economical support given to the present study.
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