Abstract:
A method for determining an elastic ratio of indentation work of an indentation of a material surface to a sampling depth is described. The method may include: placing an indenter probe on the material surface; indenting the material surface to a maximum depth greater than the sampling depth by increasing a vertical force on the probe, while recording the force as a first function of depth; retracting the probe from the material surface by decreasing the force, while recording the force as a second function of depth; and calculating the elastic ratio of indentation work for the sampling depth from the recorded functions.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of Invention 
         [0002]    This invention relates generally to a method for determining an elastic ratio of indentation work, as well as to a computer program product for performing such a method. Furthermore, the invention also relates to a nanoindenter and an apparatus for determining an elastic ratio of indentation work. 
         [0003]    2. Related Art 
         [0004]    The development of nanostructured materials, thin films, surface coatings, miniaturized electronic and engineering components etc. benefits from a detailed understanding of the mechanical properties of materials at the nanoscale. For example, in the experimental technique of nanoindentation a load is applied to an indenter probe placed against a surface of a material to be investigated. Typically, the load is increased during a loading phase until the probe has penetrated the material to a maximum depth of penetration, and decreased during an unloading phase following the loading phase. 
         [0005]    One material property obtainable by nanoindentation experiments is the elastic ratio of indentation work (ηIT), which specifies the share of the mechanical work applied to the probe during the loading phase that is recoverable as elastic energy during the unloading phase. In order to determine the elastic ratio of indentation work as a function of the maximum depth of penetration, a series of independent experiments may be performed, indenting the investigated material in each experiment to a different maximum depth of penetration. 
       SUMMARY 
       [0006]    A method for determining an elastic ratio of indentation work of an indentation of a material surface to a sampling depth is described. The method may include: placing an indenter probe on the material surface; indenting the material surface to a maximum depth greater than the sampling depth by increasing a vertical force on the probe, while recording the force as a first function of depth; retracting the probe from the material surface by decreasing the force, while recording the force as a second function of depth; and calculating the elastic ratio of indentation work for the sampling depth from the recorded functions. 
         [0007]    Additionally, an apparatus for determining an elastic ratio of indentation work of an indentation of a material surface to a sampling depth is also described. The apparatus may include: a load curve data input unit for inputting load curve data that represent a load force measured on an indenter probe indenting the material surface up to a maximum depth greater than the sampling depth, in dependence on a depth indented to; an unload curve data input unit for inputting unload curve data that represent an unload force measured on the indenter probe while retracting from the material surface from the maximum depth, in dependence on a depth retracted to; and a calculation unit for calculating the elastic ratio of indentation work for the sampling depth from the inputted data. 
         [0008]    Other systems, methods features and advantages of the invention will be or will become apparent to one with skill in the art upon examination of the following figures and detailed description. It is intended that all such additional systems, methods, features and advantages be included within this description, be within the scope of the invention, and be protected by the accompanying claims. 
     
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         [0009]    The invention can be better understood by referring to the following figures. The components in the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention. In the figures, like reference numerals designate corresponding parts throughout the different views. 
           [0010]      FIG. 1  shows a schematic view of a nanoindenter according to an approach, including a material surface to be indented; 
           [0011]      FIG. 2A  shows load and unload curves for an indentation acquired by the nanoindenter of  FIG. 1 , with a sampling depth marked; 
           [0012]      FIG. 2B  shows the load and unload curves of  FIG. 2A , with an unload curve shifted along a depth axis; 
           [0013]      FIG. 3A  shows the load and unload curves for an indentation acquired by a nanoindenter according to a further approach, with a sampling depth marked; 
           [0014]      FIG. 3B  shows the load and unload curves of  FIG. 3A , with an unload curve shifted along a force axis according to a method of an approach; 
           [0015]      FIG. 3C  shows the load and unload curves of  FIG. 3A , with the shifted unload curve extrapolated according to a method of an approach; 
           [0016]      FIG. 4  shows a flow diagram of a method for determining an elastic ratio of indentation work according an approach; 
           [0017]      FIG. 5  shows an example of a layered material surface to be investigated in an indentation experiment; and 
           [0018]      FIG. 6  shows a graph of an elastic ratio of indentation work as a function of penetration depth determined according to an approach. 
       
    
    
     DETAILED DESCRIPTION 
       [0019]    In the following description, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration one or more specific implementations in which the invention may be practiced. It is to be understood that other implementations may be utilized and structural changes may be made without departing form the scope of this invention. 
         [0020]      FIG. 1  shows a schematic view of a nanoindenter  144 ,  142  according to an approach. The nanoindenter  144 ,  142  comprises an indentation unit  144  in which a material surface  110  to be investigated by undergoing indentation is placed on a mount  112 , and a data processing unit  142  for controlling the indentation unit  144  during the investigation of the material surface  100 , as well as acquiring and processing data from the indentation unit  144 . 
         [0021]    The indentation unit  144  comprises an indentation probe  100  mounted on a shaft  146  that is held in a stable position by a pair of springs  106 . At the end of the shaft opposing the indentation probe, a coil  104  is mounted on the shaft  146  and positioned between the poles of a permanent magnet  102 . The coil  104  and magnet  102  together form an indentation force drive, which, when electric current is passed through the coil  104  exerts a force through the shaft  146  onto the indentation probe  100 . To the shaft  146 , capacitor plates  107  are attached that move with the shaft  146  between corresponding static capacitor plates  108  attached to a housing  145  of the indentation unit  144 . The moving  107  and static  108  capacitor plates form a position detector  107 ,  108  for detecting the position of the indentation probe  100  mounted to the shaft  146 . 
         [0022]    The data processing unit  142  is connected to the indentation unit  144  through data lines  148 . In alternative approaches, data processing unit  142  and indentation unit  144  are arranged in a common housing, or connected to each other via wireless data links instead of the data lines  148 . 
         [0023]    The data processing unit  142  comprises a controller unit  116  for controlling the movement of the indenter probe  100  in the indentation unit  144 . For this purpose, the controller unit  116  is connected to the coil  104  of the indentation unit. The data processing unit  142  further comprises a recorder unit  118  for simultaneously recording a position of the indenter probe  100  and the force exerted on the probe  100  by the indentation force drive  102 ,  104 . For this purpose, the recorder unit  118  is connected both to the position detector  107 ,  108  in the indentation unit  146  and to the controller unit  116 . The recording unit  118  comprises a load curve data input unit  120  for recording load curve data comprising data pairs of the depth of penetration of the indenter probe  100  into the material surface  110  during motion of the indenter probe  100  towards the material surface  110 , and a load curve data input unit  122  for recording unload curve data equally comprising data pairs of the depth of penetration of the indenter probe  100  into the material surface  110  during motion of the indenter probe  100  away from the material surface  110 . 
         [0024]    The recorder unit  118  is furthermore connected to a data console  114  from where load and unload curve data can also be received, alternatively to acquiring such data from the controller unit  116  and the position detector  107 ,  108 . For example, data acquired in earlier experiments or using different indentation units can be inputted through the data console  114 . When used in such a way, the data processing unit  142  functions as an apparatus  142  for determining an elastic ratio of indentation work in itself, without including the indentation unit  144 . 
         [0025]    The data processing unit  142  further comprises an unload curve shifting unit  124  for modifying the unload curve data recorded by the unload curve data input unit  122 . The unload curve shifting unit  124  comprises a load curve evaluator  126  for determining a sampling depth indentation force by evaluating the load curve at the sampling depth. Further, it comprises a sampling retraction depth determiner  128  for determining a sampling retraction depth at which the unload curve evaluates to the sampling depth indentation force. Further, the unload curve shifting unit  124  comprises a function redefiner  129  for redefining the unload curve such that it evaluates at a given depth to the force value of the unmodified unload curve evaluated for the sum of the depth and the difference of the sampling depth and the sampling retraction depth. 
         [0026]    The data processing unit  142  further comprises a calculation unit  130  for calculating a value of the elastic ratio of indentation work for a desired sampling depth of penetration. The calculation unit  130  receives the modified unload curve from the unload curve shifting unit  124  and the unmodified load curve from the recorder unit  118 . The calculation unit  130  comprises a load curve integrator  138  for integrating the first function in an interval up to the sampling depth, thus determining an overall indentation work of the indenter probe during the loading phase. Further, the calculation unit  130  comprises an unload curve integrator  134  for integrating the second function in the interval up to the sampling depth, thus determining an elastic indentation energy that is recovered when the indenter probe is unloaded. Further, the calculation unit  130  comprises a divider  132  for dividing the elastic indentation work by the overall indentation work, thus arriving at a value of the elastic ratio of indentation work for the sampling depth. 
         [0027]    Finally, the data processing unit  142  comprises an adjustment unit  140 , for adjusting the value of the elastic ratio of indentation work calculated by the calculation unit  130 , based in alternative approaches e.g. on experimental data of an indentation to and retraction from substantially the sampling depth, or on theoretical models of nanoindentation processes, or both. The adjustment unit  140  is connected to the data console  114  for outputting the adjusted value of the elastic ratio of indentation work. 
         [0028]    In the following, calculations carried out in the calculation unit  130  and the unload curve shifting unit  124  of the approach of  FIG. 1  will be explained in further detail by referring to  FIGS. 2A  and B. 
         [0029]      FIG. 2A  shows a coordinate system with load  200  and unload  202  curves of an indentation experiment carried out by the nanoindenter of  FIG. 1 . The load force on the indenter probe is plotted along the vertical axis  210 , and the penetration depth along the horizontal axis  208 . The indentation experiment starts at the origin  212  of the coordinate system with the indenter probe placed at the surface of the material to be investigated. The load on the probe is then gradually increased, thereby pushing the probe into the material, with force and penetration depth following the load curve  200  until a maximum penetration depth  204  is reached. From there, the load on the probe is gradually decreased, resulting in the indenter probe being pushed backwards by the elastic response of the material, following the unload curve  202 . Since not all of the indentation work of the load phase  200  is recoverable as elastic energy, the unload curve  202  lies below the load curve  200 . The unload curve reaches the depth axis at a residual depth  213 , where the elastic material force pushing the probe backward becomes zero. 
         [0030]    The overall indentation work performed by the nanoindenter during the loading phase  200  is given by the area under the load curve  200  in the interval between a depth of zero  212  and the maximum penetration depth  204 . The elastic indentation work recovered by the nanoindenter during the unloading phase  202  correspondingly is given by the area under the unload curve  202  in the interval from the residual depth  213  to the maximum penetration depth  204 , or alternatively in the interval from zero depth  212  to the maximum penetration depth  204  when assuming that the unload curve follows the depth axis  208  for depth values less than the residual depth  213 . Thus, the elastic ratio of indentation work for the particular material and maximum penetration depth  204  can be calculated by dividing the elastic indentation work by the overall indentation work. 
         [0031]    Also, the overall indentation work performed by the nanoindenter during the loading phase  200  up to an arbitrary sampling depth  206  and associated load  216  that is less than the maximum penetration depth is given by the area  214  under the load curve  200  in the interval between a depth of zero  212  and the sampling depth  206 . The area under the unload curve  202  in the same interval however is different from an area that would be obtainable in an indentation experiment with maximum penetration at the sampling depth  206 . Thus, an approach in which the unload curve  202  is adjusted to resemble an unload curve obtainable in an indentation experiment with maximum penetration at the sampling depth  206  may have an effect of enabling to calculate a value of the elastic ratio of indentation work that is a particularly close approximation of the elastic ratio of indentation work obtainable from the indentation experiment with maximum penetration at the sampling depth  206 . 
         [0032]    As shown in  FIG. 2B , in the present approach the unload curve  202  is shifted  410  to the left along the depth axis  208  to a shifted position  202 ′, by such an amount that the shifted curve  202 , intersects the load curve  200  at the sampling depth  206  and associated load  216 . Now, the area  218  under the shifted curve  202 ′ in the interval up to the sampling depth  206  is determined to obtain a value for the elastic indentation work that approximates a value obtainable from the indentation experiment with maximum penetration at the sampling depth  206 . A value for the elastic ratio of indentation work is then calculated by dividing the marked area  218  in  FIG. 2B  by the marked area  214  in  FIG. 2A . 
         [0033]      FIG. 3A  shows a coordinate system with load  200  and unload  202  curves of an indentation experiment carried out by a nanoindenter according to another approach. As in  FIG. 2A , the load force on the indenter probe is plotted along the vertical axis  210 , and the penetration depth along the horizontal axis  208 . The indentation experiment starts at the origin  212  of the coordinate system, with load force on the indenter probe and penetration depth following the load curve  200  until the maximum penetration depth  204  is reached. From there, the load on the probe is gradually decreased, following the unload curve  202 , until a load of zero is reached at the residual depth  213 . A sampling depth  206  smaller than the maximum penetration depth  204  has been marked, as well as a corresponding load  216  at the sampling depth  206  during the loading phase. 
         [0034]    In order to determine from these experimental data, a value for the elastic ratio of indentation work that approximates an elastic ratio of indentation work derivable from an indentation experiment with the sampling depth  206  as maximum penetration depth, the function  200  of depth  208  corresponding to the load curve  200  is integrated in the interval between zero depth  212  and the sampling depth  206  to obtain the overall indentation work  214  performed during the loading phase  200  up to the sampling depth  206 . 
         [0035]    In  FIG. 3B , the unload curve  202  has been shifted  410  vertically along the force axis  210 ; such that the shifted unload curve  202 ′ intersects the load curve  200  at the sampling depth  206 . In  FIG. 3C , the shifted unload curve  202 ′ is extrapolated  300  to the depth axis  208  by a suitable curve fitting algorithm. The function  200  of depth  208  corresponding to the shifted and extrapolated unload curve  202 ′,  300  is then integrated in the interval between an approximated residual depth  213 ′ at which the extrapolated portion  300  reaches the depth axis  208  and the sampling depth  216 . A value for the elastic ratio of indentation work is then calculated by dividing the marked area  218  corresponding to the result of the integration in  FIG. 3C  by the marked area  214  in  FIG. 3A . 
         [0036]      FIG. 4  shows a flow diagram of a method for determining an elastic ratio of indentation work according to an approach. In step  400 , a maximum penetration depth is chosen for an indentation experiment to be performed on a surface of a material, based e.g. on an intended application of the material. In alternative approaches, a maximum load force is chosen, thus determining implicitly the maximum penetration depth. 
         [0037]    In step  402 , an indenter probe is placed at the surface of the material such that it touches the surface but does not exert any force. In a loading phase  404 , a load on the indenter probe is gradually increased such that the indenter probe gradually penetrates the material until the maximum penetration depth (or maximum load) chosen in step  400  is reached. During the loading phase  404 , the load on the indenter probe is recorded as a first function of penetration depth. 
         [0038]    In an unloading phase  406 , the load on the indenter probe is gradually decreased again such that the indenter probe gradually retracts from the material towards the surface until a residual depth at which the load on the probe reaches zero. During the unloading phase  406 , the load on the indenter probe is recorded as a second function of penetration depth. 
         [0039]    In step  408  a sampling depth is selected for which an elastic ratio of indentation work is to be calculated in the following steps  410 - 418 . In the present approach, the range between a depth of zero and the maximum penetration depth reached in the loading phase  404  is divided into a predetermined number of intervals, from which the higher bound of the first interval is chosen as sampling depth. 
         [0040]    In step  410  the second function of depth, recorded during the unload phase  406  is shifted along the depth axis such that it evaluates at the sampling depth to the same force value as the first function of depth, recorded during the loading phase  404 . In alternative approaches, the second function is shifted along the force axis or along both axes. 
         [0041]    In step  412  the first function of depth is integrated in the interval between a depth of zero and the sampling depth, yielding a value for the overall indentation work performed up to the sampling depth during the loading phase  404 . In step  414  the second function of depth as modified in step  412  is integrated in the interval between a depth of zero and the sampling depth, yielding a value for the elastic energy portion of the overall indentation work performed up to the sampling depth during the loading phase  404 . In step  418  the value for the elastic energy portion determined in step  414  is divided by the value for the overall indentation work determined in step  412 , yielding a value for the elastic ratio of indentation work corresponding to the sampling depth selected in step  408 . 
         [0042]    In step  420  it is determined whether further values of the elastic ratio of indentation work values corresponding to further values of the sampling depth are to be calculated. If this is the case, the method branches to step  408  where a new value for the sampling depth is selected from the range between zero and the maximum penetration depth, by choosing the higher bound of the next interval as sampling depth. A further value of the elastic ratio of indentation work that corresponds to the new sampling depth selected in step  408  is then calculated in steps  410 - 418 . This is repeated until values for the elastic ratio of indentation work corresponding to the respective higher bounds of all intervals in the range between zero and the maximum penetration depth have been calculated. 
         [0043]    Step  420  then branches to step  422 , wherein the calculated values are further adjusted based on an empirical or theoretical comparison of elastic ratio of indentation work results determined according to the present approach and elastic ratio of indentation work values determined in a conventional series of separate indentation experiments in each of which the surface of the material is indented to a different sampling depth as maximum depth of penetration. In step  424 , the series of elastic ratio of indentation work values calculated in steps  408 - 420  and adjusted in step  422  is output as a function that provides the elastic ratio of indentation work as a function of sampling depth in the range between zero and the maximum penetration depth of the loading phase  404 . 
         [0044]      FIG. 5  shows an example of a layered material surface  100  to be investigated in an indentation experiment. The material  115 / 110  comprises a substrate layer  504 , an intermediate layer  502 , and a top layer  500 . An indentation probe  100  during a loading phase  404  of an indentation experiment first penetrates the top layer  500  before entering the intermediate layer  502  through the interface  604  between the top and intermediate layers. Before reaching the substrate  504 , the indentation probe  100  is retracted in an unload phase  406 . 
         [0045]      FIG. 6  shows a graph of an elastic ratio of indentation work  602  as a function  606  of penetration depth  208  determined in the indentation experiment of  FIG. 5 . The elastic ratio of indentation work  602  is displayed as a function of sampling depth  208  in the range between zero  212  and the maximum penetration depth  204  of the loading phase  404 . A distinct change  608  in the slope of the function  606  may be suggestive of a change of structure or composition of the material at or close to a corresponding depth  610 . For comparison only, the dash-dotted line marks the thickness of the top layer  500  in the non-indented material, i.e. the distance of the interface  604  between the top and intermediate layers from the surface of the non-indented material.