Method and apparatus for high resolution analysis of the composition of a material

Abstract: The present invention relates to analytical methods for determining the composition of a material that undergoes a transition as a function of a driving variable. As applied to thermogravimetric analysis (TGA), a first preferred embodiment comprises (1) decreasing the heating rate when deviations from a baseline signal are detected; (2) establishing a minimum heating rate; (3) forcing the heating rate to a predetermined maximum whenever the rate of change of the weight change with respect to the temperature falls below a predetermined value; and (4) adjusting the heating rate according to the rate of change of the weight of the sample to track a predetermined rate of change of the weight of the sample. A second preferred embodiment comprises (1) selecting either a high-productivity or a high-resolution mode; (2) in both modes, controlling the rate of temperature increase according to a function containing an exponential term, wherein the argument of the exponential term includes the percent weight change per unit time of the sample; (3) in the high-productivity mode, choosing the function so that when the percentage weight change per minute of the sample is small, the heating rate approaches the maximum allowed heating rate, and when the percentage weight change per minute is very large, the heating rate approaches zero; (4) in the high-resolution mode, choosing the function such that the heating rate is held at almost zero during a transition.

Description: BACKGROUND Field of the Invention . The present invention relates to analytical techniques for determining the composition, phase, structure, or other properties of a sample of material. Background of the Invention . Thermal analysis techniques generally comprise measuring a physical parameter as a function of the temperature of the sample. The sample temperature is strictly controlled throughout the analysis. Whenever the sample undergoes a chemical or physical transformation, phase change or other transition which affects the physical parameter being measured, the changes in that physical parameter may be interpreted to analyze the composition, structure, or thermal stability of the sample. One common thermal analysis technique is thermogravimetric analysis ("TGA"). TGA is a thermal analysis technique which measures the weight change of a material as a function of temperature, or as a function of time at a controlled temperature. The classic TGA method comprises heating the sample at a constant rate of temperature increase, typically at 10 °C to 50 °C per minute, while the sample weight change or the percent of weight change is recorded versus temperature. Other thermal analysis techniques include Differential Thermal Analysis (DTA), Differential Scanning Calorimetry (DSC), Pressure Differential Scanning Calorimetry (PDSC), Thermomechanical Analysis (TMA), Dynamic Mechanical Analysis (DMA), Dynamic Mechanical Spectrometry (DMS), Dielectric Analysis (DEA), Differential Photocalorimetry (DPC), Thermal Conductivity Analysis (TCA), and any simultaneous combination of these techniques. Differential Scanning Calorimetry measures the temperatures and the heat flow associated with transitions in materials as a function of time and temperature. These measurements provide quantitative and qualitative information about the sample transitions that involve endothermic or exothermic processes, or changes in heat capacity. Pressure Differential Scanning Calorimetry is a related technique in which the heat flow and temperature of transitions are measured as a function of temperature under controlled pressure. Differential Thermal Analysis, like DSC, measures the temperatures and heat flow associated with transitions in materials as a function of time and temperature. However, unlike DSC, DTA results are semi-quantitative. DTA is generally carried out at higher temperatures than DSC. Thermomechanical Analysis measures linear or volumetric changes in materials as a function of temperature under controlled stress or strain. Dynamic Mechanical Analysis and Dynamic Mechanical Spectrometry measure mechanical properties of a material as it is deformed under periodic stress as a function of temperature. Dielectric Analysis measures the dielectric properties of materials as a function of temperature. Differential Photocalorimetry measures the heat absorbed or released by a sample as it and an inert reference are exposed simultaneously to radiation of known wavelength and intensity. Thermal Conductivity Analysis measures the thermal conductivity of materials as a function of temperature. Conventional thermal analysis techniques have limited resolution because in conventional thermal analysis, time and temperature are changing simultaneously. Because chemical and physical transformations are time-dependent (not instantaneous), transitions that actually occur as a function of time are recorded as occurring as a function of temperature. TGA is particularly useful for observing the thermal decomposition of compounds. When individual thermal decompositions occur at well separated temperatures, quantitative information about sample composition may be obtained from the percent weight change per minute at each transition. However, due to the limited resolution of conventional TGA, in conventional TGA decomposition transitions frequently overlap or appear drawn out in temperature. This substantially reduces the ability to obtain an accurate measurement of weight change and reaction temperature. It has long been known that the use of very slow TGA heating rates will improve the separation of some overlapping transitions and, thus, increases the resolution of the technique. U.S. Patent No. 3,344,654 to Erdey, et al. ("Erdey"), discloses a quasi-static technique of reducing heating rate to limit the rate of weight change to a predetermined maximum during transitions. Although using very slow heating rates or heating rates controlled by sample weight change improves the separation of transitions, such methods also increase substantially the total time required for a measurement, thereby reducing laboratory productivity. Moreover, increasing measurement time reduces the accuracy and reliability of the analysis. For example, a sample exposed to high temperatures for a long period of time undergoes slow time-dependent changes such as oxidation, deformation, absorption and adsorption, which may introduce errors in the analysis. The long time period also implies further difficulties due to instrument drift, and to ambient temperature, humidity, and pressure variations. Mechanical vibrations and fluctuations in the voltage of the main power line also increasingly affect the accuracy of the analysis as the analysis time increases. Furthermore, the effective signal-to-noise ratio is reduced, because the signal peaks are flattened by the slow temperature ramp. Quasi-static techniques are discussed extensively in the article by F. Paulik and J. Paulik, "Thermoanalytical Examination Under Quasi-Isothermal -- Quasi-Isobaric Conditions." Thermochimica Acta , Vol. 100 (1986), pp. 23-59. The Paulik quasi-isothermal technique attempts to maintain a specific rate of weight change by controlling the temperature of the sample. DEFINITIONS "Downstream analytical technique", as used herein, means any analytical technique that is commonly used in conjunction with gas chromatography to determine quantitatively or qualitatively the presence or concentration of atomic, molecular, or ionic species in the effluent of a gas chromatograph, such as mass spectroscopy, Fourier transform infrared spectroscopy, laser diode spectrometry, flame ionization detection, electron capture detection, photoionization detection, flame photometric detection and chemiluminescent detection. "Driving variable", as used herein, means the independent physical parameter, such as temperature, pressure, time, applied stress, or wavelength of incident radiation, that is being used to drive a material through a transition. For example, in thermal analysis techniques such as TGA, temperature is the driving variable. "Characterizing physical parameter", as used herein, means the dependent physical parameter characterizing the sample, such as its weight, length, volume, dielectric or mechanical properties, or thermal conductivity. For example, in TGA the characterizing physical parameter is the percent weight change of the sample. In thermomechanical analysis, the characterizing physical parameters are the linear or volumetric changes in the material. "Resolution" of an analytical technique, as used herein, means the degree to which signals associated with different transitions can be physically separated in the analytical data produced by the technique. This quality of the analytical technique is most critical when multiple transitions occur at closely spaced values of the driving variable. "Signal baseline", as used herein, means that portion of a signal representing the value of a characterizing physical parameter obtained in a range in which there are no transitions or transformations. "Transition" or "transformation", as used herein, mean any type of physical or chemical transformation, phase change, or structural change in a material. SUMMARY OF THE INVENTION As is known from the aforementioned Paulik article the present invention provides a method of determining the composition of a material that undergoes at least one transition as a function of a driving variable, such as temperature, said method comprising the steps of: placing a sample of the material in an apparatus for detecting changes in a characterising physical parameter, such as in this case the weight of the sample, as a function of said driving variable or temperature, monitoring a signal representative of the characterising physical parameter; calculating a rate of change of the driving variable or temperature as a function of the percentage change per unit time of the characterising physical parameter and controlling the rate of change of the driving variable or temperature according to the calculated rate of change. In accordance with the present invention and in contrast to the Paulik article at least a maximum allowed value for the rate of change is selected, the control of the rate of change of the driving variable is effected continuously, the function for the calculation is a non-linear function of an exponential type with an argument and the argument includes the percentage change per unit time of the characterising physical parameter and the function is chosen such that, when the percentage change per unit time in the characterising physical parameter is small in relation to a baseline percentage change per unit time of the characterising physical parameter, the rate of change in temperature approaches the maximum selected allowed value and, when the percentage change per unit time of the characterising physical parameter is large in relation to the baseline percentage change per unit time of the characterising physical parameter, the rate of change approaches zero. The present invention can be effected by using a computer system to control the analytical method by monitoring and controlling the driving variable according to changes observed in a characterizing physical parameter, according to preselected maximum and minimum values for the rate of change of the driving variable, and/or according to preselected values for the rate of change of the characterizing parameter. For example, when the analytical technique is TGA, the driving variable is temperature, and the characterizing physical parameter is the weight of the sample. The derivative of the change in sample weight (weight percent per minute) is calculated in real time. Periodically, e.g ., every half second, the computer accepts as inputs the rate of weight change, the sample heating rate ( i.e ., the rate of increase in sample temperature), the maximum heating rate and the resolution setting. The computer then uses these inputs to calculate a new sample heating rate according to the method of the present invention. The analytical system then controls the heating of the sample material according to the new sample heating rate. Four techniques can be used to improve the resolution of thermal analysis methods and/or to reduce the total time required for analysis. Individually or in combination, they provide an analytical method applicable to all thermal analysis techniques that dramatically improves their resolution, reduces the time required for analysis, and enhances their ability to detect small transformations. These improvements are realized because these techniques sharpen the signal derived from each transition and increase the separation of closely spaced transitions. The first technique comprises monitoring the signal baseline, detecting deviations from the signal baseline, and then rapidly decreasing the sample heating rate as the sample undergoes a chemical or physical transformation. This allows the use of very high heating rates through temperature regions with no transitions (for example, greater than 50 degrees Celsius per minute) while preventing overshoot of the transition temperature. The second technique comprises constraining the furnace heating rate to a minimum rate during a transition or transformation to prevent isothermal or cooling operation, thereby driving the sample through the transition or transformation. The third technique comprises using a "heating uplift" step to force the apparatus back to the maximum heating rate when the derivative of weight change falls below a certain rate. The fourth technique comprises adjusting the heating rate as a function of the rate of sample weight change to track a desired constant rate of weight change. The present invention may be practiced using all four of the techniques and when two or more of the techniques are applied simultaneously, the heating rates derived from each technique are combined to control the heating rate of the sample. A preferred embodiment of the present invention comprises two modes of operation. The first mode continually controls the heating rate according to a function comprising an exponential type term, wherein the argument of the exponential term includes the percent weight change per minute of the sample. The function is chosen such that, when the percent weight change per minute is very small, i.e., during baseline operation, the heating rate approaches its maximum allowed value. When the percent weight change per minute is very large, i.e. , during a transition, the heating rate approaches zero. The second mode of operation in the preferred embodiment of the present invention is also based upon controlling the heating rate according to a function computing an exponential term, wherein the argument of the exponential type term includes the percent weight change per minute of the sample. The function is chosen such that, when the percent weight change per minute is very small, i.e. , during baseline operation, the heating rate is controlled close to a maximum rate as in the first mode. When the percent weight change per minute is large, i.e. , during a transition, the heating rate is held at a minimum value. Typically, the minimum value of the heating rate is 0.01°C/minute. For the majority of materials, the preferred value for the minimum value of the heating rate ranges from 0 to about 10% of the maximum value of the heating rate. Once the heating rate falls to the minimum value, it is held at that value until the percent weight change per minute of the sample falls below a selected value for the percent weight change per minute. The selected value for the percent weight change per minute required to override the "hold" at the minimum heating rate may be, for example, one order of magnitude lower than the value of the percent weight change per minute required to force the heating rate to the minimum heating rate. An alternative to the above described method for overriding the "hold" at the minimum heating rate is to override holding when the heating rate falls to a constant value, i.e. , establishes a baseline region. This is particularly useful when the baseline rate of weight change is substantially different after a transition is complete from what it was prior to the onset of the transition. The operator selects the mode of operation, and also selects the resolution of the analysis. In both modes of operation, the argument of the exponential term includes a factor based on the resolution setting, as well as the factor based upon the percent weight change per minute of the sample. In both modes of operation, the heating rate is generally either close to the maximum rate or to the minimum rate, depending upon the percent weight loss per minute, except for a transition region at intermediate values of the percent weight loss per minute, in which small changes in the percent weight loss per minute result in rapid changes in the heating rate. The relative position of the transition region, i.e. , how much percent weight loss per minute is required to increase the heating rate significantly above the minimum, or decrease it significantly below the maximum, is adjusted by the choice of the resolution setting. Both modes of operation could be used in a single analysis. For example, if the operator needs the highest possible resolution in a particular temperature region, e.g. , to resolve two closely-spaced transitions, but does not need the high resolution in other temperature regions, the operator could use one mode for the part of the analysis that does not require the highest possible resolution, and the other mode for the part of the analysis that does. A first object of the present invention is to provide a temperature control method for thermal analysis techniques which achieves substantially improved resolution of transitions, without increasing the time required for analysis, or possibly reducing the time required for analysis, compared to the times required for analysis by conventional constant heating rate methods. A second object of the present invention is to provide the analyst with a method for selecting the desired instrument resolution. A third object of the present invention is to provide a method for improving the accuracy of thermal analysis techniques by preventing thermal overshoot of transition temperatures during heating. A fourth object of the present invention is to improve the sensitivity of thermal analysis techniques. A fifth object of the present invention is to simplify the interpretation of thermal analysis data by more sharply defining the temperatures at which transitions occur, for example as demonstrated by the narrowness of the peaks of derivative signals corresponding to transitions. A sixth object of the present invention is to obtain transition temperatures which more accurately reflect the isothermal reaction temperatures of the transitions. A seventh object of the present invention is to isolate signal changes so that they can be more easily measured, integrated, compared with other results, and interpreted. These and other objects of the present invention are described in greater detail in the detailed description of the invention, the appended drawings and the attached claims. DESCRIPTION OF THE DRAWINGS FIGURE 1a is a block diagram of a thermogravimetric apparatus. FIGURE 1b is a flow chart showing some of the computing steps used in performing the invention. FIGURES 2a-2f are TGA scans of calcium oxalate obtained according to the methods described in Example 2. FIGURES 3a-3e are TGA scans of calcium oxalate obtained according to the methods described in Example 3. FIGURES 4a-4c are TGA scans of black electrical tape obtained according to the methods described in Example 4. FIGURES 5a-5c are TGA scans of candy taffy obtained according to the methods described in Example 5. FIGURES 6a-6e are TGA scans of ethylene-vinyl acetate (EVA) obtained according to the methods described in Example 6. FIGURE 7 is a flow chart showing some of the computing steps used in a second preferred embodiment of the invention. FIGURES 8a-8d are TGA scans of candy taffy obtained according to the methods described in Example 8. FIGURES 9a-9d are TGA scans of ethylene-vinyl acetate (EVA) obtained according to the methods described in Example 9. DETAILED DESCRIPTION OF THE INVENTION The present invention resides in control methods for conventional analytical apparatus, and is an improvement on the methods conventionally used to control such conventional apparatus. The following detailed description of the present invention applies specifically to thermogravimetric analysis, in which temperature is the driving variable and weight change is the characterizing physical parameter. However, although the present invention is described as it is applied to thermogravimetric analysis, it should be understood that the present invention could be used with any thermal analytical method including Differential Thermal Analysis, Differential Scanning Calorimetry, Thermomechanical Analysis, Pressure Differential Scanning Calorimetry, Dynamic Mechanical Analysis, Dynamic Mechanical Spectrometry, Dielectric Analysis, Differential Photocalorimetry, and Thermal Conductivity Analysis, as well as any combination of these techniques. The principles and methods described herein with reference to thermogravimetric analysis could be applied to any and all of the thermal analytical methods listed above, as well as to other analytical methods wherein a characterizing physical parameter is measured as a function of a driving variable. Figure 1a is a schematic representation of a conventional thermogravimetric analyzer, showing thermogravimetric balance 1, comprising a balance beam 2, a sample pan 3 and a tare pan 4, a balance null meter movement 5, photo diodes for detecting the balance null 6, sample temperature thermocouple 7, purge gas inlet 8 and purge gas outlet 9, electric furnace 10, sample temperature digitizer 11, heater controller 12, balance null controller 13, and microcomputer 14. Figure 1a also shows personal computer 15 and digital plotter 16. The thermogravimetric balance measures the weight of a sample in sample pan 3 by measuring the current required by balance null meter movement 5 to maintain the apparatus in balance, as indicated by photodiode detectors 6. The balance null controller 13 receives the signal from the photo diode detectors 6, and increases or decreases the meter current accordingly. Balance null controller 13 provides microcomputer 14 with the weight of the sample, which is calculated from the meter current. Microcomputer 14 also receives the sample temperature from sample temperature thermocouple 7 via sample temperature digitizer 11, and controls the temperature of the sample by controlling the power to furnace 10 using heater controller 12. In the first preferred embodiment of the present invention, the temperature of the sample is controlled by the microcomputer in accordance with the steps outlined below. However, the present invention can be practiced using any combination of computers, hardware and operator control. Personal computer 15 and digital plotter 16 are used to analyze, store, display and plot the analytical results. A purge gas is usually introduced via the purge gas inlet. The purge gas can be a gas that reacts with constituents of the sample being analyzed, or an inert gas, i.e. , a gas that does not react with the sample used to prevent reactions with air. Typical purge gases include dry air, oxygen, nitrogen, argon, helium, carbon monoxide and carbon dioxide. The purge gas may also be used to remove products of the reaction for additional analysis further down the gas stream, via a downstream analytical technique. Analytical techniques and detectors currently being used with gas chromatography may be used with a TGA apparatus operated according to the present invention, including mass spectrometry, atomic spectroscopy, molecular spectroscopy, dispersive infrared spectrometry, laser diode spectroscopy, flame ionization detectors (FID), thermal conductivity detectors (TCD), electron capture detectors (ECD), photoionization detectors (PID), helium ionization detectors (HID), discharge ionization detectors (DID), flame photometric detectors (FPD), chemiluminescent detectors (CD), Raman spectroscopy, and Fourier transform infrared spectrometry (FT-IR). Signals derived from the downstream analytical technique could be used to control the sample temperature, in addition to, or instead of, the thermal analysis signal. Figure 1b is a flow chart showing the computing steps used to calculate the heating rate of the sample as it is applied to thermogravimetric analysis. The flow chart shown in Figure 1b includes the four techniques described and a first method described in detail herein with reference to thermogravimetric analysis incorporates all four of the techniques. The first such technique, the BASELINE technique, comprises monitoring the signal baseline, detecting deviations from the signal baseline, and then rapidly decreasing the sample heating rate as the sample undergoes a chemical or physical transformation. This allows the use of very high heating rates during regions of no transition (for example, greater than 50 degrees Celsius per minute) while preventing overshoot of the transition temperature. The second such technique, the DRIVING technique, comprises constraining the sample heating rate to a minimum rate during a transition to prevent isothermal or cooling operation, thereby driving the sample through the transition. The minimum heating rate may be a predetermined heating rate, or it may be a dynamically determined heating rate. It could be dynamically determined based upon such factors as, for example, the rate of weight change, time spent at the minimum heating rate, the second derivative of the weight change, or other characteristics of the specific experiment. The third such technique, the UPLIFT technique, comprises using a "heating uplift" step to force the apparatus back to the maximum heating rate when the derivative of weight change falls below a certain rate. The fourth such technique, the TRACKING technique, comprises adjusting the heating rate as a function of the rate of sample weight change to track a desired constant rate of weight change. When the techniques are applied simultaneously, the heating rates derived from each technique are combined to control the heating rate of the sample. The first preferred embodiment of the present invention comprises the following steps, wherein step 11 corresponds to the BASELINE technique, step 12 corresponds to the TRACKING technique, step 13 corresponds to the UPLIFT technique, and step 16 corresponds to the DRIVING technique: STEP 1. Selecting a maximum heating rate ("rate_max"). STEP 2. Selecting a resolution setting ("res_setting"). For example, resolution settings of 1, 2, 3, 4, or 5 could set minimum heating rates of 20%, 7.4%, 2.7%, 1.0% and 0.37% of the maximum heating rate, respectively. Thus, the fastest scans would be obtained using the lowest resolution setting of 1, while scans obtained using the highest resolution setting of 5 would require increased time for analysis. STEP 3. Computing a minimum allowed heating rate ("rate_min") from the maximum heating rate and the resolution setting. This rate is constrained to be a fraction of the maximum rate such that increasingly higher resolution settings produce proportionally lower minimum heating rates. For example, at the lowest resolution setting the minimum heating rate could be 20% of the maximum rate and at the highest resolution setting the minimum heating rate could be 0.37% of the maximum rate. STEP 4. Continuously monitoring the weight change of the sample ("wgt_pct"), e.g. , by measuring the weight change every half second. STEP 5. Computing the current derivative with respect to time of the weight change of the sample ("pct_drv"). STEP 6. Computing an average derivative with respect to time of the weight change of the sample ("pct_drv_avg"). For example, the average derivative could be computed with an exponential filter in which 98% of the previous average derivative is added to 2% of the current derivative every half second. The exponential filter simulates an infinite response analog filter by taking 98% of the previous average and adding 2% of the current value ( i.e. , pct_drv_avg = 0.98 * pct_drv_avg + 0.02 * pct_drv). STEP 7. Computing the fraction of the allowed heating rate range currently being used from the selected maximum heating rate and the computed minimum heating rate. The fraction of the allowed heating rate ("range_used") is defined as the difference between the current heating rate ("rate_now") and the minimum heating rate ("rate_min") divided by the difference between the maximum heating rate ("rate_max") and the minimum heating rate ("rate_min"). This value is used to control the influence of the baseline deviation factor and weight loss tracking factor in the heating rate adjustment calculation ( see step 14, below). STEP 8. (BASELINE TECHNIQUE) Computing the baseline deviation heating rate adjustment factor ("rate_baseline_err"). This factor is equal to the baseline conversion constant ( e.g. , 1.0 °C/%) multiplied by the product of the allowed heating rate range ("range_used") and the absolute value of the difference between the average derivative ("pct_drv_avg") and the current weight change derivative ("pct_drv") (i.e., rate_baseline_err = 1.0 * range _used * ABS (pct_drv_avg - pct_drv). This factor will naturally increase at the onset of transitions when heating rates are near the maximum but it is reduced at low heating rates. This factor operates to reduce the heating rate rapidly at the onset of a transition to prevent overshooting of the transition temperature. STEP 9. (TRACKING TECHNIQUE) Computing the weight change tracking heating rate adjustment factor ("rate_tracking_err"). This factor is equal to the tracking conversion constant (0.1 °C/%) multiplied by the product of one minus the allowable heating rate range used ("range_used") and the difference between the desired constant weight change derivative ("pct_drv_target") and the absolute weight change derivative ("ABS(pct_drv)") (i.e., rate_tracking_err = 0.1 * (ABS (pct_drv) - pct_drv_target) * (1.0 - range_used)). For example, the desired derivative could be constrained to the range of 2.5%, 2.0%, 1.5%, 1.0% and 0.5% per minute corresponding to resolution settings 1, 2, 3, 4, and 5, respectively. This factor functions by tending to adjust the heating rate to maintain a constant rate of weight change during a transition in which the current heating rate is relatively low. STEP 10. (UPLIFT TECHNIQUE) Computing the uplift adjustment factor to the heating rate ("rate_uplift_err"). This factor is equal to the uplift conversion constant (0.025 °C/%) multiplied by the common logarithm of the absolute average weight change derivative ("ABS (pct_drv_avg)") ( i.e. , rate_uplift_err = 0.025 * LOG (ABS (pct_drv_avg)). However, this factor is set to zero when the average weight change derivative is greater than 1.0% per minute and to a reasonable maximum ( e.g ., -0.1) when the average weight change derivative is less than 0.0001% per minute. The principal purpose of this factor is to accelerate the rise in heating rate when the current weight change derivative is very small, thereby insuring near maximum heating rates during baseline periods and consequently minimizing the measurement time. STEP 11. Computing an adjustment to the heating rate ("rate_now") from the baseline deviation factor ("rate_baseline_err"), the weight change tracking factor ("rate_tracking_err") and the uplift heating rate factor ("rate_uplift_err"). The new heating rate ("rate_new") is defined to be the current heating rate ("rate_now"), minus the sum of the baseline deviation factor, the weight change tracking factor and the heating rate uplift factor ( i.e. , rate_new = rate_now - (rate_baseline_err + rate_tracking_err + rate_uplift_err)). STEP 12. Comparing the new heating rate ("rate_new") to the maximum allowed rate ("rate_max"). The new rate is set to the maximum allowed rate if the new rate is greater than the maximum allowed rate. This prevents thermal runaway of the furnace during baseline operation. STEP 13. (DRIVING TECHNIQUE) Comparing the new heating rate ("rate_new") to the minimum allowed rate ("rate_min"). The new rate is set to the minimum allowed rate if the new rate is less than the minimum allowed rate. This reduces analysis time and prevents reversal of the sample temperature ramp. The sample temperature thus increases throughout the thermal analysis at a rate between the minimum and maximum heating rates allowed (inclusive). STEP 14. The current heating rate is set to the new heating rate. Figures 2-6 are plots of TGA analyses practiced according to the first preferred embodiment of the present invention. These figures are discussed in detail in Examples 2-6, below. In the first preferred embodiment of the present invention, for example, the maximum heating rate used for thermogravimetric analysis could range from 1 degree/minute to 100 degrees/minute Celsius, the resolution setting could range from 1 (lowest resolution) to 5 (highest resolution), and the temperature of the sample could range from room temperature to 1000 degrees Celsius. The second preferred embodiment of the present invention is outlined in Figure 7, as it is applied to thermogravimetric analysis. The second preferred embodiment comprises two modes of operation. The first mode is chosen to maximize the productivity of the analytical instrument. The second mode is chosen to obtain the highest possible resolution and greatest analytical accuracy. In both modes of operation, the heating rate is controlled by an exponential term. The argument of the exponential term includes the percent weight change per minute of the sample and a factor based upon a resolution setting chosen by the operator. The heating rate for both modes of operation is close to the maximum heating rate during baseline operation, i.e. , in temperature regions in which no transitions occur. In the first mode of operation, the heating rate approaches zero during a transition. In the second mode of operation, the heating rate is held at 0.01°C/minute during transitions. Once the heating rate falls to 0.01°C/minute, it is held at 0.01°C/minute until the percent weight change per minute falls below a preselected value. The preselected value could be, for example, 10% of the weight percent change per minute required to drive the heating rate to 0.01 °C/minute. As an alternative, the heating rate is held at 0.01 °C/minute until the rate of weight change reaches a constant value. The second preferred embodiment comprises the following steps (1-14): STEP 1. Selecting a maximum heating rate ("rate_max"). STEP 2. Selecting a resolution setting ("res_setting"). The resolution setting ranges from - 8.0 to +8.0. Positive values of the resolution setting are used to select the first mode of operation ("mode 1"). Negative values of the resolution setting are used to select the second mode of operation ("mode 2"). Most analyses are run with a resolution setting of -5.0 to -3.0 for the second mode of operation, and +3.0 to +5.0 for the first mode of operation. STEP 3. Monitoring the percent weight change of the sample ("wgt_pct"). STEP 4. Calculating the derivative of the percent weight change with respect to time ("pct_drv"). STEP 5. Computing the resolution factor from the absolute value of the resolution setting ("res_factor"). STEP 6. Computing the minimum mode 2 percent weight change ("mode2_pct_min") from the resolution factor. STEP 7. Computing the minimum heating rate allowed. If in mode 2, then the minimum is set to 0.01°C/minute. Otherwise, the minimum rate is set to zero. STEP 8. Computing a delayed average derivative with respect to time of the weight change of the sample. The delay ("drv_delay_ctr") allows any ringing or overshoot in the weight percent per minute to dampen out. The average weight percent per minute ("pct_drv_avg") is computed by adding 1% of the current value of the weight percent derivative ("pct_drv") to 99% of the previously computed average weight percent derivative. STEP 9. Computing two new heating rates for the first mode of operation, and two new heating rates for the second mode of operation. For each mode of operation, one new heating rate is calculated using the real-time percent weight change per minute ("rate_drv_pct"), and another new heating rate is calculated using the average weight percent weight change per minute ("rate_drv_avg"). The equation used to calculate the new heating rate for the first mode of operation divides the maximum heating rate by an exponential term. The argument of the exponential term includes either the real-time or the average percent weight change per minute, as discussed above, and the resolution factor. The equation used to calculate the new heating rate for the second mode of operation multiplies the maximum heating rate by a factor comprising a constant minus an exponential term. As in the first mode of operation, the argument of the exponential term includes either the real-time or the average percent weight change per minute, and the resolution factor. STEP 10. Choosing the new heating rate ("rate_new") calculated from the largest weight change per minute. STEP 11. In mode 2, checking the current heating rate ("rate_now"). If the current heating rate is equal to or below the minimum heating rate and the average weight change per minute is greater than the minimum weight change per minute for mode 2 operation, then the new heating rate is held at the minimum heating rate. If the average weight change per minute is less than the minimum weight change per minute for mode 2 operation, the computed new heating rate is used to control the sample heating rate. STEP 12. Limiting the new heating rate to the maximum heating rate allowed. If the new heating rate is greater than the maximum heating rate allowed, the new heating rate is set at the maximum heating rate. STEP 13. Limiting the new heating rate to the minimum heating rate allowed. If the new heating rate is less than the minimum heating rate, the new heating rate is set at the minimum heating rate. STEP 14. Setting the current heating rate ("rate_now") equal to the new heating rate. The steps listed above for the second preferred embodiment use the first alternative for overriding a "hold" at the minimum temperature. The second alternative, overriding the "hold" when the rate of weight change reaches a constant value, can be implemented by, for example, detecting when the rate of weight loss per minute does not change for a preselected period of time. The present invention can be practiced at higher temperatures, by using materials resistant to higher temperatures, and high temperature equipment. For example, a temperature of 1600 degrees Celsius can be achieved using a high temperature furnace. Different temperature ranges could be used for particular applications, depending on the characteristics of the apparatus used. The present invention can also be applied to chemical or physical transformations that occur upon cooling as well as upon heating, or upon any combination of heating and cooling cycles, whether or not separated by constant temperature periods or constant-rate heating or cooling periods. When the technique is used to analyze materials that undergo transitions upon cooling, the methods described above are still applied, but with the sample cooling rate substituted for the sample heating rate, and with a cooling apparatus substituted for or used in conjunction with the furnace. The sample could be heated or cooled using hot or cold air, lasers, fluid baths, microwave energy, chemical reactions, or other appropriate experimental techniques. For certain reactions, e.g. , for explosive reactions, the sample could be self-heated by the chemical reactions, in which the reaction rate is controlled by controlling another variable, such as a fluid flow rate, a pressure or partial pressure of a reacting gas, or by exposure to electromagnetic radiation or to a magnetic field. The preferred embodiments of the present invention, as described above, control the heating rate of the sample. However, the present invention could be practiced by controlling the sample temperature itself instead of by controlling the heating rate. FURTHER EMBODIMENTS OF THE INVENTION The present invention may be employed in other analytical techniques in which temperature is not the driving variable to achieve a similar improvement in the resolution, sensitivity, accuracy and efficiency of those techniques. For example, the present invention could be applied to chemical transformations that occur as a function of applied or ambient pressure, or as a function of the intensity or wavelength of electromagnetic radiation directed at the sample. The present invention may be applied whenever a chemical or physical transformation produces a change in a measurable physical characteristic of a sample as a function of a driving variable. The present invention may be implemented using hardware, software, or a combination of hardware and software. For example, the maximum heating rate may be limited by adjusting the physical characteristics of the furnace, or by programming a computer that controls the furnace heating rate. Analog devices such as amplifiers could be used for multiplication, integrators for summation, inverters for subtraction, and comparators for decision-making. The following examples are provided to illustrate certain embodiments of the present invention. They are not to be construed as limiting the invention in any way. EXAMPLE 1 : THERMOGRAVIMETRIC ANALYSIS This example describes the experimental apparatus and procedures used in Examples 2-6 and Examples 8-9. The TGA system used for these analyses was the TA Instruments TGA 2950 Thermogravimetric Analyzer connected to the TA Instruments Thermal Analyst 2100 computer/thermal analyzer. The instruction manuals for this apparatus are hereby incorporated by reference. All samples used in the experiments were readily available commercial products. Sample sizes varied from 5 to 22 milligrams of material. The samples were loaded into standard 50 microliter platinum sample pans. The purge gases used were either compressed air or nitrogen (99.998% pure), as noted in the Figures for the specific examples. The TGA was set up and leveled on a solid non-vibrating work surface. Before operation of the TGA apparatus, a steady flow of purge gas was established. The purge gas was connected through pressure regulators and flow meters to the balance and furnace purge inlet ports on the instrument. The flow meters were adjusted to a flow rate of 40 ml/minute to the balance and 60 ml/minute to the furnace for a combined total of 100 ml/min. purge flow rate over the sample pan. When nitrogen was used, the purge gas flow was allowed to purge the system for one hour before starting the experiment. The sample thermocouple was adjusted to a distance of 5 millimeters above the sample in the analyses described Examples 4, 5, 6, 8 and 9. The distance was 1 millimeter in Examples 2 and 3. The longer distance (5 mm) produced a more stable heating control because the thermocouple is not overly influenced by sudden changes in sample temperature due to endothermic or exothermic reactions of the sample material. The sample pan was cleaned by first heating the sample pan in a bunsen burner flame until the pan was glowing orange due to its temperature to burn off any residual materials leaving only ash. Any ash remaining was dumped from the pan and the pan was brushed out if necessary. The sample pan was then placed on the loading platform and the TGA was commanded to perform an automatic tare operation by pressing the Tare button. This procedure entailed loading and weighing the empty sample pan and storing the empty weight of the sample pan as a negative offset to the measured experimental weight. Samples were prepared by cutting off or pouring out small portions of the material of the approximate weight needed and placing them in the center of the sample pan on the platform. In Example 4, the adhesive side of the sample tape was placed facing upwards. Powdered samples were distributed uniformly inside the sample pan. Sample identification, the desired maximum heating rate, the final temperature to be reached, and the resolution setting were programmed into the computer/thermal analyzer. The TGA experimental sequence was started by pressing the "Start" button on the TGA 2950. The computer/thermal analyzer then automatically performed the following steps: (1) the sample platform, with the loaded sample pan riding on the platform, was moved to the balance load position and the balance arm was lowered electronically to engage a hook on the balance hang-down wire with the bale of the sample pan; (2) once engaged, the sample pan was lifted from the platform by the balance arm and the platform was moved aside; the sample pan and sample were held motionless by the nulling current in the balance meter movement; (3) the furnace was raised via a motor and drive screw mechanism to surround the sample and the sample thermocouple and seal the furnace chambers against the bottom of the balance housing with an O-ring; (4) the sample pan and sample were weighed for approximately 30 seconds to allow the system to stabilize and establish the initial (100%) sample weight; (5) the sample was heated according to the present invention, as described specifically in Examples 2-6 and Examples 8-9; (6) sample temperature and sample weight were continuously monitored at a rate of 10 times/second; temperature and weight information were averaged, filtered and stored on hard disk in the computer at a rate of one data point every two seconds; (7) when the specified final temperature was reached, the computer turned off power to the furnace and stopped data storage; (8) the furnace cooled back to ambient temperature aided by a heat exchanger and a water cooling jacket around the outside of the furnace; and (9) the sample was unloaded automatically by reversing steps 1 through 3. The data collected on the computer hard disk was then analyzed and plotted on the digital x-y plotter. Derivative curves were generated mathematically from the temperature, time and weight change data stored on disk. EXAMPLE 2 : EFFECT OF RESOLUTION SETTINGS ON TEMPERATURE ACCURACY, TRANSITION SEPARATION AND ANALYTICAL TIME FOR CALCIUM OXALATE Samples of calcium oxalate monohydrate, CaC 2 O 4 :H 2 O were examined in air by conventional TGA using the first preferred embodiment of the present invention to compare the results obtained using different resolution settings. The experimental procedures followed in this example for preparing the apparatus, loading the samples, and running analyses were described in Example 1. The results of the analyses are shown in Figures 2a-2f. Figure 2a is a TGA scan obtained according to the conventional method, at a constant heating rate of 50 °C/minute from ambient temperature to above 800 °C. Figures 2b, 2c, 2d, 2e and 2f are TGA scans obtained according to the first preferred embodiment (steps 1-14) of the present invention at resolution settings of 1, 2, 3, 4 and 5, respectively. Each of the scans shows three curves. Curves 201 (Figure 2a), 211 (Figure 2b), 221 (Figure 2c), 231 (Figure 2d), 241 (Figure 2e) and 251 (Figure 2f) are plots of the percent weight change of the sample as a function of sample temperature. Curves 202 (Figure 2a), 212 (Figure 2b), 222 (Figure 2c), 232 (Figure 2d), 242 (Figure 2e) and 252 (Figure 2f) are plots of the derivative with respect to time of the sample temperature versus sample temperature. Curves 203 (Figure 2a), 213 (Figure 2b), 223 (Figure 2c), 233 (Figure 2d), 243 (Figure 2e) and 253 (Figure 2f) are plots of the derivative with respect to temperature of the percent weight change of the sample as a function of sample temperature. Calcium oxalate is a well understood material which produces three well-separated and easily observed weight loss transitions between 100 °C and 800 °C. In the first transition, water is released. The first transition appears as features 204 and 207 in Figure 2a; 214 and 217 in Figure 2b; 224 and 227 in Figure 2c; 234 and 237 in Figure 2d; 244 and 247 in Figure 2e; and 254 and 257 in Figure 2f. In the second transition, calcium carbonate is formed with the release of carbon monoxide gas. The second transition appears as appears as features 205 and 208 in Figure 2a; 215 and 218 in Figure 2b; 225 and 228 in Figure 2c; 235 and 238 in Figure 2d; 245 and 248 in Figure 2e; and 255 and 258 in Figure 2f. In the third transition, calcium oxide is formed with the release of carbon dioxide gas. The third transition appears as features 206 and 209 in Figure 2a; 216 and 219 in Figure 2b; 226 and 229 in Figure 2c; 236 and 239 in Figure 2d; 246 and 249 in Figure 2e; and 256 and 259 in Figure 2f. As shown in figures 2b-2f, increasing the resolution setting from 1 (lowest resolution) to 5 (highest resolution) results in progressively sharper weight loss transitions and higher, narrower derivative peaks. The approach to isothermal transition temperature can be observed at all resolution settings, but is most pronounced at the higher resolution settings. Contrasting the TGA scan at resolution setting 5 (Figure 2f) with the constant heating rate TGA scan (Figure 2a) shows the dramatic improvement in reaction temperature determination afforded by the present invention. The total time required for the experiments of Figures 2b-2f were 28, 37, 46, 53, and 63 minutes for resolution settings of 1-5 respectively. This demonstrates the trade-off between minimizing the total time required for analysis and obtaining higher resolutions. EXAMPLE 3 : IMPROVED TRANSITION SEPARATION AND REDUCED ANALYTICAL TIME AT DIFFERENT MAXIMUM HEATING RATES FOR CALCIUM OXALATE Samples of calcium oxalate monohydrate, CaC 2 O 4 :H 2 O were examined in air by conventional TGA using the first preferred embodiment of the present invention to compare the results obtained at a given resolution setting at different maximum heating rates. The experimental procedures followed in this example for preparing the apparatus, loading the samples, and running analyses were described in Example 1. The results of the analyses are shown in Figures 3a-3e. Figure 3a is a TGA scan obtained according to the conventional method, at a constant heating rate of 20 °C/minute from ambient temperature to above 800 °C. Figures 3b, 3c, 3d and 3e are TGA scans obtained according to the first preferred embodiment (steps 1-14) of the present invention described above, with a resolution setting of 5, and maximum heating rates of 100 °C/minute, 50 °C/minute, 20 °C/minute, and 10 °C/minute, respectively. Each of the scans shows three curves. Curves 301 (Figure 3a), 311 (Figure 3b), 321 (Figure 3c), 331 (Figure 3d) and 341 (Figure 3e) are plots of the percent weight change of the sample as a function of sample temperature. Curves 302 (Figure 3a), 312 (Figure 3b), 322 (Figure 3c), 332 (Figure 3d) and 342 (Figure 3e) are plots of the derivative with respect to time of the sample temperature as a function of sample temperature. Curves 303 (Figure 3a), 313 (Figure 3b), 323 (Figure 3c), 333 (Figure 3d) and 343 (Figure 3e) are plots of the derivative with respect to temperature of the percent weight change of the sample as a function of sample temperature. Calcium oxalate is a well understood material which produces three well-separated and easily observed weight loss transitions between 100 °C and 800 °C, as explained in Example 2. The first transition (water is released) appears as features 304 and 307 in Figure 3a; 314 and 317 in Figure 3b; 324 and 327 in Figure 3c; 334 and 337 in Figure 3d; and 344 and 347 in Figure 3e. The second transition (calcium carbonate is formed and carbon monoxide is released) appears as features 305 and 308 in Figure 3a; 315 and 318 in Figure 3b; 325 and 328 in Figure 3c; 335 and 338 in Figure 3d; and 345 and 348 in Figure 3e. The third transition (calcium oxide is formed and carbon dioxide is released) appears as features 306 and 309 in Figure 3a; 316 and 319 in Figure 3b; 326 and 329 in Figure 3c; 336 and 339 in Figure 3d; and 346 and 349 in Figure 3e. Figures 3d and 3e show apparent anomalies at the tail ends of features 338 and 339 (Figure 3d), and features 348 and 349 (Figure 3e), which are not expected in constant heating rate TGA scans of calcium oxalate. These features are due to plotting the weight loss versus temperature, as the heating rate of the sample is changing. The magnitude of the apparent anomalies is due to the specific implementation and parameters used in this example. These apparent anomalies may be removed by selecting other values for the maximum heating rate and resolution settings. As shown in Figures 3b-3e, decreasing the maximum heating rate while maintaining a constant resolution setting of 5 results in progressively less transition temperature overshoot, except for the setting of 10 °C/minute, which shows no significant improvement over the scan at 20 °C/min. In each case, a much sharper weight loss transition is obtained compared to the transition obtained using the conventional constant rate scan shown in Figure 3a. Weight loss baseline is excellent in all scans. EXAMPLE 4 : IMPROVED SEPARATION DEMONSTRATION USING THREE TGA SCANS OF BLACK ELECTRICAL TAPE Samples of commercially available black electrical tape, a co-polymer sheet with an inorganic adhesive coating on one surface, were examined using conventional TGA (TGA scans shown in Figures 4a and 4b) and using the first preferred embodiment of the present invention (TGA scan shown in Figure 4c), to compare the results obtained by each method. The experimental procedures followed in this example for preparing the apparatus, loading the samples, and running analyses were described in Example 1. In the conventional constant heating rate TGA scan shown in Figure 4a, a 6.34 milligram sample of tape was analyzed in air from ambient temperature to 650 degrees Celsius at a constant heating rate of 50 °C/minute. In the conventional constant heating rate scan shown in Figure 4b, a 9.59 milligram sample of tape was analyzed in air from ambient temperature to 650 degrees Celsius at a constant heating rate of 1 °C/minute. In the TGA scan obtained according to the first preferred embodiment (steps 1-14) of the present invention shown in Figure 4c, an 8.80 milligram sample of tape was analyzed in air from ambient temperature to 650 degrees Celsius at a heating rate of 50 °C/minute and a resolution setting of 4. The results of the analyses are shown in Figures 4a-4c. Curves 401 (Figure 4a), 411 (Figure 4b) and 421 (Figure 4c) are plots of the percent weight change of the sample as a function of sample temperature. Curves 402 (Figure 4a), 412 (Figure 4b) and 422 (Figure 4c) are plots of the derivative with respect to time of the sample temperature as a function of sample temperature. Curves 403 (Figure 4a), 413 (Figure 4b) and 423 (Figure 4c) are plots of the derivative with respect to temperature of the percent weight change of the sample as a function of sample temperature. Features 404 and 405 in Figure 4a and 414 and 415 in Figure 4b appear with much greater resolution in Figure 4c as features 424 and 425. Moreover, peak 426 which is clearly resolved in Figure 4c, is an unresolved component of peak 405 in Figure 4a and peak 415 in Figure 4b. Features 406 and 407 in Figure 4a and 416 and 417 in Figure 4b appear as features 427 and 428 in Figure 4c. As shown by Figures 4a and 4b, several temperature-overlapped transitions occur between 100 and 350 °C, and again between 370 and 550 °C. Several of these transitions are difficult to discern in the 50 °C/minute scan. In the 1 °C/minute scan the overlapped transitions at 240 °C and at 400 °C are still not resolved even though the heating rate has been reduced dramatically. In sharp contrast to these scans is the 50 °C/minute scan obtained according to the present invention, which cleanly separates four of the transitions, and shows a significant indication of additional minor transitions. It is evident that the conventional technique of running a fast heating rate survey scan followed by a slow scan to separate transitions at critical points would not have produced results comparable to the scan shown in Figure 4c. Thus, the present invention affords resolution enhancement that cannot be achieved by constant rate TGA regardless of the heating rate selected. EXAMPLE 5 : IMPROVED SEPARATION DEMONSTRATION USING TWO SCANS OF CANDY TAFFY Samples of commercially available candy taffy, a complex mixture of organic compounds, were examined using conventional TGA and using the first preferred embodiment (steps 1-14) of the present invention, to compare the results obtained by each method. The experimental procedures followed in this example for preparing the apparatus, loading the samples, and running analyses were described in Example 1. In Figure 5a, a 12 milligram sample of taffy was analyzed in air from ambient temperature to 670 degrees Celsius at a constant heating rate of 20 °C/minute. In Figure 5b, a 12 milligram sample of taffy was analyzed in air, from ambient temperature to 670 degrees Celsius, according to the first preferred embodiment (steps 1-14) of the present invention at a maximum heating rate of 50 °C/minute and at a resolution setting of 4. Curves 501 (Figure 5a) and 521 (Figure 5b) are plots of the percent weight change of the sample as a function of sample temperature. Curves 502 (Figure 5a) and 522 (Figure 5b) are plots of the derivative with respect to time of the sample temperature as a function of sample temperature. Curves 503 (Figure 5a) and 523 (Figure 5b) are plots of the derivative with respect to temperature of the percent weight change of the sample as a function of sample temperature. As can be seen in Figures 5a and 5b, many temperature overlapped transitions occur between 100 and 550 °C. These transitions are much better resolved in the scan shown in Figure 5b. Features 504, 505, 506, 507 and 508 in Figure 5a appear as features 524, 525, 526, 528 and 529, respectively, in Figure 5b. Feature 527 in Figure 5b does not have a readily discernable counterpart in Figure 5a. Features 509, 510, 511, 512, and 513 in Figure 5a appear as features 530, 531, 532, 534 and 535 in Figure 5b. Thus, transitions which were barely perceptible in the conventional scan at 350 °C and 430 °C (not identified in Figure 5a) are well defined at 315 °C (features 527 and 533) and 365 °C (shoulder on feature 534) in the scan obtained according to the present invention. The final weight loss transition, features 529 and 535, is sharply defined in the scan shown in Figure 5b, but appears as a gradual sloping decomposition in the conventional scan shown in Figure 5a, as features 508 and 513. In all cases the derivative of weight change (percent/°C) curve in figure 5b is sharper and better resolved with respect to the baseline compared to the same curve shown in Figure 5a. There is also excellent correlation between the size and location of transitions in the scans shown in Figure 5a and 5b. Figure 5b shows lower transition temperatures than Figure 5a, especially at higher temperatures. This is because the technique of the present invention reduces transition temperature overshoot, thereby providing a more accurate determination of transition temperatures than the conventional constant heating rate technique. It is also worth noting in Figure 5a the sudden increase in the apparent heating rate caused by sample self-heating during combustion at 410 degrees, followed by a reduction in rate as the furnace catches up to the now warmer sample. EXAMPLE 6 : IMPROVED SEPARATION AND RUN TIME DEMONSTRATION USING THREE SCANS OF ETHYLENE-VINYL ACETATE (EVA) Samples of ethylene-vinyl acetate (EVA) copolymer with 40% vinyl acetate were analyzed using the conventional TGA method and using the first preferred embodiment of the present invention, to compare the results obtained using these methods. This example shows the improved quantitative accuracy of the present invention, compared to the conventional method. The experimental procedures followed in this example for preparing the apparatus, loading the samples, and running analyses were described in Example 1. In the analyses shown in Figures 6a, 6b, 6e and 6f, EVA samples were analyzed in nitrogen from ambient temperature to 880 degrees Celsius using the conventional TGA method at a constant heating rate of 20 °C/minute (Figures 6a and 6b) and 5 °C/minute (Figures 6e and 6f). Figures 6c-6d show TGA scans obtained according to the first preferred embodiment (steps 1-14) of the present invention, using a maximum heating rate of 50 °C/minute. Figures 6a, 6c, and 6e show the TGA data plotted versus temperature. Curves 601, 621, and 641 in Figures 6a, 6c, and 6e are plots of the percent weight change of the sample as a function of sample temperature. Curves 602, 622, and 642 in Figures 6a, 6c, and 6e are plots of the derivative with respect to time of the sample temperature as a function of sample temperature. Curves 603, 623, and 643 in Figures 6a, 6c, and 6e are plots of the derivative with respect to temperature of the percent weight change of the sample as a function of sample temperature. Figures 6b, 6d, and 6f show the TGA data plotted versus time. Curves 611, 631, and 651 in Figures 6b, 6d, and 6f are plots of the percent weight change of the sample as a function of time. Curves 612, 632, and 652 in Figures 6b, 6d, and 6f are plots of the derivative with respect to time of the sample temperature as a function of time. Curves 613, 633, and 653 in Figures 6b, 6d, and 6f are plots of the sample temperature as a function of time. Curves 613 and 653 are straight lines, because that data was obtained using a constant heating rate, whereas curve 633 shows steep high heating rate regions and almost-flat low heating rate regions. Figure 6d shows that when thermogravimetric analysis is practiced according to the present invention, a greater proportion of the analysis time is generally spent obtaining data during a transition, compared to the conventional method. Figures 6a and 6e show two overlapped transitions between 250 and 500 degrees C. Figure 6c shows that the same two transitions are clearly separated when thermogravimetric analysis is performed according to the present invention. The first transition is the result of the decomposition of the 40% vinyl acetate in the sample. The first transition appears as features 604 and 606 (Figure 6a), 614 (Figure 6b), 624 and 626 (Figure 6c), 634 (Figure 6d), 644 and 646 (Figure 6e), and 654 (Figure 6f). The second transition, in which the remainder of the sample decomposes, appears as features 605 and 607 (Figure 6a), 615 (Figure 6b), 625 and 627 (Figure 6c), 635 (Figure 6d), 645 and 647 (Figure 6e), and 655 (Figure (6f). In Figure 6a, the weight loss from the first transition is difficult to determine because baseline is not re-established before the onset of the second transition. In Figure 6c the weight loss is much easier to determine because the baseline region between transitions is longer and flatter. The baseline in Figure 6e is much better established than in Figure 6a, but is still not as well-defined as in Figure 6c. This separation permits increased accuracy and increased reproducibility of analyses by decreasing the effect of selecting peak limits and peak overlaps. In this example, the amount of vinyl acetate in the sample can be determined from the weight loss attained at the point of lowest slope in the weight change curve (i.e., the bottom of the valley between peaks in the derivative of weight change curve), from the following simple relation, described in Jen Chiu, "Applications of Thermogravimetry to the Study of High Polymers", Applied Polymer Symposia No. 2, pp. 25-43 (1966): % vinyl acetate = % weight loss x 1.43 According to this equation, the percent vinyl acetate in each sample is given by: sample-1 (Figure 6a): 30.5 weight% x 1.43 = 43.6% vinyl acetate sample-2 (Figure 6c): 28.4 weight% x 1.43 = 40.6% vinyl acetate sample-3 (Figure 6e): 28.8 weight% x 1.43 = 41.2% vinyl acetate These calculations show the improvement in accuracy obtained with the first preferred embodiment of the present invention (sample-2, Figure 6c) compared to conventional TGA scans at 20 °C/minute (sample-1, Figure 6a) and 5 °C/minute (sample-3, Figure 6e). The total time required to analyze the first sample using the conventional method at 20 °C/minute was 44 minutes (excluding sample preparation, loading and after-test cool down). The time required to analyze the third sample using the conventional method at 5 °C/minute was 177 minutes. The time required to analyze the second sample according to the present invention at a maximum heating rate of 50 °C/minute was 51 minutes. These results demonstrate that the improved accuracy of the present invention over the conventional method is attained without a significant decrease in laboratory productivity. A comparison of the constant heating rate time base plots in Figures 6b and 6f with the present invention time base plot in Figure 6d shows the operating modes of the present invention. In Figures 6b and 6f, the weight loss curve appears the same as the weight loss curve in Figures 6a and 6e, respectively. This is because time and temperature are both increasing linearly in the constant heating rate technique, as shown by the straight lines of curves 613 and 653. The weight loss versus time curve in Figure 6d differs drastically from the weight loss curves in Figures 6b and 6f. This is due to the variable heating rate of the present invention. In the present invention, the weight loss per minute is not held constant as in thermogravimetric analysis practiced according to the Paulik quasi-isothermal method. Key aspects of the present invention are shown by curve 632 of Figure 6d. During the first few minutes of the experiment, the UPLIFT technique pushes the heating rate to the maximum value set by the operator since no weight is being lost by the sample. At about 5 minutes into the scan, the BASELINE technique detects the beginning of the first transition and rapidly reduces heating rate to the minimum allowable. The heating rate is held at minimum as the weight loss per minute reaches maximum and then subsides to a level within the range of the TRACKING technique. At this point, 13 minutes into the scan, the heating rate is gradually increased primarily due to the influence of the TRACKING technique which strives to maintain the decaying weight loss per minute. At 18 minutes into the scan, the BASELINE technique once again detects the onset of a transition and drives the heating rate to the minimum. The heating rate is held at the minimum for more than 20 minutes while the second transition peaks and subsides. At this point, the TRACKING technique once again raises the heating rate in an effort to maintain the rate of weight loss. At 42 minutes into the scan, the sample has not yet completely decomposed, but the weight loss per minute has fallen to near zero. This reduction in the weight loss per minute triggers the uplift technique to rapidly raise the heating rate to the maximum heating rate set for the analysis, where it remains throughout the remainder of the scan. EXAMPLE 7 : COMPUTER PROGRAM FOR IMPLEMENTING THE FIRST PREFERRED EMBODIMENT OF THE PRESENT INVENTION The following computer program is an example of a computer program used to control the main steps of the first preferred embodiment of the present invention. Examples 1-6 were implemented using a computer program following an algorithm similar to the algorithm used in this example. However, the computer programs used in Examples 1-6 were not identical to the program disclosed in the present invention. The program is written in PL/M. PL/M is a high-level computer programming language developed by Intel Corporation. The PL/M language is one of many block structured programming languages (similar to PL/1, Pascal and C) which are based on the ALGOL-60 programming language introduced in the early 1960's. The PL/M language was used to program the control system and high resolution software for the TGA 2950 Thermogravimetric Analyzer used in the experimental apparatus for the first preferred embodiment of the present invention. The steps listed in the first preferred embodiment of the invention described above are referenced in the listing of the computer program herein. EXAMPLE 8: IMPROVED SEPARATION DEMONSTRATION USING TWO SCANS OF CANDY TAFFY Samples of commercially available candy taffy (the same material as was used in Example 5) were examined by the method (steps 1-14) of the second preferred embodiment of the present invention to compare the results obtained by each mode of the present invention with the scans of Example 5. The experimental procedures followed in this example for preparing the apparatus, loading the samples, and running analyses were described in Example 1. In Figures 8a and 8b, a 22 milligram sample of taffy was analyzed in air, from ambient temperature to 570 degrees Celsius, using the first mode of operation (steps 1-14) of the second preferred embodiment of the present invention at a maximum heating rate of 50 °C/minute and at a resolution setting of 4.0. In Figures 8c and 8d, a 15 milligram sample of taffy was analyzed in air, from ambient temperature to 480 degrees Celsius, using the second mode of operation (steps 1-14) of the second preferred embodiment of the present invention at a maximum heating rate of 20 °C/minute and at a resolution setting of -3.5. Figures 8a and 8c show the TGA data plotted versus temperature. Curves 801 (Figure 8a) and 831 (Figure 8c) are plots of the percent weight change of the sample as a function of sample temperature. Curves 802 (Figure 8a) and 832 (Figure 8c) are plots of the derivative with respect to time of the sample temperature as a function of sample temperature. Curves 803 (Figure 8a) and 833 (Figure 8c) are plots of the derivative with respect to temperature of the percent weight change of the sample as a function of sample temperature. Figures 8b and 8d show the TGA data plotted versus time. Curves 821 (Figure 8b) and 851 (Figure 8d) are plots of the percent weight change of the sample as a function of time. Curves 822 (Figure 8b) and 852 (Figure 8d) are plots of the derivative with respect to time of the sample temperature as a function of time. Curves 823 (Figure 8b) and 853 (Figure 8d) are plots of the sample temperature as a function of time. The scans in Figures 8a and 8c can be readily compared to the constant heating rate scan in Figure 5a and the high resolution scan in Figure 5b. Features 504, 505, 506, 507 and 508 in Figure 5a appear as features 804, 805, 806, 808 and 809, respectively, in Figure 8a, and as features 834, 835, 836, 838 and 839, respectively, in Figure 8c. Features 807 and 837 in Figures 8a and 8c, respectively, do not have a readily discernable counterpart in Figure 5a. Features 509, 510, 511, 512 and 513 in Figure 5a appear as features 810, 811, 812, 814 and 815, respectively, in Figure 8a, and as features 840, 841, 842, 844 and 845, respectively, in Figure 8c. As can be seen, the many temperature-overlapped transitions occurring between 100 and 550 °C are much better resolved in the scans shown in Figures 8a and 8c than in Figure 5a, and are similar to the scan shown in Figure 5b. The weight loss transitions at 834, 835, 836 and 839 in Figure 8c are extremely sharp thereby allowing very accurate determination of the respective decomposition temperatures of those transitions. In all cases, the derivative of weight change (percent/°C) curves in Figures 8a and 8c are sharper and better resolved with respect to the baseline compared to the same curves shown in Figure 5a. As in Figure 5b, there is excellent correlation between the size and location of transitions in the scans shown in Figures 8a and 8c. Figures 8b and 8c may be compared to better understand the difference between mode 1 operation (Figure 8b) and mode 2 operation (Figure 8d). Curve 822 in Figure 8b and curve 852 in Figure 8d show the heating rate (°C/minute) of the sample as a function of time. In Figure 8b it can be seen that the heating rate changes slowly and continuously in response to changes in weight loss. This is reflected in the temperature curve, 823, which maintains a positive but varying slope. This contrasts with the heating rate curve, 852, in Figure 8d which changes rapidly from approximately 70% of maximum to near zero every time a major transition is encountered. The heating rate is fixed at a constant 0.01 °C/minute for the duration of each major transition after which it rises again rapidly. This results in the stair-step temperature versus time curve 853, and the very sharp derivative of weight loss peaks shown in Figure 8c. EXAMPLE 9 : IMPROVED SEPARATION DEMONSTRATION USING TWO SCANS OF ETHYLENE-VINYL ACETATE (EVA) Samples of ethylene-vinyl acetate (EVA) copolymer with 40% vinyl acetate (the same material as was used in Example 6) were examined by the method (steps 1-14) of the second preferred embodiment of the present invention to compare the results obtained by each mode of the present invention with the scans of Example 6. The experimental procedures followed in this example for preparing the apparatus, loading the samples, and running analyses were described in Example 1. In Figures 9a and 9b, a 9.9-milligram sample of EVA was analyzed in nitrogen, from ambient temperature to 680 °C, using the first mode of operation (steps 1-14) of the second preferred embodiment of the present invention at a maximum heating rate of 50 °C/minute and at a resolution setting of 5.0. In Figures 9c and 9d, an 8.5 milligram sample of EVA was analyzed in nitrogen, from ambient temperature to 660 degrees Celsius, using the second mode of operation (steps 1-14) of the second preferred embodiment of the present invention at a maximum heating rate of 50 °C/minute and at a resolution setting of -4.0. Figures 9a and 9c show the TGA data plotted versus temperature. Curves 901 (Figure 9a) and 921 (Figure 9c) are plots of the percent weight change of the sample as a function of sample temperature. Curves 902 (Figure 9a) and 922 (Figure 9c) are plots of the derivative with respect to time of the sample temperature as a function of sample temperature. Curves 903 (Figure 9a) and 923 (Figure 9c) are plots of the derivative with respect to temperature of the percent weight change of the sample as a function of sample temperature. Figures 9b and 9d show the TGA data plotted versus time. Curves 911 (Figure 9b) and 931 (Figure 9d) are plots of the percent weight change of the sample as a function of time. Curves 912 (Figure 9b) and 932 (Figure 9d) are plots of the derivative with respect to time of the sample temperature as a function of time. Curves 913 (Figure 9b) and 933 (Figure 9d) are plots of the sample temperature as a function of time. The scans in Figures 9a, 9b, 9c, and 9d can be readily compared in the same manner as in Example 8 to the constant heating rate scans in Figures 6a, 6b, 6e, and 6f, respectively, and to the high resolution scans in Figures 6c and 6d. It is clear from this comparison that the two weight loss transitions, features 904 and 905 in Figure 9a, and features 924 and 925 in Figure 9c are better resolved than their respective features in Figures 6a and 6e. In all cases the derivative of weight change (percent/°C) curves are sharper and better resolved with respect to the baseline compared to the same curves shown in Figures 6a and 6e. EXAMPLE 10 : COMPUTER PROGRAM FOR IMPLEMENTING THE SECOND PREFERRED EMBODIMENT OF THE PRESENT INVENTION The following computer program is an example of a computer program used to control the main steps of the second preferred embodiment of the present invention. Examples 8 and 9 were implemented using a similar computer program. The program used to implement Examples 8 and 9 differed from the program listed here only in minor respects, such as the names of some variables. The program is written in PL/M. The foregoing examples of computer programs for controlling the heating rate of the sample during thermogravimetric analysis are two of many algorithms that could be used to control the TGA apparatus. These algorithms could reside in the thermal analysis module or in a separate computer. The algorithms would be executed concurrently with other algorithms for controlling the system and analyzing data. The foregoing disclosure of embodiments of the present invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Many variations and modifications of the embodiments described herein will be obvious to one of ordinary skill in the art in light of the above disclosure. The scope of the invention is to be defined only by the claims appended hereto, and by their equivalents.

Claims: An analytical method of determining the composition of a material that undergoes at least one transition as a function of a driving variable, said method comprising the steps of: (a) placing a sample of the material in an apparatus for detecting changes in a characterizing physical parameter as a function of said driving variable; (b) selecting at least a maximum allowed value for the rate of change of the driving variable; (c) monitoring a signal representative of the characterizing physical parameter; (d) calculating a rate of change of the driving variable as a function of the percentage change per unit time of the characterizing physical parameter and (e) controlling the rate of change of the driving variable according to the calculated rate of change; wherein the control of the rate of change of the driving variable is effected continuously, the function for step (d) is a non-linear function of an exponential type which comprises an argument and the argument includes the percentage change per unit time of the characterizing physical parameter and this function is chosen such that, when the percentage change per unit time in the characterizing physical parameter is small in relation to a baseline percentage change per unit time of the characterizing physical parameter, the rate of change of the driving variable approaches the maximum selected allowed value and, when the percentage change per unit time of the characterizing physical parameter is large in relation to the baseline percentage change per unit time of the characterizing physical parameter, the rate of change of driving variable approaches zero. A method according to claim 1, wherein the driving variable is temperature and the method effects thermal analysis. A method according to claim 1 or 2, and further comprising the step of controlling the resolution of the analytical method by selecting a resolution number and wherein said function further includes a term based upon the resolution number A method according to claim 2 or claim 3 when appended to claim 2, wherein the thermal analytical method is thermogravimetric analysis, and the characterizing physical parameter is the weight of the sample being analyzed. A method according to claim 4 further comprising the step of analyzing constituents evolving from the sample using a downstream analytical technique. A method according to claim 5, wherein the downstream analytical technique is selected from the group consisting of mass spectroscopy and Fourier transform infrared spectropscopy. A method according to claim 6, wherein the downstream analytical technique is selected from the group consisting of infrared spectroscopy, atomic spectroscopy, molecular spectroscopy, computer-assisted dispersive infrared spectroscopy, laser diode spectroscopy, flame ionization detection, Raman spectroscopy, thermal conductivity detection, electron capture detection, photoionization detection, helium ionization detection, discharge ionization detection, flame photometric detection, and chemiluminescent detection. A method according to claim 2 or claim 3 when appended to claim 2, wherein the thermal analytical method is selected from the group consisting of Differential Thermal Analysis (DTA), Differential Scanning Calorimetry (DSC), Pressure Differential Scanning Calorimetry (PDSC), Thermomechanical Analysis (TMA), Dynamic Mechanical Analysis (DMA), Dynamic Mechanical Spectrometry (DMS), Dielectric Analysis (DEA), Differential Photocalorimetry (DPC), and Thermal Conductivity Analysis (TCA). A method according to any one of the preceding claims and further comprising selecting a minimum allowed value for the rate of change in the driving variable or temperature with the calculated rate of change being less than the minimum value when the percentage change per unit time of the characterizing physical parameter is large. A method according to any one of claims 1 to 8 and further comprising selecting a minimum allowed value for the rate of change of the driving variable or temperature so that, when the percentage change per unit of time in the characterizing physical parameter is large, the calculated rate of change is less than the minimum value and controlling the rate of change according to the calculated rate of change, when the calculated rate of change is greater than the selected minimum allowed value, and according to the selected minimum rate of change when the calculated rate of change is less than the selected minimum rate of change. A method according to claim 10 and further comprising the step of selecting a value of the percentage change per unit of time in the characterizing physical parameter, wherein once the calculated rate of change falls below the selected minimum allowed value, the rate of change is controlled at the minimum allowed rate of change until the percent change per unit of time in the characterizing physical parameter falls below the selected value of the percentage change per unit of time in the characterizing physical parameter. A method according to claim 11 wherein the selected value of the percentage change per unit of time in the characterizing physical parameter is a fraction of the percentage change per unit time in the characterizing physical parameter required for the proposed rate of change calculated according to said function to equal the minimum allowed value for the rate of change. A method according to claim 12 wherein the fraction of the percentage change per unit of time in the characterizing physical parameter is approximately one order of magnitude lower than the value of the percentage change per unit of time in the characterizing physical parameter required for the proposed rate of change to equal the minimum allowed value for the rate of change. A method according to any one of claims 1,2 or 3 and further comprising selecting a minimum allowed value for the rate of change wherein said controlling step e) is performed in a first mode of operation and in a second mode of operation said function is chosen such that, when the percentage change per unit of time in the characterizing physical parameter is small in relation to the baseline percentage change per unit time of the characterizing physical parameter, the calculated rate of change of temperature of the driving variable approaches the maximum selected allowed value for the rate of change and, when the percentage change per unit of time in the characterizing physical parameter is large in relation to the baseline percentage change per unit time, the calculated rate of change is less than the minimum value and the rate of change is controlled according to the calculated rate of change, when the calculated rate of change of temperature is greater than the selected minimum rate of change and the rate of change is controlled at the selected minimum rate of change of temperature, when the calculated rate is less than the selected minimum rate of change of the driving variable. A method according to claim 14 further comprising the step of selecting a value of the percentage change per unit of time in the characterizing physical parameter, wherein in the second mode of operation once the calculated rate of change of the driving variable falls below the selected minimum rate of change the rate of change is controlled at the minimum allowed rate of change until the percentage change per unit of time in the characterizing physical parameter fall below the selected value of the percentage change per unit of time in the characterizing physical parameter.

Inventors: CROWE BENJAMIN S, SAUERBRUNN STEVEN R, CROWE, BENJAMIN S., SAUERBRUNN, STEVEN R.
Applicants: TA INSTR INC, TA INSTRUMENTS, INC.
Application year: 1991
Main classification: G01N5
IPC codes: G06F17, G01N25, G01N5, G01N21