Abstract:
Method and apparatus for damping oscillatory behavior and reducing the effective time constant incident to reattainment of equilibrium pressure and flow following a change in fluid pressure or flow rate in a liquid chromatography system. The system may include a chromatographic column, a reservoir for a slightly compressible liquid mobile phase, and piston means cooperating with the reservoir and normally driven at a velocity which is constant over a given time interval, for enabling pumping of the mobile liquid phase through the chromatographic column. In accordance with the invention, a transient velocity component is superimposed upon the constant velocity component of the mobile liquid phase caused by the piston motion. The transient velocity component is proportional to the reservoir volume and to the time derivative of pressure in the reservoir. The additionally imparted motion damps the transient oscillations that arise from any pressure change imposed on the slightly compressible liquid phase. Such pressure changes may e.g., be induced during operation of the chromatography system in a gradient elution mode.

Description:
BACKGROUND OF INVENTION 
     This invention relates generally to chromatography systems and more specifically to liquid chromatography systems, particularly those in which so-called gradient elution techniques are employed. 
     In the course of carrying out liquid chromatography methods, particularly where high-pressure liquid chromatography is practiced with relatively large reservoirs (e.g., in excess of about 10 ml), it is found that the time constant incident to reattainment of equilibrium following a sudden pressure change in the system, can be of the order of several minutes, e.g., typically 1 to 6 minutes. The sudden pressure change noted can arise e.g., in consequence of a programmed change in flow rate, or in consequence of practicing gradient elution techniques, i.e., the use of a solvent mixture with continuously changing concentration ratio -- where the source of the pressure change phenomenon is the variation in viscosity with change in relative concentration of the components. These oscillatory effects are all driven by the expansion and compression of the solvents in the reservoir or reservoirs, i.e., the oscillatory effects are ultimately caused by the fact that the solvents, although not always so though of, are indeed slightly compressible. 
     The normal and usual arrangement in chromatography apparatus of the type considered herein entails use of one or more reservoirs, which are basically in the nature of syringe pumps. A given reservoir thus may comprise a cylindrical tube or the like, having a volume V. A piston of circular cross-section is mounted for axially directed movement in the cylinder, and is normally driven by motor means at a velocity v, which is constant over a given time interval to provide a constant average flow rate Q o  for the liquid mobile phase present within the reservoir. Assuming, however, that for the reasons mentioned above a sudden pressure change is effected, the compressibility k of the liquid causes a transient change from the average flow rate Q o  for the liquid phase flowing out of the reservoir, the transient flow rate Q t  of the flow being in accordance with the equation: 
     
         Q.sub.t = (-KV) (dP/dt),                                   (1) 
    
     where k is the fluid compressibility, V is the reservoir volume, and dP/dt is the time derivative of pressure in the reservoir. It can be seen from equation (1) that a sudden increase or decrease in P cause the volume of the liquid phase to transiently contract or expand, due to the compressibility of the liquid phase thereby causing the transient change in the flow rate. 
     The net effect of the foregoing phenomena is one of providing erroneous concentration variations at the output of the fluid mixer. These concentrations variations can interact with the chromatographic column to produce additional pressure changes which reinforce the initial pressure disturbance, thereby generating a continuing instability or oscillation. 
     In accordance with the foregoing, it may be regarded as an object of the present invention to provide method and apparatus for use with liquid chromatography systems, which reduce the effectivetime constant incident to reattainment of equilibrium pressure and flow, and thereby damp the oscillatory behavior following a change in fluid pressure of the system. 
     SUMMARY OF INVENTION 
     In accordance with the present invention, the foregoing objects, and others as will become apparent in the course of the ensuing specification, are achieved by introducing modifications to a liquid chromatography system which function to reduce the effective time constant incident to reattainment of equilibrium pressure and flow, and thereby damp oscillatory behavoir following a change in fluid pressure in the system. 
     The systems of the type considered herein are normally characterized by the inclusion of a chromatographic column, a reservoir for a slightly compressible liquid mobile phase, and piston means that cooperates with the reservoir and is normally driven at a velocity, v, which in general is such as to reproduce a desired gradient--and which can be regarded as constant over a time interval of appropriate duration so as to enable pumping of the mobile liquid phase at a constant average flow rate Q o  through the chromatographic column. In accordance with the present invention, an additional velocity component which is proportional to the volume of the reservoir and to the time derivative of pressure at the reservoir, is superimposed upon the piston motion. The additionally imparted motion serves to damp the transient oscillations that arise from pressure changes imposed on the slightly compressible liquid phase, e.g., in consequence of operation of the chromatography system in a gradient elution mode. 
     In one apparatus embodiment of the invention, an electronic feedback loop is provided for the motor drive of the aforementioned piston, the feedback signal being of such character as to enable the desired superimposed velocity component. The loop may thus include means for generating a signal proportional to the time derivative of the system pressure acting upon the reservoir, means for amplifying the signal in proportion to the reservoir volume, and means for coupling the amplified signal in feedback relationship to the speed control of the aforementioned motor means. where (as is usual in gradient liquid chromatography systems) a pair of such reservoirs and piston pumps are provided, the signal indicative of the time derivative of the pressure may thus be taken from a point between the fluid mixer and the chromatographic column, with the velocity of each pump piston being modified in the manner indicated. 
    
    
     BRIEF DESCRIPTION OF DRAWINGS 
     The invention is diagrammatically illustrated, by way of example, in the drawing appended hereto, in which: 
     FIG. 1 schematically illustrates key elements of a signal reservoir liquid chromatography system incorporating the features of the present invention; and 
     FIG. 2 schematically illustrates a gradient liquid chromatography system, wherein each pump is driven as in the FIG. 1 depiction. 
    
    
     DESCRIPTION OF PREFERRED EMBODIMENT 
     In FIG. 1 a schematic diagram is set forth which illustrates in a highly simplified fashion the manner in which the present invention may be applied to a single reservoir liquid chromatography system 10. The system 10, which can be of the so-called high-pressure liquid chromatography (HPLC) type, includes basic elements which are well-known in the prior art. In particular, a syringe pump 11 is provided, which includes a reservoir 12, which may be a cylindrical tube. The reservoir 12 may have a relatively large volume V, exceeding 100 ml. A liquid mobile phase 14 is contained within reservoir 12, and is pumped or expelled through tubing 16 processing from reservoir outlet 18 in consequence of axially directed movement 30 of a piston 20 into the reservoir. The piston 20 is normally driven at a velocity v by a motor means 22, the drive gear 23 of which engages a rack 24 on an axially projecting piston rod 25. Assuming for purposes of analysis that v is constant over a time interval of suitable duration, the liquid 14 is thus normally pumped from reservoir 12 at an average constant flow rate Q o , and thence passes through tubing 16 to a conventional chromatographic column 28. 
     It will, of course, be understood that liquid 14 constitutes the mobile liquid phase, i.e., a solvent utilized in a chromatographic separation process; and further, as is also known in the present art, that the output from column 28 may be provided to a detector 31 and a collector 32; and that the detector 31 may be associated with a suitable recorder 34. 
     For purposes of simplifying explanation of the present invention, the depiction of FIG. 1 illustrates use of a simple one-component mobile phase. It will be understood, however, that the invention is applicable and indeed will be widely used with chromatography systems operating in a so-called gradient elution mode, and that in such instances various arrangements known in the art may be utilized. For example, a first solvent from reservoir 12 may be fed into an inline mixer wherein the first solvent is mixed with a second solvent in a desired concentration ratio, which ratio may be made to vary in accordance with a predetermined program. In this sort of arrangement, for example, reservoir 12 may hold water, and methenol or the like may be fed into the mixer. 
     The two reservoir gradient liquid chromatography (L.C.) systems will be further discussed in connection with FIG. 2 hereinbelow. Regardless, however, of whether one reservoir (as in FIG. 1) is utilized, or whether a pair of reservoirs (as in FIG. 2) is employed, the principles of the invention are such that each piston velocity is modified as indicated hereinafter. In particular, and continuing to refer to FIG. 1, the &#34;normal&#34; velocity v of forward movement 30 of piston 20 is modified by imposition on such motion of an additional component, which is not constant, but rather is proportional to the time derivative of pressure at reservoir 12, and also to the volume V of reservoir 12. A simple electronic arrangement enabling the aforementioned result is illustrated in FIG. 1. 
     In particular, a pressure gauge 36 is provided, which is seen to be so positioned as to be responsive to the pressure at the output of reservoir 12 -- or alternatively in 1 this may be viewed as the pressure at the input of chromatographic column 28. Pressure gauge 36 may, for example, be a strain gauge operating on piezoelectric princples. Other types of sensitive pressure gauges may similarly be utilized, which yield outputs in an electrical form suitable for further manipulation. 
     The electrical signal proceeding from gauge 36 at line 38 is, in accordance with the invention, provided to a differentiating network 40 consisting of a capacitor 42 and a resistor 44, the latter being connected to ground at 46. Thus, the electrical signal at 38 is proportional to pressure, and the differentiated signal proceeding from network 40 at line 48, is proportional to the time derivative of pressure i.e. dP/dt. The ensuing differentiated signal thence passes through a slide wire resistor 50, which is connected to ground at 52. 
     The contact arm 54 for slide wire resistor 50 is seen directly linked to piston rod 25 (for movement therewith), so that the contact arm 54 is displaced along with the movement of the piston. Thus, a resistance is placed in series with line 48, which is in accordance with the axial position of piston 20; The inserted series resistance is thus proportional to the volume V of reservoir 12. Accordingly, the signal in line 56 proceeding from contact arm 54 is of the form V (dP/dt). This signal is then passed to an amplifier 58, and thence proceeds through a further slide wire resistor 60. 
     The contact arm 62 for resistor 60 is adjusted to provide a suitable proportionality constant. In particular, the arm 62 is so set that the signal furnished at line 64, which constitutes the final portion of a feedback loop to motor means 22, is equal to the expression kV (dP/dt). Thus, the arrangement illustrated in FIG. 1 is such that the velocity of piston 20 in the direction 30 is modified by the feedback signal -- which, being of the form indicated, precisely cancels out the transient flow generated in accordance with equation (1) above. 
     Consideration of the physical phenomena occurring in system 10, further illustrates the underlying mechanism of the invention, and the general mode of its application. In particular, and as previously indicated by equation (1), a sudden pressurre change, induced, e.g., by viscosity changes where gradient elution is practiced, effects a transient flow rate rate Q t  as. As it well-known in the present art, the impedance R 0  of column 28, in analogy to an electrical system, may be expressed as: 
     
         R.sub.0 = P.sub.0 /Q.sub.0                                 (2) 
    
     where P 0  is the system pressure at the constant average flow rate Q 0 . 
     Similarly, the time constant T of the system 10 is given by the expression: 
     
         T = R.sub.0 kV                                             (3) 
    
     where R 0  is the aforementioned impedance of column 28, k is the compressibility of the liquid 14 in reservoir 12, and V is the volume of reservoir 12. 
     The &#34;time constant&#34; herein refers to the time for the transient flow rate Q t  to decay to 1/e of its maximum value. 
     In order to appreciate the result achieved by the invention, assume that the velocity v of the piston is altered by Δv. Then the change in Q t , i.e. ΔQ t , is 
     
         ΔQ.sub.t = AΔv                                 (4) 
    
     where A is the cross-sectional area of piston 20. Since Δv is proportional to V and dP/dt, 
     
         Δv = C.sub.1 V dP/dt                                 (5) 
    
     and hence 
     
         ΔQ.sub.t = AC.sub.1 V dP/dt.                         (6) 
    
     Hence, the &#34;new&#34; transient flow Q&#39; t  = Q t  + ΔQ t  is given by: 
     
         Q&#39;.sub.t = - k V dP/dt + C.sub.2 V dP/dt 
    
                                                                (7) 
     where C 2  = AC1, or 
     
         Q&#39;.sub.t = kV dP/dt [C.sub.2 /k + 1] 
    
     By comparing equations (1) and (3) with equation (8), it can be seen that the &#34;new&#34; time constant T&#39; for the system with modified piston velocity, becomes 
     
         T&#39; = T (1 - c/k),                                          (9) 
    
     where c is a proportionality constant which may be set by positioning arm 62 in relation to the compressibility k of liquid 14. Thus, in accordance with the invention, the system time constant may be arbitraritly reduced, subject, limitations in measuring the derivative dP/dt -- as will be hereinbelow discussed. 
     As has previously been indicated, the present invention is particularly applicable to an L.C. system of the type adapted to operate in a gradient elution mode. Thus in FIG. 2, a highly schematic showing is set forth, depicting a system 70 of the type indicated, i.e., one operating in a gradient elution mode. The system 70 differes in its mode of operation from that of FIG. 1, primarily in that instead of a single cylinder and piston pump as in FIG. 1, a pair of such pumps 72 and 74 are utilized. These pumps, in each instance, may be deemed similar to the pump described in connection with FIG. 1, i.e., they are &#34;syringe pumps&#34; comprising cylindrical reservoirs 76 and 78 of volume V 1  and V 2 , in which pistons 80 and 82 unidirectionally advance, to provide their respective solvents through lines 84 and 86 to a mixer 88. Mixer 88 is conventional, and as is known in this art serves to thoroughly blend the two solvents together, with the mixed solvents then being furnished via a line 90 to the chromatographic column 28. In this FIG. 2 elements corresponding to those previously discussed (as, for example, the column 28) are identified by corresponding reference numerals. 
     Thus again, in FIG. 2, a pressure sensor 36 is provided, which may be of the type discussed in connection with FIG. 1. Sensor 36 in this instance is seen to be placed in the line 90 between mixer 88 and the input to column 28. The output signal from sensor 36 is differentiated at 91 and then furnished by a line 92 to amplifier control logic 94, which is also provided with inputs via lines 96 and 98 from piston position indicator means 100 and 102. The signals thus provided through lines 96 and 98 may be regarded as proportional to the volumes V 1  and V 2  of reservoirs 76 and 78 at a given time. Thus, these signals may be derived in the manner discussed in connection with FIG. 1. 
     Amplifier control logic 94 furnishes control signals through lines 108 and 110 to two amplifiers 104 and 106, with the feedback signals from the amplifiers then being furnished to the piston drives 112 and 114 for each syringe pump. Again, this operation is analogous to the mechanism that has been described in connection with FIG. 1. It may be noted further, however, that the precise control scheme for the amplifiers as set forth in FIG. 1 need not be utilized. For example, amplifiers 104 and 106 may be of the variable gain type, with the gain being programmed to follow the volumes V 1  and V 2  of the reservoirs, as such volumes are indicated by the signals in lines 96 and 98. The volume signal, again, need not be derived precisely as set forth in connection with FIG. 1. For example, the drives 112 and 114 for the pumps may consist of stepping motors, in which event the volumes may be determined by summing the pulses preceding to the stepping motor drive. Such techniques are quite well-known in the art. 
     It may also be pointed out that the schematic depiction of FIG. 2 does not explicitly show certain well-known elements normally present in gradient elution systems of the present type. For example, and as is well-known in this art, the piston drives 112 and 114 may also be under the control of a solvent control logic block adjusts the advance rates of the pistons for successive time intervals, as aforementioned to provide desired solvent ratios in accordance with a pre-selected program. 
     For the single pump sytem of FIG. 1, if the flow rate from the piston is changed from Q 1  to q 2  in a single step, the flow rate through column becomes 
     
         Q= Q.sub.1 + (Q.sub.2 -Q.sub.1) (1-e.sup.- t/T.spsb.&#39;). 
    
     if T&#39; is sufficiently short, the flow rate through the column approximates the programmed step, and the compressibility of the fluid is overcome. 
     For the dual-pump gradient system of FIG. 2, the same consideration holds, where now 
     
         T = T.sub.1 + T.sub.2                                      (11) 
    
     such that T 1  = k 1  V 1  R 0 , and T 2  = k 2  V 2  Rhd 0, where k 1  and k 2  are the compressibilities of the solvents in reservoirs 76 and 78, respectively. In general, 
     
         T&#39; + T.sub.1 [1- C.sub.3 /k.sub.1 ] + T.sub.2 [1- C.sub.4 /k.sub.2 ], 
    
     where C 3  and C 4  are the velocity feedback proportionality constants for the two pistons. If C 3  /k 1  + C 4  /k 2  .tbd. c/k, then T&#39; = T [1-c/k] as in equation (9). In addition, as explained above, when the fluid visocity varies with concentration of the mixture, unstable oscillations may result. It may then be shown that the system 70 will be stable if 
     
         τ.sub.m /T.sup.1 &gt; | 1/P δP/δγ | - 1 
    
     where τ m  is the volume exchange time of mixer 88, and γ and P are the steady state concentration and pressure taken from the curve of pressure vs. Concentration (i.e. viscosity) at the column, at constant flow rate Q 0 . Equation (12 ) shows that oscillations can occur if | 1/P δP/δγ | &gt; 1. For example, with a mixture of 99% methanol and 1% water, | 1/PδP/δγ | ≈ 4.6. In this case, the system 70 is stabilized if τ m  /τ&#39; &gt; 3.6, according to this invention. 
     The form of equation (12) arises from a perturbation analysis of system 70. In the water-methanol system, the instability is sinusoidal with a period about equal to twice the fluid transit time from the mixer to the column. In the hexane-isopropanol system, the instability has a different form, such that the period is approximately equal to (T&#39;τ m ) 1/2  /2. In both cases, equation (12) applies. Since the instability is driven by the compression in the reservoirs, a reduction in T&#39; as indicated in equaton (12) must lead to stability. 
     Experimentally, it is not possible to measure the instantaneous value of dP/dt. Two adjacent points on the P(t) curve are required, and this entails a small time delay τ e . This delay limits the smallest value of T&#39;, such that T&#39;≳√T τ e . It is easy to make τ e  ≈ 1 sec., whereas T is typically several minutes in value. Thus, it is possible to obtain a substantial reduction in the dynamic time constant T&#39;. 
     While the present invention has been particularly set forth in terms of specific embodiments thereof, it will be understood in view of the present disclosure, that numerous variations upon the invention are now enabled to those skilled in the art, which variations yet reside within the teaching of the invention. 
     Thus, while the invention has been particularly described in the context of L.C. systems based upon syringe pumps, the invention is applicable to other types of pumps and pumping systems wherein a pump drives fluid through a reservoir. In particular, principles of the invention remain applicable in these further cases, i.e., the derivation dP/dt is determined for the system reservoir and a feedback signal proportional to dP/dt is provided to the pump drive to alter the flow rate from the reservoir, thereby effectively reducing the compressibility of the liquid in the reservoir. 
     Accordingly the invention is to be broadly construed, and limited only by the scope and spirit of the claims now appended hereto.