Methods for designing and processing a microcantilever-based probe with an irregular cross section applied in an ultra-low friction coefficient measurement at a nanoscale single-point contact

A method for designing a microcantilever-based probe with an irregular cross section applied in an ultra-low friction coefficient measurement at a nanoscale single-point contact, wherein an axis of a microcantilever-based probe with the irregular cross section is a straight line, and a uniform cross section is provided along a direction of the axis;

• the uniform cross section is the irregular cross section composed of a plurality of rectangles having an equal wall thickness;
• the method for designing the probe comprises the following steps:
• step (1):
• establishing a universal theoretical model of a friction coefficient measurement using the microcantilever-based probe, wherein
• a friction force and a positive pressure are respectively measured by a torsion and a bending of the microcantilever-based probe according to a measurement principle of an atomic force microscope, namely,
• FL=KT×InvOLSL×UL (1),
• FN=KN×InvOLSN×UN (2),
• wherein FL and FN refer to the friction force and the positive pressure, respectively;
• KT and KN refer to a torsional elastic coefficient and a normal elastic coefficient of the microcantilever-based probe, respectively;
• InvOLSL and InvOLSN refer to a reciprocal of a lateral optical lever sensitivity and a reciprocal of a normal optical lever sensitivity, respectively;
• UL and UN refer to a lateral output voltage and a normal output voltage of a photodetector;
• and
• the universal theoretical model of the friction coefficient measurement using the microcantilever-based probe is then established, as shown in the following equation:
• μ
• =
• F
• L
• F
• N
• =
• K
• T
• ×
• InvOLS
• L
• ×
• U
• L
• K
• N
• ×
• InvOLS
• N
• ×
• U
• N
• ,
• (
• 3
• )
• wherein μ is the friction coefficient;
• step (2):
• establishing a specific theoretical model of the friction coefficient measurement using the microcantilever-based probe with the irregular cross section, wherein
• according to a measurement principle of an optical path system, the reciprocal of the lateral optical lever sensitivity InvOLSL and the reciprocal of the normal optical lever sensitivity InvOLSN are expressed as follows:
• InvOLS
• L
• =
• H
• /
• α
• L
• 4
• 
• dU
• sum
• /
• α
• sum
• ,
• (
• 4
• )
• InvOLS
• N
• =
• Hl
• /
• α
• N
• 6
• 
• dU
• sum
• /
• α
• sum
• ,
• (
• 5
• )
• wherein H is a length of a photosensitive surface of the photodetector;
• d is a length of an optical path;
• l is a length of the micro-cantilever;
• Usum is a total voltage generated by a laser in four quadrants of the photodetector;
• and αsum, αL, and αN respectively refer to magnification times of a total output current, a lateral output current, and a normal output current of the photodetector subjected to a current/voltage converter, with a unit of V/A;
• based on an elastic mechanics and material mechanics theory of thin plates, the torsional elastic coefficient KT of the microcantilever-based probe with the irregular cross section is expressed as follows:
• K
• T
• =
• F
• L
• ϕ
• tip
• =
• G
• ×
• I
• t
• (
• h
• tip
• +
• e
• 1
• )
• ×
• l
• ,
• (
• 6
• )
• wherein ϕtip is a torsion angle of the microcantilever-based probe with the irregular cross section under an action of the friction force FL;
• G is a shear modulus of a material of the micro-cantilever;
• It is a torsional inertia moment of the irregular cross section, htip is a tip height, e1 is a distance from a centroid of the irregular cross section to a bottom side;
• the normal elastic coefficient KN of the microcantilever-based probe with the irregular cross section is expressed as follows:
• K
• N
• =
• F
• N
• υ
• tip
• =
• 3
• 
• E
• ×
• I
• x
• C
• l
• 3
• ,
• (
• 7
• )
• wherein νtip is a deflection of the microcantilever-based probe with the irregular cross section under an action of the positive pressure FN;
• E is an elastic modulus of the material of the micro-cantilever;
• IxC is an inertia moment of the irregular cross section to a neutral axis xC;
• the equations (4)-(7) are put into the universal theoretical model (3) established in the step (1) to calculate and obtain the specific theoretical model for measuring the friction coefficient of the microcantilever-based probe with the irregular cross section, as shown in the following equation:
• μ
• =
• G
• ×
• l
• 2
• 
• E
• ×
• (
• h
• tip
• +
• e
• 1
• )
• ×
• I
• t
• I
• x
• C
• ×
• I
• L
• I
• N
• ,
• (
• 8
• )
• wherein IL is the lateral output current of the photodetector, and IL=UL/αL;
• IN is the normal output current of the photodetector, and IN=UN/αN;
• and
• step (3):
• designing the microcantilever-based probe with the irregular cross section meeting measurement requirements for an ultra-low friction coefficient, wherein
• according to the step (2), assuming that μ is a friction coefficient resolution μmin, and FN is a loadable maximum positive pressure FNmax, or FL is a measurable minimum friction force FLmin;
• combined with constraint conditions of the friction coefficient resolution μmin, the loadable maximum positive pressure FNmax or the measurable minimum friction force FLmin, characteristics of the atomic force microscope, the equations (1)-(8) are considered to calculate and designed dimensions of the microcantilever-based probe with the irregular cross section to meet the measurement requirements;
• the designed dimensions comprise the length l of the micro-cantilever, a width wi of an i-th sub-rectangle constituting the irregular cross section, and a wall thickness t.