HK1222462B - Silicon hairspring - Google Patents

Description

silicon balance spring

Technical Field

The present invention relates to spiral balance springs, in particular timepiece balance springs made of silicon and their design.

Background Art

The hairspring is a key component of a mechanical clock. It is one of the two main components of an oscillator, the other being the balance wheel. A clock's oscillator provides time regulation through its simple harmonic motion. The balance wheel acts as an inertial element and is attached to the inner terminal of the spiral hairspring. The outer terminal of the hairspring is typically rigidly attached to a fixed stud. In an ideal oscillator, the hairspring applies a restoring torque to the balance wheel that is proportional to its angular displacement relative to its equilibrium position, and the equations of motion describe this as a linear second-order system.

The equilibrium position is defined as the angular position of the balance wheel such that the net torque exerted on the balance wheel by the hairspring is zero when the oscillator is at rest. An oscillator is considered isochronous when its natural frequency is independent of its amplitude and other external factors such as temperature variations and magnetic fields. Since the accuracy of a timepiece is largely determined by the stability of the oscillator's natural frequency, isochronism is one of the most important properties of a mechanical timepiece.

Historically, the hairspring has been considered one of the most challenging components to manufacture in timepieces, particularly those used in mechanical movements. For the entire lifespan of a timepiece, often several years before servicing, the hairspring is required to flex continuously at a frequency typically ranging from 3 to 5 Hz, the frequency range of modern mechanical watch oscillators. The hairspring is also one of the smallest components in the mechanism, with a coil width typically ranging from 30 to 40 microns.

The balance spring must also be formed from a material that resists the effects of temperature variations on mechanical properties, particularly Young's modulus, in order to maintain correct timekeeping and minimize fluctuations.

Furthermore, with the proliferation of consumer electronics leading to an increase in the amount of electric and magnetic fields, modern hairsprings must also be able to resist or substantially minimize the effects of these magnetic fields. This is because the precision and consistency of a hairspring's stiffness are highly demanding parameters, and even a 0.1% variation in stiffness can result in a timepiece inaccuracy of up to one minute per day, which is unacceptable in the horological and watchmaking industry. Historically, watch industry technicians have expended considerable effort to design and manufacture hairsprings that minimize these effects.

Traditional hairsprings are made from metal alloys, starting with hardened steel used by John Harrison about 300 years ago, to Elinvar, invented by Charles Guillaume in 1919, and more recently, Nivarox, invented by Dr. H.C. Reinhard Straumann. Almost all modern hairsprings are made from some variation of Nivarox, an alloy based on iron and nickel. Hairsprings are made using a drawing process, in which a strip of material is drawn into a thin filament. The straight strip is then rolled into a spiral, which is then treated to stabilize the spiral's shape. This process has several known disadvantages, including:

(a) The drawing process is generally not a high-precision technique, with tolerances in the range of several microns, and a significant percentage of balance springs typically have widths in the range of 30 to 40 microns, resulting in inconsistent stiffness.

(b) metal alloys such as Nivarox inherently have a tendency to creep and deform slightly in use after prolonged stress, so that a metal balance spring may not retain its original spiral shape after more than one year of continuous operation, which may require adjustment and inevitably affect timekeeping accuracy, and

(c) Although the thermoelastic constants and magnetic susceptibility of Nivarox alloy materials have been significantly reduced compared to early metal balance springs by heavily doping with trace elements such as chromium, these problems and shortcomings have not been completely eliminated.

In order to address or minimize the aforementioned problems with Nivarox and other metal alloys for balance springs and their methods of manufacture, the past decade has seen the introduction of the use of silicon and micromachining techniques in balance spring manufacture.

Silicon balance springs are produced using micromachining processes that achieve sub-micron precision with significantly greater accuracy than traditional metal forming techniques such as drawing. The advantages of using silicon include:

(a) Compared to most metal alloys, this material does not creep or oxidize over time, thus maintaining its mechanical properties and integrity.

(b) the material is completely non-magnetic, and

(c) By providing a balance spring with a silicon core and a thin silicon dioxide layer so that the net thermoelastic constant of the balance spring is close to zero, the temperature sensitivity of normal operating parameters can be minimized or substantially eliminated.

Thus, the technology used to manufacture silicon balance springs has seen numerous advances over the past decade, including the one disclosed in document DE 101 27 733 (June 7, 2001), which discloses the use of a silicon micromechanical balance spring in which single-crystal silicon is in either the <100> or <111> plane, with both orientations disclosed as equally suitable. This balance spring is a spiral balance spring with good resistance to a wide range of thermal stresses and good dimensional stability. The disclosed balance spring can also be coated with a silicon dioxide coating.

Document EP 1422436 (June 25, 2002) describes a method for reducing the thermal drift of a single spiral hairspring for a timepiece in order to achieve a near-zero temperature coefficient. This method and device utilize a hairspring intended for use in a mechanical timepiece balance wheel and formed from a spiral rod cut from a <100> single-crystal silicon wafer with first- and second-order thermoelastic constants. The hairspring's turns have a width w and a thickness t, whereby a silicon dioxide coating minimizes the thermal coefficient of the hairspring's spring constant. Consequently, the hairspring ideally includes a modulation of its width.

EP 2224293 (April 29, 2004) discloses a timepiece movement comprising a regulating device comprising a balance wheel and a flat balance spring that may be formed from silicon. The flat balance spring includes a stiffened portion on its outer coil, arranged so that the deformation of the multiple coils is substantially concentric. The stiffened portion terminates before the outer end of the balance spring and is characterized by a sufficient width between the terminal portion of the outer coil and the penultimate coil of the balance spring to maintain radial freedom during expansion of the balance spring in the movement up to an amplitude substantially corresponding to the maximum rotation angle of the balance wheel. This helps maintain the concentricity of the balance spring during use, thereby contributing to good timekeeping. The document discloses that a silicon balance spring can be formed using the method of EP 0732635. This patent, whose inventor is known as Patek Philippe, discloses the structure of its balance spring as described below.

In 2006, Patek Philippe publicly released a balance spring made of . This balance spring is obtained through a vacuum oxidation process that allows compensation for temperature variations. As described and claimed in EP2224293, the concentricity (symmetrical expansion and contraction of the balance spring about its center) is achieved by bending the ends, which are not rolled up but have a significantly thicker area at the outer ends.

Document EP 2 215 531 (November 28, 2007) describes a mechanical oscillator for watchmaking, comprising a spiral balance spring formed from single-crystal silicon (Si) oriented along the crystal axis <111>, with a coating selected to ensure that the elastic torque of the balance spring varies with temperature, thereby compensating for temperature variations in the moment of inertia of the balance wheel. This document uses single-crystal silicon material for axis <111> in the same manner as the <001> single-crystal silicon used in EP 1 422 436, and employs a coating to provide a temperature-insensitive balance spring, similar to a Spiromax balance spring formed from Silinvar with an oxide coating.

More recently, silicon balance springs have also become available, however, such use is commercially limited and, given their limited use over a significantly shorter period of less than a decade and their lack of widespread use compared to metal alloy balance springs that have been in large quantities for decades, the long-term stability and integrity of balance springs formed from silicon have not yet had the opportunity to be assessed and compared to industry standard metal or metal alloy balance springs such as the Nivorax metal alloy balance spring.

Summary of the Invention

Objective of the invention

An object of the present invention is to provide a silicon balance spring that overcomes or at least ameliorates at least some of the disadvantages associated with those of the prior art.

Summary of the Invention

In a first aspect, the present invention provides a torque restoring element for a mechanical timepiece oscillator, the oscillator having an oscillation frequency, the torque restoring element comprising a hairspring body having N turns, the hairspring body having an inner terminal end for engaging a rotating inertia element via a hairspring collet and an outer terminal end for engaging a fixed bridge element, and the hairspring body having a width, a height, and a total arc length; wherein the hairspring body comprises a core formed from a single-crystal silicon wafer oriented along a crystal axis <110>; and wherein the hairspring body comprises at least one peripheral material coating having a different thermoelastic constant from that of the core of the hairspring body so as to maintain the oscillation frequency of the oscillator including the torque restoring element substantially insensitive to changes in ambient temperature.

Preferably, the spiral spring body has a substantially rectangular cross-section. Preferably, the spiral spring body has a width ranging from 20 μm to 60 μm, a height ranging from 100 μm to 400 μm, and a total arc length ranging from 50 μm to 200 μm. Preferably, the spiral spring body has a number of turns ranging from 5 to 20.

The peripheral coating of the balance spring is formed of silicon dioxide and preferably has a thickness ranging from 3 μm to 6 μm.

In an embodiment, the <110> orientation angle of the single-crystal silicon wafer produces a small variation in the overall balance spring stiffness that is used in fine-tuning the stiffness. Preferably, the spiral balance spring body has a width that varies periodically along at least a portion of the total arc length based on the number of turns, in order to compensate for the varying sectional stiffness of the balance spring due to the anisotropic Young's modulus in the plane of the silicon wafer. The width of the <110> single-crystal silicon wafer can be varied according to the following equation:

Here, S ₁₁ , S ₁₂ , and S ₄₄ are elements of the single crystal silicon compliance matrix, defined as 7.68, -2.14, and 12.6, respectively, in units of 10 ⁻¹² Pa ⁻¹ . The term θ is the orientation angle in the wafer plane. The spiral spring body can be formed using dry etching techniques, including deep reactive ion etching (DRIE) techniques.

In another embodiment, a longitudinal vertex formed at the intersection of the height plane and the width plane of the spiral spring body has an inclined surface extending along at least a portion of the total arc length. The inclined surface reduces structural stress concentration at the vertex during elastic deformation of the spiral spring body in use. The inclined surface can be formed by wet etching.

In a second aspect, the present invention provides an oscillator for a timepiece, the oscillator comprising the torque recovery element according to the first aspect, and a balance wheel attached to an inner terminal end of the torque recovery element.

In a third aspect, the present invention provides a method for forming a torque restoration element according to the first aspect, wherein the spiral spring body is formed by a dry etching manufacturing technique, including a deep reactive ion etching (DRIE) manufacturing technique.

In a fourth aspect, the present invention provides a method for forming a torque recovery element according to the first aspect, wherein the width of the <110> single crystal silicon wafer can be varied according to the following equation:

In a fifth aspect, the present invention provides a mechanical oscillator for a mechanical timepiece, the mechanical oscillator comprising: a torque restoring element comprising a spiral hairspring body having N turns, the spiral hairspring body having an inner terminal end and an outer terminal end, and having a width, a height and a total arc length, the inner terminal end being used to engage a rotating inertia element rotating around an axis through a hairspring inner post, and the outer terminal end being used to engage with a fixed bridge element; and a rotating inertia element engaging the inner terminal end of the spiral hairspring element and rotatable around the axis; wherein the spiral hairspring body comprises a core formed from a single crystal silicon wafer oriented along a crystal axis <110>; and wherein the spiral hairspring body comprises at least one peripheral material coating having a different thermoelastic constant from that of the core of the spiral hairspring body, so as to maintain the oscillation frequency of the mechanical oscillator comprising the torque restoring element to be substantially insensitive to changes in ambient temperature.

Preferably, the torque recovery element is a torque recovery element according to the first aspect.

BRIEF DESCRIPTION OF THE DRAWINGS

Examples of the present invention will now be described with reference to the accompanying drawings, in which:

1 shows a diagram of a general torque-restoring element according to the invention as a balance spring for a mechanical timepiece with a constant pitch, in a relaxed state, except for the outermost coil consisting of an Archimedean spiral;

2 shows a cross-sectional view of a first embodiment of a silicon balance spring according to the invention, consisting of a silicon core with a silicon dioxide coating of constant thickness according to the invention;

3 shows a cross-section of a second embodiment of a silicon balance spring according to the invention, consisting of a silicon core having a silicon dioxide coating of constant thickness and with beveled longitudinal edges according to the invention;

FIG4 shows the relationship between Young's modulus and orientation angle in the silicon wafer plane for <100>, <110> and <111> single crystal silicon;

FIG5 shows a graph of the oscillation rate versus temperature for five different silicon dioxide thicknesses; and

FIG. 6 shows the variation of the width of the silicon balance spring according to the angular position to compensate for the variation of the bending curvature caused by the anisotropic Young's modulus of <110> single crystal silicon according to the present invention.

DETAILED DESCRIPTION

The invention utilizes <110> single crystal silicon to form a torque restoring element, in particular a balance spring for a mechanical timepiece. The invention and details are described with reference to Figures 1 to 3 and the advantages provided by the invention and its parameters are described and explained with reference to Figures 4 to 6.

With reference to FIG. 1 , a typical torque-restoring element, a balance spring 100 according to the present invention, is shown. Balance spring 100 is used in an oscillator having an oscillating frequency and comprises a spiral balance spring body 110, having a main body portion with an inner terminal end 115, an outer segment 120, and an outer terminal end 125. The balance spring has a width 140, a height into the page, and a total arc length of turns 170.

The body of spiral hairspring body 110 forms an Archimedean spiral of constant pitch, with an inner terminal end 115 for coupling to a hairspring collet 117. Hairspring collet 117 is in turn rigidly connected to a balance wheel which, although not shown, will be understood and appreciated by those skilled in the art as a rotating inertial element.

The outer segments 120 have a significantly increased pitch to allow space for press stud placement, and outer terminal ends 125 for engaging fixed clamping elements.

The balance spring 100 is formed from a single crystal silicon wafer oriented along the crystal axis <110> and can be formed by microfabrication techniques such as dry etching fabrication techniques, including deep reactive ion etching (DRIE) fabrication techniques.

Spiral spring body 110 is formed with a substantially rectangular cross-section. According to the present invention, spiral spring body 110 may have a width 140 ranging from 20 μm to 60 μm, a height 150 ranging from 100 μm to 400 μm, and a total arc length 160 ranging from 50 μm to 200 μm, with a number of turns ranging from 5 to 20, and as shown, 13.5 turns in this particular example. These dimensions and sizes make spiral spring 100 suitable for use in timepieces, such as wristwatches.

Referring to FIG2 , a cross-sectional view of the first embodiment of a balance spring 100 a is shown at an arbitrary position on the coils of the balance spring, whereby balance spring 100 a has a width 140 a and a height 150 a. The features and dimensions of the general torque restoring element of FIG1 are similar to those of the present embodiment, and likewise, like identifiers indicate like features.

Spiral spring body 110a includes a core 180a formed from a single-crystal silicon wafer oriented along the crystal axis <110>. Spiral spring body 110a also includes a coating 190a on its periphery, made of a material having a different thermoelastic constant than the core of the spiral spring body. This peripheral coating allows hairspring 100a to be thermally compensated and insensitive to changes in ambient temperature. Furthermore, providing this peripheral coating allows hairspring 100a to be used in a timepiece regulator having a hairspring attached to a balance wheel, thereby maintaining the oscillation frequency of the oscillator and making the oscillator substantially insensitive to changes in ambient temperature.

It will be appreciated that a balance wheel may have its own thermal expansion curve, that the balance wheel may change its moment of inertia as the ambient temperature varies, and that a suitable choice of the material of the peripheral coating 190 a for the balance spring 110 a allows any heat-related variations of the balance wheel to be taken into account, so that the regulator is substantially insensitive to variations in the ambient temperature, so as to maintain a constant oscillation frequency and thus provide good and consistent timekeeping of a timepiece incorporating such a regulator.

According to the present invention, peripheral coating 190a may be a silicon oxide coating applied by thermal oxidation. The silicon balance spring is heated to approximately 1000°C in an oxidizing environment to achieve peripheral coating 190a. Typically, a peripheral coating 190a having a thickness ranging from 3 μm to 6 μm is employed according to the present invention. When used with a <110> single-crystal silicon balance spring 100a having the geometric parameters described with reference to FIG1 , this peripheral coating 190a allows for thermal regulation and compensation of a timepiece regulator comprising a balance spring and a balance wheel. Thus, when incorporated into an oscillator for a timepiece, balance spring 100a of the present invention provides temperature stability and precise timekeeping independent of temperature fluctuations and variations.

According to the present invention, the cross section of the spiral hairspring body 110a of the hairspring 100a may be:

(i) is of constant cross-section over substantially the entire arc length, or

(ii) may vary along at least a portion of the total arc length.

By employing the <110> single crystal silicon wafers defined in the present invention, slight variations in the stiffness of the overall balance spring are allowed to be used in stiffness fine-tuning due to variations in Young's modulus as discussed further below, providing an advantage over prior art techniques such as those employing <100> and <111> single crystal silicon wafers, also discussed below.

In embodiments of the present invention where the cross-section of the spiral spring body is not constant throughout its length, the cross-section of the spiral spring body 110a of the hairspring 100a can vary depending on desired design parameters. Because <110> single-crystal silicon wafers are anisotropic and are incorporated into the present invention, the width, and therefore the cross-section, of the spiral spring body 110a can vary. In embodiments of the present invention, the cross-section of the spiral spring body 110a can have a periodically varying width 140a. This periodically varying width can apply along at least a portion of the main portion of the spiral spring body 110a, or substantially along the entire arc length, though not necessarily along the perimeter. The width 140a of the spiral spring body 110a can vary according to the following equation:

Referring to FIG. 3 , a cross-sectional view of a second embodiment of a balance spring 100b according to the present invention is shown. As shown, core 180b includes an inclined surface 185b located along a longitudinal vertex formed at the intersection of a plane defining the height 150b and a plane defining the width 140b of spiral balance spring body 110b. The inclined surface may extend along at least a portion of the total arc length.

Bevel 185b provides a reduction in structural strain concentration at said bevel during elastic deformation of the spiral spring body in use, which reduces the likelihood of fatigue failure of balance spring 100b in use. Bevel 185 may be formed by wet etching and is discussed further below.

By way of background, the materials employed in the present invention are discussed and explained with reference to other similar materials, and the advantages provided by employing <110> single crystal silicon in a balance spring for a timepiece are demonstrated.

There are two different types of silicon wafers: single crystal silicon and multicrystalline silicon. Single crystal silicon consists of a single crystal that is uniformly arranged throughout the wafer. Multicrystalline silicon can be further categorized as <100>, <110>, and <111>, depending on the crystal orientation. Multicrystalline silicon consists of many tiny (nanometer to micrometer scale) crystals that are randomly arranged within the wafer. Generally speaking, single crystal silicon wafers have more consistent material properties throughout the wafer than multicrystalline silicon wafers.

The uniform arrangement of single-crystal silicon crystals provides consistent material properties throughout the material, particularly in terms of orientation. However, the random arrangement of tiny crystals in polycrystalline silicon means that the material properties are highly dependent on the uniformity of the mixture, resulting in a roughly uniform macroscopic effect. Furthermore, polycrystalline silicon has visible grain boundaries that depend on the size of the individual crystals, negatively impacting the aesthetics of mechanical timepieces, where visual appearance is highly valued.

The key difference between <100>, <110>, and <111> single crystal silicon wafers lies in the Young's modulus, which depends on the in-plane orientation shown in Figure 4, as well as other material properties. <111> silicon wafers can be described as isotropic within the plane of the wafer, as the Young's modulus is independent of orientation. In contrast, <110> and <100> silicon wafers are anisotropic within the plane of the wafer, as the Young's modulus varies from 130.2 to 187.5 GPa for <110> wafers and from 130.2 to 168.9 GPa for <100> wafers. Each wafer type has its strengths and weaknesses as a material used in silicon balance spring processing.

<111> The isotropic properties of single-crystal silicon significantly simplify the design of silicon balance springs, a fact generally acknowledged in the theoretical literature on micromechanical systems. Because a silicon balance spring never bends out of the plane of the silicon, the uniform in-plane material properties make it easy to predict how it will deform when subjected to stress.

However, the special orientation of <111> silicon wafers also makes them more difficult to cut and polish during manufacturing, and more labor-intensive, so <111> silicon wafers are the most expensive and rare of the three types of single crystal silicon wafers.

In contrast, <100> and <110> single crystal silicon wafers are simpler and cheaper to produce because the crystal directions are easier to align with the cutting and polishing planes, especially <100> silicon wafers, whose crystal structure is perfectly aligned with Cartesian coordinates relative to the wafer plane.

However, the anisotropic properties of <100> and <110> silicon wafers make silicon design more complicated because Young's modulus and therefore sectional strip stiffness are directionally sensitive.

The design of a silicon balance spring is a complex process that takes into account several requirements. At its core, the spring's stiffness must be matched to the balance's inertia to produce the desired natural frequency according to the following equation.

Here, ω _n , E , b , h , L , I , k , and I _b are the oscillator's natural frequency, the Young's modulus of the hairspring, the hairspring's cross-sectional width, the hairspring's cross-sectional height, the hairspring's total arc length, the hairspring's second-order surface moment (rectangular cross-section), the hairspring's stiffness, and the balance's moment of inertia. Additional constraints also exist on the radial positions of the hairspring's inner and outer terminals.

The balance spring's pitch-to-bar width ratio has a lower limit to prevent collisions between the bars as the spiral contracts during rotation, particularly with conventional constant cross-section designs that lack concentricity during deformation.

The difference in the angular position of the inner and outer terminals also has an impact on the stability of the stiffness with varying oscillator amplitudes. The cumulative requirements impose extremely strict constraints on the degrees of freedom in the design of the hairspring, so that almost all hairsprings with similar stiffness have roughly the same spring geometry.

The aforementioned constraints on hairspring design freedom do not pose a significant problem for metal hairsprings, as the metal’s tendency to creep and deform over time also applies to its robustness and flexibility. Despite the relative lack of precision in the manufacturing process, metal hairsprings can be easily bent and cut as part of post-processing, allowing for fine-tuning of their rigidity to match the respective balance wheel on a case-by-case basis.

Silicon balance springs, on the other hand, offer greater structural stability over time, and micromachining is a highly precise manufacturing technique. However, the downside is limited flexibility in post-processing to adjust the length and geometry of the balance spring during fine-tuning of the spring's rigidity. Any deviation from the desired spring rigidity typically requires a complete redesign, which is technically difficult and inefficient.

The use of <100> and <110> single-crystal silicon wafers allows for fine-tuning of the balance spring stiffness by rotating the design in the wafer so that the orientation of the wafer is different. This is possible because the Young's modulus of silicon depends on the orientation angle.

Assuming that the geometry of the hairspring is primarily that of an Archimedean screw, the hairspring stiffness can be described by the following equation:

Here, θ ₀ , θ _F , and α are the angular positions of the inner and outer terminals of the balance spring and the pitch of the Archimedean screw, respectively. Note that Young's modulus is a function of the orientation angle θ, and that a change in the balance spring's orientation means that θ ₀ and θ _F change while keeping their difference constant. Thus, this equation demonstrates that changing the balance spring's orientation on the wafer can alter its stiffness.

Comparing the use of <100> and <110> single crystal silicon wafers, the present invention has confirmed that <110> single crystal silicon wafers are preferred for fine adjustment of balance spring stiffness.

The Young's modulus of <100> and <110> silicon wafers as an in-plane function can be defined as follows.

Here, S ₁₁ , S ₁₂ , and S ₄₄ are elements of the single crystal silicon compliance matrix defined as 7.68, -2.14, and 12.6, respectively, in units of 10 ⁻¹² Pa ⁻¹ . Over 90-degree intervals, E _<100> varies between 130.2 and 168.9 GPa, compared to E _<110> which varies between 130.2 and 187.5 GPa.

This means that for the <110> single crystal silicon wafer provided by the present invention, the hairspring stiffness is more sensitive to its orientation angle on the silicon wafer, thereby providing greater flexibility in fine-tuning the stiffness.

Another advantage of the present invention confirmed by using <110> single crystal silicon wafers over <100> single crystal silicon wafers is that <110> single crystal silicon wafers tend to form bevels under wet etching technology, and the bevels act as stress reduction mechanisms.

A typical hairspring in a mechanical timepiece experiences heavy cyclic loading, as it must flex at 3 to 5 Hz for several years. Even a small reduction in the hairspring's load stress can significantly extend its lifespan. This is particularly true for hairsprings coated with a thermal oxide for thermal compensation, as shown in Figures 2 and 3.

One of the weakest points of a coated silicon balance spring is the boundary between the silicon core and the silicon dioxide coating. The two materials differ in their crystal structures and experience severe adhesion stresses, exacerbated by thermal stresses caused by differing coefficients of thermal expansion as the balance spring cools from the coating process temperatures exceeding 1000°C. If these stresses are severe enough, the thermal oxide may debond from the silicon core, resulting in a loss of thermal compensation and, consequently, poor regulation and timekeeping.

The effective technique provided by the present invention is described with reference to FIG3 , which reduces stress by eliminating sharp corners that cause stress concentration.

Silicon balance spring processing typically involves deep reactive ion etching (DRIE) to create the bulk silicon structure. However, depending on the type of single-crystal silicon wafer, the chamfers provided by the present invention can be applied to the balance spring edge using a wet etching process. Wet etching involves using an alkaline solution to remove silicon by immersion.

Depending on the exact type of solution used, the etching results can be highly anisotropic based on crystal orientation, with selectivity for the <100> crystal orientation being up to 400 times greater than for the <110> or <111> directions. On <110> or <111> single-crystal silicon wafers, wet etching alone is not capable of producing structures with trapezoidal cross-sections.

Wet etching, used as a post-process after DRIE, can create bevels on otherwise rectangular cross-sections, thereby eliminating sharp edges and reducing stress concentrations. This is not possible with <100> single-crystal silicon wafers, as wet etching would primarily etch into the sides of the silicon balance spring cross-section and thus would not provide the benefits offered by the present invention.

The silicon balance spring must have a silicon dioxide coating to achieve thermal compensation, and this mechanism has been shown to work for the <110> single-crystal silicon wafers employed according to the invention.

The purpose of the silicon dioxide coating is to compensate for the decrease in oscillation frequency with increasing temperature caused by the positive thermal expansion coefficient of the balance wheel and the negative thermoelastic constant of the silicon core of the hairspring. The temperature dependence of the moment of inertia of the balance wheel (also known as the balance wheel) can be described as follows.

I _b (ΔT) = I _b0 (1 + αΔT) ²

Here, I _b0 , α, and ΔT are the balance moment of inertia at the rated temperature, the thermal expansion coefficient of the balance material, and the temperature difference from the rated temperature, respectively.

The thermal expansion coefficient of beryllium copper alloy, which is commonly used as a balance wheel material, is approximately 16 ppm/K. The watch industry usually uses a temperature range of 23 +/- 15 ° C to check thermal expansion.

In order to derive the equations for the stiffness of a silicon balance spring with a temperature-sensitive silica coating, it is necessary to first derive the equivalent Young's modulus for the composite structure at the rated temperature.

Here, ζ, E ₀ , E _Si,0 , E _{SiO 2,0} , b, and h are the oxide thickness, the nominal Young's modulus of the composite balance spring, the nominal Young's modulus of silicon, the nominal Young's modulus of silicon dioxide, the total width, and the total height of the balance spring cross section, respectively.

Note that E _Si,0 depends on the orientation angle on the silicon wafer and varies for <100> and <110> silicon wafers. The value of E _SiO2,0 is approximately 72.4 GPa. If temperature is taken into account, the equation becomes as follows.

E _Si (θ, ΔT) = E _{Si, 0} (θ) + ε _Si ΔT

Here, ε _Si and ε _{SiO 2} are the thermoelastic constants of silicon and silicon dioxide, and their values are approximately -60 ppm/K and 215 ppm/K, respectively. Then, the hairspring stiffness can be defined as follows.

Combining this with the equation for the balance's moment of inertia yields the following equation for the frequency of oscillation.

If the oscillation frequency is orthogonalized to the desired frequency for the rated temperature, the oxide thickness required to achieve thermal compensation to the necessary tolerance, typically set at +/- 1 sec/day/K, can be determined numerically.

FIG. 5 shows a graph showing the relationship between oscillation frequency and temperature difference for five different oxide thicknesses using a <110> single crystal silicon wafer according to the present invention.

The width and height of the balance spring vary from 35 to 40 μm and 200 to 210 μm respectively, depending on the thickness of the oxide used. The total balance spring arc length is approximately 130 mm, and the balance moment of inertia is approximately 1.65 g*mm ² .

It can be seen that when using the <110> single crystal silicon wafer according to the present invention, for an oxide thickness of 4.5 μm, the oscillation frequency varies by less than 0.1 s/day/K over the temperature range of 5°C to 40°C, which is within the standard tolerance of thermal compensation.

The anisotropic material properties of <110> single-crystal silicon wafers pose a challenge in controlling the bending of the balance spring at each cross-section because the cross-sectional stiffness depends on the orientation angle.

In order to maintain a constant cross-sectional stiffness throughout the entire arc length, in embodiments of the present invention, the balance spring width may be varied as a function of Young's modulus so as to mitigate the net stiffness variation.

Considering that the cross-sectional stiffness of the balance spring is proportional to the Young's modulus and the cube of the strip width, for <110> silicon wafers, the width can be calculated according to the following equation.

Using binomial expansion, the above equation can be approximated as follows.

Here, _b0 is the width of the silicon balance spring. This width varies periodically for each half turn of the balance spring. The equation assumes a nominal Young's modulus of 1/ _s⁻¹ , or 130.2 GPa. Figure 6 graphically illustrates the variation in width of a <110> silicon balance spring using a <110> single crystal silicon wafer according to the present invention as a function of orientation angle.

By using <110> single crystal silicon wafers according to the present invention, the present invention provides several advantages over prior art technologies including <111> and <100> single crystal silicon, including:

(i) The present invention uses materials that are easier to manufacture and lower in cost than <111>, providing easy and cost-effective manufacturing;

(ii) The present invention provides materials that allow for greater tuning sensitivity than <100>, allowing for more precise tuning;

(iii) Using a material that allows a bevel to be formed on the peripheral edge provides the following advantages:

- Reduce the interfacial stress between the apex of the silicon balance spring and the silicon dioxide layer, thus reducing debonding and thermal compensation losses;

- Reduce stress concentration factors and localized stress rise, reducing the possibility of failure due to hairspring fatigue;

- providing a balance spring with greater resistance to failure from inherent imperfections at the apex and thus with a longer fatigue life, parameters required for a long life of a balance spring in a timepiece;

- by providing reduced local stresses, and therefore a lower likelihood of fatigue failure, such a balance spring may be employed in high frequency applications, such as 8 to 10 Hz applications; and

Silicon balance springs have not been used for the extended periods of time that conventional metal or metal alloy balance springs have. Thus, historically, the long lifespan of such items, such as 50 years or more, has been uncertain. Consequently, providing the balance spring with lower local stresses provides the balance spring with a greater tendency to achieve the long-life requirements of prior art balance springs, such as those comprising <100> and <111> single-crystal silicon.

The present invention overcomes the shortcomings of both <111> and <100> single crystal silicon balance springs through selection of manufacturing, design parameters and long life/fatigue parameters, including long life of thermal compensation layer adhesion and long life of the silicon core of the balance spring.

Claims (18)

1. A torque recovery element for a mechanical watch oscillator, the oscillator having an oscillation frequency, the torque recovery element comprising: A spiral hairspring body with N turns, the spiral hairspring body having an inner terminal end for engaging with a rotating inertial element via a hairspring stud and an outer terminal end for engaging with a fixed clamp element, and the spiral hairspring body having a width, a height and a total arc length; The spiral hairspring body includes a core formed of a monocrystalline silicon wafer, wherein the monocrystalline silicon wafer is oriented along the crystal axis <110>; and The spiral hairspring body includes at least one peripheral material coating, the thermoelastic constant of which differs from that of the core of the spiral hairspring body, so as to maintain the oscillation frequency of the oscillator including the torque recovery element as substantially insensitive to changes in ambient temperature. 2. The torque recovery element according to claim 1, wherein the spiral hairspring body has a substantially rectangular cross-section. 3. The torque recovery element according to claim 1 or claim 2, wherein the torque recovery element has a width ranging from 20 μm to 60 μm, a height ranging from 100 μm to 400 μm, and a total arc length ranging from 50 mm to 200 mm. 4. The torque recovery element according to claim 1 or 2, wherein the torque recovery element has a number of revolutions in the range of 5 to 20. 5. The torque recovery element according to claim 1 or 2, wherein the at least one peripheral coating of the hairspring is formed of silicon dioxide. 6. The torque recovery element according to claim 5, wherein the at least one peripheral coating has a thickness ranging from 3 μm to 6 μm. 7. The torque recovery element according to claim 1 or 2, wherein the orientation angle of the <110> monocrystalline silicon wafer produces a small change in the overall hairspring stiffness used in rigidity fine-tuning. 8. The torque recovery element according to claim 1 or 2, wherein the spiral hairspring body has a periodically varying width along at least a portion of the total arc length based on the number of turns, in order to compensate for the varying hairspring segment stiffness due to the anisotropic Young's modulus on the silicon wafer plane. 9. The torque recovery element according to claim 8, wherein the width of the <110> monocrystalline silicon wafer varies according to the following equation: Where b is the width of the spiral hairspring body at an angular position on the spiral hairspring body, where _b0 is the nominal width of the spiral hairspring body, and where _S11 , _S12 and _S44 are elements of the compliance matrix of the <110> single crystal silicon wafer. 10. The torque recovery element according to claim 1 or 2, wherein the spiral hairspring body is formed by a dry etching manufacturing technique, including deep reactive ion etching (DRIE) manufacturing technique. 11. The torque recovery element according to claim 1 or 2, wherein the longitudinal vertex formed at the intersection of the height plane of the spiral hairspring body and the width plane of the spiral hairspring body has an inclined surface extending along at least a portion of the total arc length. 12. The torque recovery element of claim 11, wherein the inclined surface provides a reduction in structural stress concentration at the apex during elastic deformation of the spiral hairspring body during use. 13. The torque recovery element of claim 11, wherein the bevel is formed by wet etching. 14. An oscillator for a clock, the oscillator comprising a torque recovery element according to any one of the preceding claims, and a balance wheel attached to an inner terminal end of the torque recovery element. 15. A method for forming a torque recovery element according to any one of claims 1 to 13, wherein the spiral hairspring body is formed by a dry etching manufacturing technique, including deep reactive ion etching (DRIE) manufacturing technique. 16. A method for forming a torque recovery element according to any one of claims 1 to 13, wherein the width of the <110> monocrystalline silicon wafer varies according to the following equation: Where b is the width of the spiral hairspring body at an angular position on the spiral hairspring body, where _b0 is the nominal width of the spiral hairspring body, and where _S11 , _S12 and _S44 are elements of the flexibility matrix of the <110> single crystal silicon wafer. 17. A mechanical oscillator for use in mechanical clocks, the mechanical oscillator comprising: A torque recovery element comprising a spiral hairspring body having N turns, the spiral hairspring body having an inner terminal and an outer terminal, and the spiral hairspring body having a width, a height, and a total arc length; the inner terminal is used to engage a rotational inertial element rotating about an axis via a hairspring stud; and the outer terminal is used to engage with a fixed clamping element; and A rotational inertial element, which engages with the inner terminal of the spiral hairspring element and is rotatable about the axis; The spiral hairspring body includes a core formed of a monocrystalline silicon wafer, wherein the monocrystalline silicon wafer is oriented along the crystal axis <110>; and The spiral hairspring body includes at least one peripheral material coating whose thermoelastic constant differs from that of the core of the spiral hairspring body, so as to maintain the oscillation frequency of the mechanical oscillator, including the torque recovery element, as substantially insensitive to changes in ambient temperature. 18. The mechanical oscillator of claim 17, wherein the torque recovery element is the torque recovery element of any one of claims 2 to 13.