Four brackets evaluated

Two types of metal-insert ceramic brackets were evaluated: a stainless steel-lined polycrystalline alumina bracket (Clarity®, 3M/Unitek, Monrovia, Calif) and an 18-kt gold-lined polycrystalline alumina bracket (Luxi®, RMO Corp, Denver, Colo). A stainless steel Mini-Taurus® (RMO Corp) and an Ormco Mini-Twin bracket (SDS/Ormco, Glendora, Calif) served as controls. All brackets had 22 mil (ie, 0.022″) slots. Representative gross morphologies of the 4 bracket products were obtained by first sputter-coating a 150Å layer of gold-palladium and then viewing using a JEOL 6300 Scanning Electron Microscope at an accelerating voltage of 15 kV. Preangulation, pretorque, and lot numbers are also provided in Table 1. Opposing each bracket was a 0.019″ × 0.026″ stainless steel archwire (RMO Corp). Dead soft 0.010″ stainless steel ligature wires (item 47-010-05, GAC International, Commack, NY) provided the normal forces.

TABLE 1. Materials Evaluated

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Experimental method 1

Resistances to sliding (RS) were determined by 2 experiments. In the first experiment, the wire was drawn relative to the bracket at a fixed normal or ligature force (N) of 0.3 kg (300 cN) as the angulation was varied from −12° to +12°. This is the same apparatus as that reported earlier17,18 except that poly(tetrafluoroethylene) bearings are now used instead of steel bearings to simulate the 2 contiguous brackets. These were set to the maximum distance of 18 mm to simulate an extraction site in which the archwire stiffness and resilience have little interaction with the bracket being investigated.

Having obtained the drawing force (P) as a function of the distance traversed (δ) (Figure 1, lower left) for each test angle, each static drawing force was obtained from the initial force maximum at which motion is imminent, and each kinetic drawing force was obtained from the average of the 150 data points during which motion occurred. After dividing this average value of P by 2 to obtain its RS (circled in Figure 1, upper), other kinetic RS data were obtained and plotted against angulation. A second-order polynomial18 was applied to the collective 32 points (Figure 1, upper) that were based on 32 δ-P plots (Figure 1, lower left). Using numerical analyses, the true position of zero angulation (ie, when θ = 0°) was established without expending unnecessary effort to try to initially align the bracket on its mounting post. After adjusting all θ's according to this true θ = 0°, the data for the negative angles were virtually folded at true zero onto the positive angle data (Figure 1, bottom right). By running each bracket through both the positive and negative angles, both sets of diagonals were evaluated, in effect replicating each run without removing the archwire-bracket couple.

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Using 2 linear regression analyses, the passive and active regions were defined by RS as a function of θ along with the boundary conditions at which elastic binding first occurs, ie, at the so-called critical contact angle (θ_c) (Figure 1, bottom right). For this experiment, 2 operators each tested an as-received couple of each archwire-bracket combination. From those combined outcomes, the magnitudes of RS could be determined at N = 0.3 kg, from which the coefficients of friction (μ_FR) and the coefficients of binding (μ_BI) could be calculated from the RS against θ plots in the passive and the active regions, respectively, ie, when θ ≤ θ_c and θ ≥ θ_c.

Experimental method 2

In the second experiment, the wire was drawn relative to the bracket at a fixed θ = 0° or θ = 11° as N was varied from 0.1 kg to 0.9 kg in 0.2 kg increments. The first angle represented the passive region (θ ≤ θ_c; Figure 1, lower right), where only classical friction (FR) occurred, whereas the second angle represented the active region (θ ≥ θ_c; Figure 1, lower right) in which elastic binding (BI) dominates. From these plots of RS as a function of N, μ_FR could be calculated from the regression slopes at both θ = 0° and θ = 11° along with BI at θ = 11°, which for each value of θ is different but invariant with N.

Testing of sliding resistance

After all couples were wiped down with 95% ethanol, each frictional test was conducted at 34°C either under prevailing atmospheric humidity or in the wet state using fresh whole human saliva at a flow rate of 3 cm3/min. The details of gathering and evaluating the saliva have been presented previously.19,20 In both experiments, static (μ_FR-S) and kinetic (μ_FR-K) coefficients of friction were calculated from the appropriate drawing force data (Figure 1, lower left).6,21 Clearly, the kinetic value should be more reliable in terms of its substantial database.

Data analyses

All aforementioned regression analyses were performed using Excel 97 (Microsoft, Redmond, Wash). From the correlation coefficient and the number of data points, a level of confidence was expressed as a probability, P < .001.22 Unless otherwise stated, this P-value represents the level of confidence of all plots of RS as a function of θ or N. An analysis of variance (ANOVA) was performed to determine the significant differences between the brackets and the dry and wet states (SYSTAT Version 5, SYSTAT, Inc, Evanston, IL). The static and kinetic data were combined prior to performing these ANOVA's.

General morphology

Clarity and Luxi brackets illustrate the geometry of an insert slot relative to the monolithic structure of Mini-Taurus and Mini-Twin brackets (Figure 2). In the case of Clarity (upper left), the stainless steel insert neither extends to the top of the tie-wings nor maintains a constant slot width at the upper extent of the insert. This could present problems when torquing teeth. In the case of Luxi (lower left), the 18-kt gold insert is quite thick and substantially extends to the full height of the slot; although at the top of the slot, a gap appears between the alumina material and the insert slot. This could host bacteria. Both Mini-Taurus and Mini-Twin brackets (upper and lower right, respectively) appear to have parallel slot walls with rather square corners at the roots of their slots.

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RS against θ experiment

The first experiment illustrates the resistance to sliding (RS) as a function of angulation (θ) for Clarity, Luxi, and Mini-Taurus in the dry state (Figure 3) and the wet state (Figure 4) in which static (left-hand columns) and kinetic (right-hand columns) conditions prevail. Since a common bracket slot size, archwire alloy, archwire size, and ligature force (N = 0.3 kg) were utilized, the brackets that had metal inserts could be directly compared with one another and with the frictional standard of the profession—stainless steel on stainless steel. In the dry state (Figure 3), the regression lines of the static and kinetic plots that were associated with each bracket superposed. The same was generally true for the data in the wet state (Figure 4), too, although the regression lines of the kinetic plots were just slightly lower than those of the static plots, particularly for the Mini-Taurus bracket.

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In the passive region (θ ≤ θ_c and μ_FR = RS/N) and in the dry state, the RS values for Clarity and Luxi were comparable, averaging about 0.045 kg (Table 2). Given that μ_FR = RS/N, where N was fixed at 0.3 kg, the mean coefficient of friction averaged about 0.15 (Table 3). This value is comparable with that of the present conventional stainless steel couple and with previous observations reported in the literature.6,15,16,21 The experimental values of θ_c range from 2.1 to 2.5° for Clarity, from 2.6 to 2.9° for Luxi, and from 3.0 to 3.2° for Mini-Taurus. Since the wire portion of the couple is invariant, a larger value of θ_c means that the quotient of the bracket's width to its slot (ie, WIDTH/SLOT) is greater in magnitude.23 This could mean that the bracket is wider or that the actual slot dimension is smaller or that both are true.

TABLE 2. Magnitudes of the Resistance to Sliding (RS) in kg^a as Measured in the Passive Region (θ ≤ θ_c) When N = 0.3 kg

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TABLE 3. Coefficients of Friction (μ_FR) as Derived From μ_FR = RS/N in the Passive Region (θ ≤ θ_c) When N = 0.3 kg

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In the active region (θ ≥ θ_c and μ_BI = RS/θ), Clarity and Luxi are comparable, the former having a lower coefficient of binding (μ_BI) in the dry state and the latter being some 11% less (and hence better) in the wet state (Table 4). Both metal-insert brackets are comparable to the Mini-Taurus bracket in the wet state, whereas their μ_BI values are from 31% to 38% less (better) than the conventional bracket in the dry state. An ANOVA indicated that differences between these 3 brackets were statistically significant (P < .001). Likewise, the interaction between the bracket and the dry and wet states was also significant (P < .001), although the influence of the state alone was not significant.

TABLE 4. Coefficients of Binding (μ_BI) in kg/° as Determined From the Slopes of the RS against Angulation (θ) in the Active Region (θ ≥ θ_c) When N = 0.3 kg^a

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RS against N experiment

The second experiment shows RS as a function of normal or ligature force (N) in the dry state (Figure 5) and the wet state (Figure 6) in which the 2 metal-insert brackets (left-hand columns) are compared with 2 stainless steel brackets (right-hand columns). For each product, the linear regression lines of the grouped static and kinetic data show similar slopes in both the passive (θ = 0°) and active (θ = 11°) regions, separated only by a constant value of RS that represents BI. The encircled data points, most of which just happen to occur for the Clarity product, were not used in the linear regression lines presented. Tabulation of μ_FR and BI for the individual runs that were grouped in Figures 5 and 6 are shown in Tables 5 and 6.

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TABLE 5. Coefficients of Friction (μ_FR) as Determined from the Slopes of the RS against N Plots in the Passive Region (θ ≤ θ_c) When θ = 0° and in the Active Region (θ ≥ θ_c) When θ = 11°^a

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In the case of μ_FR, mean values were lowest in the passive region (θ = 0°) for the gold-inserted appliance and equaled 0.13 ± 0.01 and 0.15 ± 0.02 in the dry and wet states, respectively (Table 5). Among those regression lines that were statistically significant (see footnotes, Table 5), Luxi and Mini-Twin appeared most consistent, their overall standard deviations equaling 0.02 and 0.01, respectively, vs 0.04 and 0.03 for Clarity and Mini-Taurus, respectively. An ANOVA indicated that the influence of the bracket was significant (P < .001) for the 0° data but not for the 11° data. For both of these sets of data, the differences between the dry and wet states were not significant.

In the case of BI (Table 6), values ranged widely from 0.24 kg to 0.51 kg and had a mean value of 0.36 kg. Although the variance was great, the mean value fell within the general envelope observed for the 4 bracket materials—that of 0.34–0.37 kg. Only Clarity in the wet state and Luxi in the dry state were above and below this benchmark, respectively. ANOVA showed that the results were highly influenced by both the bracket (P < .001) and the dry and wet states (P < .002).

TABLE 6. Magnitudes of Binding (BI) in kg as Determined from the Intercepts of the RS against N Plots in the Active Region (θ ≥ θ_c) When θ = 11°

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Overview of classical friction and binding

In principle, binding (BI) and classical friction (FR) can occur at any time during treatment. Although the practitioner may be confronted with FR and BI in the occlusal-gingival, mesial-distal, or labial-lingual directions, the present work considers only second-order angulation and the presence of adjacent teeth. Future work will need to consider other directions and utilize the superposition principle to determine the total contributions to FR and BI. And although in vitro tests can be a valid indication of in vivo performance,24 the complexity that is associated with the coordinated movement of potentially 2 arches poses substantive challenges. Nonetheless, a 3-dimensional (3-D) mapping can now be proposed to facilitate the understanding of, at least, second-order angulation.

General elements of 3-D friction maps

As the normal or ligature force (N) and/or the angulation (θ) are varied, the resistance to sliding (RS) changes (Figure 7). When all N-θ combinations are considered, 2 surfaces are generated: 1 for classical friction (FR) and 1 for elastic binding (BI) (see Figure 7a). The FR surface appears as a 3-D wedge upon which the BI contribution rests (RS = FR + BI).^6,25–27 And once BI occurs, that magnitude is constant for a given value of θ, equaling zero for θ ≤ θ_c and equaling a unique nonzero value for each successive angle of θ > θ_c.

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More specifically, if the RS values of the right-hand side of a 3-D map are considered (see Figure 7b), RS remains constant (see portion labeled 1) until the demarcation line labeled 2 is reached. Here the vertical dashed line corresponds to the critical contact angle (θ_c), below which only classical friction exists and above which binding additionally occurs.26–28 Thereafter, BI increases linearly (see portion labeled 3) to some point, which is yet to be identified, that is dependent on material (eg, metal, ceramic, or composite), mechanical (eg, yield strength, hardness, or modulus), dimensional (eg, archwire, bracket, or ligature), and anatomical (eg, interbracket distance, degree of aligning and leveling, or arch curvature) parameters. The slope of the portion labeled 3 is independent of normal or ligature force.29 Consequently, binding occurs to the same extent even when no ligature force is present if θ > θ_c (see portion labeled 3 on left-hand side of 3-D map of Figure 7b). Such general behavior coincides with method 1, of which Figures 3 and 4 are representative.

The lines that are marked with a 4 are derived from method 2, such as those depicted in Figures 5 and 6. As Figure 5 has shown, the passive region starts at the origin and increases linearly in accordance with its value of μ_FR. Like the μ_BI discussed as the slope of the line labeled 3 in the preceding paragraph, μ_FR is the slope of the line labeled 4 whether it be in the passive region at θ = 0° (Figure 7b, foreground line labeled 4), the boundary between the passive and active regions at θ = θ_c (Figure 7b, intermediate line labeled 4 at the demarcation line labeled 2), or the terminus of elastic binding— θ_z in reference 26 (Figure 7b, background line labeled 4). Thus, although the classical friction that is associated with RS systematically increases with increasing N, its contribution to RS is independent and constant for all θ's at a given N. For example, with no normal or ligature force (N = 0) at θ ≤ θ_c, RS = 0 (see Figure 7b, the line labeled 5), wherein the magnitude thereafter equals only BI as defined by the slope of the line labeled 3 (see Figure 7b, left-hand line labeled 3). As you move from left to right across the diagram, these same magnitudes exist except that the constant magnitude of FR must be superposed, which is dependent only on N. At the far right-hand terminus, this magnitude of FR corresponds to the hatched area below the line labeled 1.

As a consequence of these superposition capabilities and the common slopes of the lines labeled 3 and also the lines labeled 4, a 3-D map can be generated for a given archwire-bracket-ligature wire system at a specified set of material, mechanical, dimensional, and anatomical parameters—if 3 parameters can be determined: θ_c, μ_FR, and μ_BI. After analyzing a vertical section of the 3-D map (foreground-to-background) at a constant value of N (method 1), Figures 3 and 4 along with their accompanying analyses of Tables 2–4 provide all 3 parameters. At θ ≤ θ_c, RS = FR = μ_FRN, whereas at θ ≥ θ_c, RS = FR +BI = μ_FRN + μ_BIθ. The intersection of these 2 regression lines yields the experimental value of θ_c. In contrast, Figures 5 and 6 along with their accompanying analyses in Tables 5 and 6 provide 2 of the required parameters after analyzing vertical sections of the 3-D maps (left-to-right) at, at least, 2 constant θ's (method 2): one at θ ≪ θ_c and one at θ ≫ θ_c. The first θ establishes the value of μ_FR (Table 5), which is shown in the foreground of Figure 7 as the line labeled 4. The second θ establishes the value of BI from the intercepts of its RS versus N plot, which corresponds to a point on the left-hand line labeled 3. For that line, RS =FR +BI = BI = μ_BIθ. To use Figures 5 and 6, θ_c must be determined by measuring 3 geometric dimensions: the size of the archwire (SIZE), the width of the bracket (WIDTH), and the slot of the bracket (SLOT). From the approximation, θ_c = 57.32[1 − (SIZE/SLOT)]/(WIDTH/SLOT).23,26

Application of 3-D friction maps to present work

Using both methods for Clarity, Luxi, and Mini-Taurus brackets and method 2 along with its calculation of θ_c = 57.32[1 − (SIZE/SLOT)]/(WIDTH/SLOT) for the Mini-Twin brackets, 3-D maps were determined for a wide range of N and θ (where N = 0–1.0 kg and θ = 0–15° in Figures 8 through 11, respectively). In the first 3 of these figures (Figures 8 through 10), redundancy was intrinsic to both μ_FR and μ_BI. Here θ_c was determined experimentally from the RS against θ plot when N = 0.3 kg via method 1. In the fourth of these figures (Figure 11), θ_c was calculated to be 2.4°, having measured the SIZE, SLOT, and WIDTH to be 0.01875″, 0.0241″, and 0.1264″ (see Materials and Methods section and Table 1 for some nominal dimensions). Having observed no significant difference between static and kinetic values of RS, the regression lines (— ••) were pooled for the dry and the wet states. To summarize then, Figures 8 through 11 illustrate the overall influence of normal or ligature force (N) and second-order angulation (θ) on the resistance to sliding (RS) when (1) no first- or third-order effects are present, (2) contiguous brackets are identically represented by low friction contacts at an interbracket distance (IBD = 18 mm) far from the subject tooth, (3) static and kinetic data are pooled, (4) a common nominal slot size is used (22 mil), (5) the same archwire (19 × 26 mil stainless steel) and ligature wire (10 mil stainless steel) are used, and (6) 1 operator is used. Indeed, the redundant analyses afforded by methods 1 and 2 (see Figures 8 through 10) corroborate that the intraoperator variations are inconsequential.

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From such 3-D maps, an amplification of the trends occurs in very few tests that could only be otherwise achieved by exhaustive measurements and analyses. Although the Clarity bracket (Figure 8) has a similar μ_FR in both the dry and wet states, its μ_BI is more severe in the presence than in the absence of saliva. Qualitatively, these same effects are seen for the Luxi bracket; however, the magnitudes of both μ_FR and μ_BI are lower for Luxi than for Clarity. In the dry state, the Mini-Taurus bracket appears identical to the Clarity bracket. Little difference exists between the Mini-Taurus bracket in either its dry or wet states. In the dry state, the magnitudes of FR for the Mini-Twin bracket appear comparable with the Luxi bracket. And although the FR of the Mini-Twin bracket in the wet state increases over its values in the dry state, the overall RS at high values of θ remains fairly comparable with the Luxi bracket in the wet state. Therefore, Clarity and Luxi appear comparable with the 2 conventional stainless steel brackets in the dry and wet states, respectively. In the dry state, moreover, Luxi is superior among the 4 brackets evaluated.