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Chapter 3 Detector Characteristics that Affect Column Performance
CHAPTER 3 Detector Characteristics that Affect Column Performance Factors That Directly Affect Band Dispersion Separations are achieved in liquid chromatography by employing a mobile and stationary phase system that will move the individual solute bands apart during development and by designing the column to keep the individual solute bands narrow.
Obviously, the more narrow the bands and the further they are
moved apart, the better the separation.
The detector and its connecting
tubes cannot affect the degree to which the solute bands are moved apart, as this depends solely on the characteristic of the two phases, but they can affect the width of the solute bands.
Band spreading in the connecting
tubes or cell volume itself results from poor radial transfer of the solute in the liquid and the parabolic velocity profile of the mobile phase that exists in the connecting tubes and the detector cell itself.
As the
technique develops, columns will provide higher and higher efficiencies, which means
the bands will become more narrow and provide
improved
resolution.
At present the base width of the peak eluted from a high
efficiency colymn in terms of volume of mobile phase may only be about 100 microliters, and it follows, therefore, that a detector having a cell volume of 8 microliters will significantly contribute to the band width and thus the real efficiency and resolving power of the column not realized.
Peaks
already separated in the column will be merged together due to the solute bands being broadened in the detector.
In a similar way the time constant
of the amplifier and the recorder can contribute to the apparent band width as recorded on the chart.
Most amplifiers and recorders have filter
circuits associated with their outputs to eliminate high frequency electronic noise, but if the filter frequency limit approaches that of the eluted peak then the peak will be reduced in height and broadened.
As the
design of the detector and its electronics can significantly affect the performance of the column with which it is associated those characteristics
22
Liquid Chromatography Detectors of a detector that control this effect need to be known and stated in the detector specifications. Theory It is not the intent of this book to give a treatment of the theory of chromatography and for those wishing to study the subject more fully are recommended to read the app5opriate chapters in Contemporary Liquid Chromatography by the same author and published by J. Wiley.
However, the basic
principles of the theoretical approach that must be used to investigate dispersion in detector cells and detector connecting tubes will be outlined and the pertinent equations given. The dispersion processes that occur in the column connecting tubes and detector cells are all random in nature and therefore broaden any Gaussian distribution of solute concentration but maintains its Gaussian form. Thus the Gaussian curve relating solute concentration to volume flow of mobile phase or time as measured by the detector is made up of the elution curve resulting fran the dispersion effects in the column, together with the curve resulting from the dispersion effects occurring in the connecting detector tubes and the detector cell itself.
It can be shown that the variances of
each dispersion effect can be summed to give the variance of the final curve 2 T. Thus Tc 2 + TT2 + To2 = T2
where T is the standard deviation of the solute band eluted from the C column TT is the standard deviation of the curve resulting from dispersion in the connecting tubes and T
D
is the standard deviation of the curve resulting from dispersion in the detecting cell
It is generally accepted that the band width of a peak can be increased by 5% (1) -vithout seriously impairing the efficiency and resolution of the column. Thus T = 1.05 Tc Hence Tc
2
+ TT 2 + TD2
=
(1.05Tc)
2
Dispersion in Detector Cell
+ TD
or T
T
=
23
2 0.103 Tc
Thus the sum of the variances resulting from dispersion in the connecting tubes and the detector cell should not exceed 10% of the variance of the band leaving the column.
Scott and Kucera (2) examined the effect
2r
of band dispersion in capillary tubes and derived the following expression:
'
(
=
2.4 DMVR nnQl
Where r is the radius of the tube in cm 2
D is the diffusivity of the solute in the mobile phase in cm /sec M V is the retention volume of the solute in rnl R n is the efficiency of the attached column in theoretical plates
Q is the volume flow rate through the column in ml/sec and 1 is the length of the capillary tube in cm This equation can be used to calculate the dimensions of a detector cell or the connecting tubes for a column of given dead volume and efficiency. However, the detector comprises of a cell and connecting tubes, and, therefore, both must be taken into account.
As
the detector cell is the
most important part of the detector and controls the overall sensitivity that can be achieved, the cell dimensions must be allowed to be as large as possible and account for the major proportion of acceptable band dispersion. If a 4.5% increase in band width is allowed to occur in the detector cell, I
and 0 . 5 % increase in band width is allowed to occur in the connecting tubes, then two equations can be obtained giving the dimensions of both the detector cell and the connecting tubes.
(
2.21 DMVR
For the detector cell
r
=
nnQl
where r and 1 are the radius and length of the detecting cell respectively and the other symbols have the meaning previou~l.ir ascribed to them.
For the connecting tubes
24
Liquid Chromatography Detectors
r
=
IT nQ1
where r and 1 are the radius and length of the connecting tube respectively. The dimensions of a satisfactory detecting cell and connecting tubes can now be calculated for a given column and given detector combination.
The
following are very common column and detector characteristics. Column Length 25 an
, I.D.
4.6
Flow Rate 1 ml/min D, =
mm
, V0 (Dead Volume) 3 ml,
= 0.0167 ml /sec, 2 cm /sec.
The results obtained using this data in the above equation are shown in table I taking column efficiency values of 4,000; 8,000: 12,000; 16,000; and 20,000 theoretical places.
As the efficiency decreases as the retention
volume of the solute increases, Vr will be taken as both the value of Vo (3 ml) the dead volume which will have the maximum efficiency for a given column and at k'=2 where V will be 9 ml.
At k'=2 the position of a peak in
the chromatogram is that which provides the highest resolution in the minimum time. The detector will be taken to have connecting tubes 5 cm long and the detecting cell will be assumed to have a path length of 1 cm.
The
values of the radius of the cell and connecting tubes together with their respective volumes are given in Table I. It is seen from Table I that for efficiencies ranging from 4,000 to 20,000 theoretical plates, the maximum cell volume ranges from 3.06 microliters down to 1.37 microliters respectively for a dead volume peak and from 9.18 down to 4.11 microliters respectively for a peak eluted at k'=2.
From
the results it would appear that the efficiencies of present day microparticulate columns can never be realized for a dead volume peak due to the relatively large detector volumes associated with available detectors. The detector dispersion becomes so significant that efficiency values obtained from them will be much lower than that actually provided by the column. As the equations show, the length of the cell and its radius are interrelated, a cell with a long path length but small radius may meet the same column requirements in an identical manner to an alternative cell of short path length and relatively large radius. In Table I1 a range of cell diameters
Detector Cell Dimensions
Peak Eluted at Dead Volume (3 ml) I Cell Cell Tube Tube I.D. Volume I.D. Volume 4,000 0.621 mm 3.1 p1 0.240 mm 2.3 p1 0.025 in 0.010 in
Efficiency
8,000
12,000 16,000 20,000
I
0.525 mm 2.2 pl 0.021 in
25
Peak Eluted at kb2 (V = 9 ml) Cell Cell Tube Tube I.D. Volume I.D. Volume 1.08 mm 9.2 pl 0.416 mm 6.8 pl 0.043 in 0.076 in
0.202 mm 1.6 p1 0.008 in
0.909 mm 6.5 p l 0.036 in
0.349 mm 4.8 p l 0.014 in
0.474 mm 1.8 0.019 in
p l 0.182 mm 1.3 pl 0.821 mm 5.3 pl 0.316 mm 3.9 pl
0.441 mm 1.5 0.017 in
p l 0.170 mm 1.1 pl 0.764 mm 4.6 p l 0.294 mm 3.4 pl
0.032 in
0.007 in 0.007 in
0.417 mm 1.4 pl 0.016 in
0.030 in
0.161 nun 1.0 0.006 in
pl
0.012 in 0.012 in
0.727 mm 4 . 1 ~ 1 0.278 mm 3.0 0.029 in 0.011 in
P1
Column Length 25 cm Column Diameter 4.6 mm 2 Diffusivity of Solute in the Mobile Phase lo-’ cm /sec Plow Rate 1 ml/min Path Length of Cell, 1 cm Length of Connecting Tube 5 cm.
and cell lengths that will meet the requirements for minimum dispersion for a given column system are given.
It is seen that at one extreme a cell
d
length 0.25 cm, I.D. 0.59 mm and a volume of 0.68 ~1 can be used and at the other extreme a cell of length 2.5 cm, I.D.
0.33 mm and a volume of 2.167
microliters could meet the same column requirements.
The choice of the-best
combination of path lengths and cell I.D. that will meet both the minimum dispersion requirements of the column and at the same time provide the maximum sensitivity is somewhat complex.
It will depend on the intensity
of the light source, the characteristics of the photo cell employed and on the detector electronics.
Increasing the path length will increase the
response of the detector to a given solute concentration as predicted by Bee&
Law.
However, as the I . D . of the cell must also be reduced as shown in
Table 11, this results in a decrease in light intensity falling on the
It follows that the effect on detector
I
26
Liquid Chromatography Detectors sensitivity which has already been shown to depend on the signal to noise ratio may be increased or decreased. The choice of the optimum cell length and I.D. for maximum sensitivity has to be left to the manufacturer who must carefully select and design the light source, optics, photo cell and electronics such that the maximum sensitivity can be realized while employing cell geometries that provide the minimum chromatographic band Table I1 Cell Dimensions that will Provide Minimum Dispersion for a Given Column System Column System
Column Length 25 cm Column Diameter 4.6 mm Dead Volume 3 ml Flow Rate 1 ml/sec Diffusivity of Solute in Mobile Phase cm2/sec
Cell Length (an) 0.25 0.50 1.00 1.50 2.00 2.50
Cell I.D. (mm) 0.59 0.50 0.42
0.38 0.35 0.33
Cell Volume
P 0.68 0.97 1.37 1.68 1.94 2.16
dispersion demanded by the column. Dispersion effects between column and detector can be extremely important where post column reactors are employed. In such systems a reagent is mixed with the column eluent that reacts with the eluted solute to render it detectable by the detecting system employed.
Under such conditions band
dispersion in the reactor volume between column and detector can be very serious. chapter.
The design of column reactors will be discussed in the next
Turbulence Turbulence in either the detector connecting tube or the cell itself can significantly increase the diffusivity of the solute by convective mixing and reduce band dispersion. However, turbulence in the detector cell itself
Other Factors Affecting Band Dispersion 27 also results in serious noise and thus reduces the detector sensitivity. It fcllows that turbulence in the detector should be avoided. Turbulence can be employed in the detector connecting tube to reduce dispersion by interfering with the regular geometry of the tube.
Providing the lamina nature
of the flow patterns in the tube are broken, this will result in convective mixing and reduce dispersion.
This can be achieved by crushing the tube
almost flat every 2 mm along its length between the column and detector. However, such a system easily becomes blocked, provides significant back pressure on the column and for these reasons an open tube of the correct dimensions is to be preferred. Factors that Indirectly Affect Band Dispersion The band width as drawn by the recorder on the chart paper may be significantly broader than that actually sensed by the detector due to a spreading effect that results from the time constant of the detector amplifier and the recorder itself.
Amplifiers and recorders have an
inherent time constant that arises from their respective circuit design but a further time constant is often purposely introduced to remove high frequency noise.
If this time constant is of commensurate period to the
time standard deviation of the eluted peak then the peak will suffer significant broadening. The effect of amplifier time constant on the shape of detected chromatographic peaks has been elegantly treated by Vandenheuvel ( 3 ) , Schmauch ( 4 ) , and Sternberg (5). For those readers wishing to study the effect of time constants on peak distortion in detectors they are recommended to read the work of Sternberg. Sternberg developed the following equation to describe the pcak shape after being distorted by an amplifier with a time constant of T'.
x
=
Xoe
-t/T'
t/T
- ( t-to) /2Tt2
/~lJe e
where X = the voltage output of the amplifier to the recorder Xo = a constant to = time at the peak maximum Tt = the time standard deviation of the eluted peak t = the elapsed time The explicit solution of the above equation is rather complicated, but the
28
Liquid Chromatography Detectors distortion of a normal Gaussian peak can be determined from the above equation by the use of a computer.
Consider a column 25 cm long and 4.6 mm in
diameter having a dead volume (V
0
)
of 3 ml, an efficiency (n) of 12,000
theoretical plates and operated at a flow rate of 1 ml/min (Q). From the plate theory the standard deviation of the dead volume peak in ml of mobile phase will be
-vO nk
3
-
(12,000)k
=
0.0274 ml
Thus the time standard deviation Tt
=
Q
=
0.0274 x 60
=
1.64 sec.
Thus taking values for the time constant of the amplifier of 0.6 and 1.5 secs and replacing T in the equation by the calculated value of 1.64 sec, t
= 1.5 sec
, T '
5
0.6 sec
Time Figure 1
Curves Demonstrating Peak Distortion Resulting from Significant Amplifier Time Constant
Effect of Amplifier Time Constant the shape ,.f
tiz
resultant peaks can be calculated.
Using a computer the
resultant curves were determined over the range of t = 4 . 9 2 sec to t = 8.2 sec
and the results obtained are shown in figure 1 together with the
original undistorted peak.
It is seen that the effect of a 1.5 second time
constant is to serioilsly distort the peak both by increasing the peak width and reducing the peak height.
This has resulted in a reduction of the
resolving power of the column and the sensitivity of the detector.
It is,
20
c 4J
‘c1
.,+
6
Y
m
a
10
C
.,+ 0)
m LI
U
n dp
0
1
2
Time Constant in sec Column Length 2 5 cm Column Diameter 4.6 nun I. D. Dead Volume 3 ml Figure 2
Curves Relating the Percentage Increase in Peak Width to the Amplifier Time Constant
therefore, important to determine the maximum time constant that can be employed with any given column. Using the same equation and again with the aid of a computer the percentage increase in peak width that will occur for a range of different time constants can be calculated. The results obtained
29
30
Liquid Chromatography htectors for the dead volume peak from colunms of the same dimensions as given previously but with efficiencies of 4,000; 12,000; and 20,000 theoretical plates are given
in
figure
If an increase in hand width of 5% is
2.
tolerated which will result in a reduction in peak height of about 5%, then the maximum permitted amplifier time constant will be 0.48 sec, 0.58 sec and 1 sec for peaks having time standard deviations of 1.26 sec, 1.64 sec and 2.85 sec
respectively.
Thus the maximum time constant that can be
tolerated is seen to be consistently about 35% of the time standard deviation of the peak. IF figure 3 the same calculations have heen carried out to provide similar curves but for a solute eluted at k' = 2. It is seen that the maximum time co!i;tants
that can be tolerated are now 1.32 sec,
1.76 sec and 3.04 sec for peaks having standard deviations of 3.82 sec,
4.93 sec and 8.54 sec respectively. It should also be noted that the maximum amplifier time constant is again consistently 35% of the time standard deviation of the peak. The problem can be approached in an entirely different inanner by using the principle of the summation of variances. The effect of amplifier time constant is to combine two functions, an exponential function and a Gaussian function. As these two functions describe physical phenomena that are not interacting in the sense that they proceed independently of one another, the variance of the combined function is equivalent to the sum of the variances of each individual function. The time variances of an exponential function of the form t 2 e- Tl is T1 and that of a Gaussian function 2
-t 2 T~~
is T~2
It follows that
+
Tt2 = T2
where T2 is the time variance of the resulting pcs4 as described by the recorder.
Now Tt2 is the time variance of the solute band leaving the
column and thus if a band width increase of 5% is considered acceptable (T = 1.05 Tt)
Effect of Amplifier Time Constant then Tt2 + T12 =
(1.05 T )2 = t
thus T12 =
1.103 T
2
t
0.103 T: 2
and T1 = 0.32 Tt Thus the maximum value of T1 that can be tolerated to maintain column resolution will be 32% of the time standard deviation of the eluted peak. The discrepancy between the value of 32% derived in this manner to the value of 35% derived from computer integration results partly from the fact that
in digital integration finite and not infinitesimal steps are used during
a,
m LI
V
Y
Time Constant in sec Column Length 25 cm,Column Diameter 4.6 mm I. D. Dead Volume 3 ml Figure 3
Curve Relating the Percentage Increase in Peak Width to the Amplifier Time Constant
31
32
Liquid Chromatography Detectors
integration procedure and partly due to the assumption that the variance of the distorted peak is at 0.607 of the peak height.
The more correct value
is, therefore, given by the second treatment, namely, 32%.
It follows that
for any column the maximum permitted time constant T' will be given by T'
0.32 VR
-
=
4
n Q Where V and n are the retention volume and efficiency of the peak and Q is R the flow rate through the column. VR should normally be taken for the dead
0
20
I0 Retention Volume ml 1. Column Efficiency
Figure 4
5,000 Theoretical Plates
2.
Column Efficiency
10,000 Theoretical Plates
3.
Column Efficiency
15,000 Theoretical Plates
Curves Relating Retention Volume to Amplifier Time Constant for Columns of Different Efficiencies
33
Recorder Time Constant volume peak as it has the smallest retention volume and the highest efficiency.
The above equation for T is applicable to columns of all
dimensions packed with any stationary phase and employed with any mobile phase.
In figure 4 curves are shown relating the minimum amplifier time
constant that can be permitted to maintain column resolution, to solute retention volume.
The individual curves are for columns of different
efficiency operated at a flow rate of 1 ml/min.
The appropriate time
constant for any column system, within the range given, can be obtained by interpolation. The Time Constant of the Recorder The potentiometric recorder does not have a time constant of the form normally associated with the amplifier which, in general, results from a capacity resistance network intrinsic in the amplifier circuit.
The
response of an amplifier to an instantaneous applied constant voltage is normally an exponential function of time, whereas for a
potentiometric
recorder the response is usually linearly related to time.
The linear
response results from the feed back circuitry incorporated in the sensor system which is necessary for stability. In figure 5 the recorded reading
is plotted against time for an instantaneous applied constant 9 mv signal. The recorder was the Honeywell Electronik 196, 10 mv potentiometric recorder operated with a chart speed of 1 cm/sec and having a specified balancing time of 0 . 5 sec.
It is seen that the response is approximately linear.
Now for a linear function the time variance T’, is given by
where tR is the time taken for the recorder to reach the applied voltage. In the example given T2 =
w2 =
0.035 sec2
and thus the time standard deviation, T = 0.19 sec. It is seen that if a recorder is employed having a balancing time of about 0 . 5 sec the contribution in time variance to the eluted peak is relatively small and by itself would not significantly increase the width of the eluted peak and reduce resolution.
However, it is yet another
34
Liquid Chromatography Detectors contribution to band dispersion and it is the sum of all the dispersive effects that take place subsequent to the column that ultimately limits the chromatoqrapher from realizing the actual efficiency that the column provides.
Figure 5 Response Curve of a Potentiometric Recorder
1
0.5
0.79
1.0
Adsorption Effects in the Detector When using a mobile phase that contains a small percentage of a polar solvent contained in a nonpolar solvent, for example ether or ethanol in heptane, a layer of the more polar solvent is often adsorbed on the surface of the quartz or glass detector cell.
If a solute is eluted that is more
polar than the ether or ethanol then the polar solvent is displaced from the
Summary of Dispersion Effects cell walls and is replaced by the solute.
This displacement results in a
spurious peak or, more often, a distorted peak shape.
The effect of
adsorption on the walls of the detecting cell cannot be treated quantitatively and when it occurs it should be eliminated by choosing an alternative mobile phase. Summary There are a number of band dispersing processes that occur in various parts of the detector and its associated electronics.
By careful design,
any one of these processes can usually be reduced to a level where it does not significantly affect column performance.
However the total dispersion
that takes place is a combination of all these effects and thus the T will be given by resulting time variance of an eluted peak '
2
T
Tc
=
2
+ T12 + T22 + T32 +
T4
2
where 2 . is the time variance of the peak leaving the column. Tc T12 is the time variance resulting from dispersion in the detector con-
necting tubes. T22 is the time variance resulting from dispersion in the detector cell. Tf
is the time variance resulting from the time constant of the amplifier.
2 is . the time variance resulting from the time constant of the recorder. T4
If a 5% increase in band width is tolerated then T2 = T12
+
(T12
+ T22 + T3
T22
+
T32
and
(1.05 C)t
+ T42
+
=
0.103
T42)t =
TC2
0.32 Tc.
If it is assumed that in the future a 25 cm long column 4.6 mn I.D. with a
35
36
Liquid Chromatography Detectors dead volume of 3 ml and operated at a flow rate of 1 ml/min will provide 20,000 theoretical plates then the time standard duration of the peak, Tc, will be about 1.27 sec. It follows that 2 (T1
+
2 T2
+
T32
+
=
T42)”
0.41 sec.
It is seen from the above equation that the detecting system has to be very carefully designed if the efficiency of 20,000 theoretical plates is to be realized.
It is an unfortunate fact that with many present day detecting
systems for liquid chromatography, an efficiency of even 10,000 theoretical plates from such a column would never be seen.
If 10,000 theoretical plates
were indeed achieved then the column would, in fact, be providing very much more than the 10,000 theoretical plates measured.
It is hoped that the
detector manufacturers will, in the future, design detectors having specifications that are appropriate to the column efficiencies presently obtainable and also be suitable for even higher efficiencies that will result from future developments in column technology.
REFERENCES 1.
A.
Klinkenberg, Gas Chromatography 1960, (Ed. R. P. W. Scott), Butter-
worth, London, 1960, p. 182. 2.
R. P. W. Scott and P. Kucera, J. Chromatogr. Sci., 9 (1971) 641.
3.
F. A. Vandenheuvel, Anal. Chem.,
35 (1963) 1193.
4.
L.
5.
J. C. Sternberg, Advances in Chromatography, (Ed. J. C. Giddings
S.
Schmauch, Anal. Chem.,
31 (1959) 225.
and R. Keller), Marcel Dekker, New York, Vol. 11, 1966, p. 206.
Obviously, the more narrow the bands and the further they are
moved apart, the better the separation.
The detector and its connecting
tubes cannot affect the degree to which the solute bands are moved apart, as this depends solely on the characteristic of the two phases, but they can affect the width of the solute bands.
Band spreading in the connecting
tubes or cell volume itself results from poor radial transfer of the solute in the liquid and the parabolic velocity profile of the mobile phase that exists in the connecting tubes and the detector cell itself.
As the
technique develops, columns will provide higher and higher efficiencies, which means
the bands will become more narrow and provide
improved
resolution.
At present the base width of the peak eluted from a high
efficiency colymn in terms of volume of mobile phase may only be about 100 microliters, and it follows, therefore, that a detector having a cell volume of 8 microliters will significantly contribute to the band width and thus the real efficiency and resolving power of the column not realized.
Peaks
already separated in the column will be merged together due to the solute bands being broadened in the detector.
In a similar way the time constant
of the amplifier and the recorder can contribute to the apparent band width as recorded on the chart.
Most amplifiers and recorders have filter
circuits associated with their outputs to eliminate high frequency electronic noise, but if the filter frequency limit approaches that of the eluted peak then the peak will be reduced in height and broadened.
As the
design of the detector and its electronics can significantly affect the performance of the column with which it is associated those characteristics
22
Liquid Chromatography Detectors of a detector that control this effect need to be known and stated in the detector specifications. Theory It is not the intent of this book to give a treatment of the theory of chromatography and for those wishing to study the subject more fully are recommended to read the app5opriate chapters in Contemporary Liquid Chromatography by the same author and published by J. Wiley.
However, the basic
principles of the theoretical approach that must be used to investigate dispersion in detector cells and detector connecting tubes will be outlined and the pertinent equations given. The dispersion processes that occur in the column connecting tubes and detector cells are all random in nature and therefore broaden any Gaussian distribution of solute concentration but maintains its Gaussian form. Thus the Gaussian curve relating solute concentration to volume flow of mobile phase or time as measured by the detector is made up of the elution curve resulting fran the dispersion effects in the column, together with the curve resulting from the dispersion effects occurring in the connecting detector tubes and the detector cell itself.
It can be shown that the variances of
each dispersion effect can be summed to give the variance of the final curve 2 T. Thus Tc 2 + TT2 + To2 = T2
where T is the standard deviation of the solute band eluted from the C column TT is the standard deviation of the curve resulting from dispersion in the connecting tubes and T
D
is the standard deviation of the curve resulting from dispersion in the detecting cell
It is generally accepted that the band width of a peak can be increased by 5% (1) -vithout seriously impairing the efficiency and resolution of the column. Thus T = 1.05 Tc Hence Tc
2
+ TT 2 + TD2
=
(1.05Tc)
2
Dispersion in Detector Cell
+ TD
or T
T
=
23
2 0.103 Tc
Thus the sum of the variances resulting from dispersion in the connecting tubes and the detector cell should not exceed 10% of the variance of the band leaving the column.
Scott and Kucera (2) examined the effect
2r
of band dispersion in capillary tubes and derived the following expression:
'
(
=
2.4 DMVR nnQl
Where r is the radius of the tube in cm 2
D is the diffusivity of the solute in the mobile phase in cm /sec M V is the retention volume of the solute in rnl R n is the efficiency of the attached column in theoretical plates
Q is the volume flow rate through the column in ml/sec and 1 is the length of the capillary tube in cm This equation can be used to calculate the dimensions of a detector cell or the connecting tubes for a column of given dead volume and efficiency. However, the detector comprises of a cell and connecting tubes, and, therefore, both must be taken into account.
As
the detector cell is the
most important part of the detector and controls the overall sensitivity that can be achieved, the cell dimensions must be allowed to be as large as possible and account for the major proportion of acceptable band dispersion. If a 4.5% increase in band width is allowed to occur in the detector cell, I
and 0 . 5 % increase in band width is allowed to occur in the connecting tubes, then two equations can be obtained giving the dimensions of both the detector cell and the connecting tubes.
(
2.21 DMVR
For the detector cell
r
=
nnQl
where r and 1 are the radius and length of the detecting cell respectively and the other symbols have the meaning previou~l.ir ascribed to them.
For the connecting tubes
24
Liquid Chromatography Detectors
r
=
IT nQ1
where r and 1 are the radius and length of the connecting tube respectively. The dimensions of a satisfactory detecting cell and connecting tubes can now be calculated for a given column and given detector combination.
The
following are very common column and detector characteristics. Column Length 25 an
, I.D.
4.6
Flow Rate 1 ml/min D, =
mm
, V0 (Dead Volume) 3 ml,
= 0.0167 ml /sec, 2 cm /sec.
The results obtained using this data in the above equation are shown in table I taking column efficiency values of 4,000; 8,000: 12,000; 16,000; and 20,000 theoretical places.
As the efficiency decreases as the retention
volume of the solute increases, Vr will be taken as both the value of Vo (3 ml) the dead volume which will have the maximum efficiency for a given column and at k'=2 where V will be 9 ml.
At k'=2 the position of a peak in
the chromatogram is that which provides the highest resolution in the minimum time. The detector will be taken to have connecting tubes 5 cm long and the detecting cell will be assumed to have a path length of 1 cm.
The
values of the radius of the cell and connecting tubes together with their respective volumes are given in Table I. It is seen from Table I that for efficiencies ranging from 4,000 to 20,000 theoretical plates, the maximum cell volume ranges from 3.06 microliters down to 1.37 microliters respectively for a dead volume peak and from 9.18 down to 4.11 microliters respectively for a peak eluted at k'=2.
From
the results it would appear that the efficiencies of present day microparticulate columns can never be realized for a dead volume peak due to the relatively large detector volumes associated with available detectors. The detector dispersion becomes so significant that efficiency values obtained from them will be much lower than that actually provided by the column. As the equations show, the length of the cell and its radius are interrelated, a cell with a long path length but small radius may meet the same column requirements in an identical manner to an alternative cell of short path length and relatively large radius. In Table I1 a range of cell diameters
Detector Cell Dimensions
Peak Eluted at Dead Volume (3 ml) I Cell Cell Tube Tube I.D. Volume I.D. Volume 4,000 0.621 mm 3.1 p1 0.240 mm 2.3 p1 0.025 in 0.010 in
Efficiency
8,000
12,000 16,000 20,000
I
0.525 mm 2.2 pl 0.021 in
25
Peak Eluted at kb2 (V = 9 ml) Cell Cell Tube Tube I.D. Volume I.D. Volume 1.08 mm 9.2 pl 0.416 mm 6.8 pl 0.043 in 0.076 in
0.202 mm 1.6 p1 0.008 in
0.909 mm 6.5 p l 0.036 in
0.349 mm 4.8 p l 0.014 in
0.474 mm 1.8 0.019 in
p l 0.182 mm 1.3 pl 0.821 mm 5.3 pl 0.316 mm 3.9 pl
0.441 mm 1.5 0.017 in
p l 0.170 mm 1.1 pl 0.764 mm 4.6 p l 0.294 mm 3.4 pl
0.032 in
0.007 in 0.007 in
0.417 mm 1.4 pl 0.016 in
0.030 in
0.161 nun 1.0 0.006 in
pl
0.012 in 0.012 in
0.727 mm 4 . 1 ~ 1 0.278 mm 3.0 0.029 in 0.011 in
P1
Column Length 25 cm Column Diameter 4.6 mm 2 Diffusivity of Solute in the Mobile Phase lo-’ cm /sec Plow Rate 1 ml/min Path Length of Cell, 1 cm Length of Connecting Tube 5 cm.
and cell lengths that will meet the requirements for minimum dispersion for a given column system are given.
It is seen that at one extreme a cell
d
length 0.25 cm, I.D. 0.59 mm and a volume of 0.68 ~1 can be used and at the other extreme a cell of length 2.5 cm, I.D.
0.33 mm and a volume of 2.167
microliters could meet the same column requirements.
The choice of the-best
combination of path lengths and cell I.D. that will meet both the minimum dispersion requirements of the column and at the same time provide the maximum sensitivity is somewhat complex.
It will depend on the intensity
of the light source, the characteristics of the photo cell employed and on the detector electronics.
Increasing the path length will increase the
response of the detector to a given solute concentration as predicted by Bee&
Law.
However, as the I . D . of the cell must also be reduced as shown in
Table 11, this results in a decrease in light intensity falling on the
It follows that the effect on detector
I
26
Liquid Chromatography Detectors sensitivity which has already been shown to depend on the signal to noise ratio may be increased or decreased. The choice of the optimum cell length and I.D. for maximum sensitivity has to be left to the manufacturer who must carefully select and design the light source, optics, photo cell and electronics such that the maximum sensitivity can be realized while employing cell geometries that provide the minimum chromatographic band Table I1 Cell Dimensions that will Provide Minimum Dispersion for a Given Column System Column System
Column Length 25 cm Column Diameter 4.6 mm Dead Volume 3 ml Flow Rate 1 ml/sec Diffusivity of Solute in Mobile Phase cm2/sec
Cell Length (an) 0.25 0.50 1.00 1.50 2.00 2.50
Cell I.D. (mm) 0.59 0.50 0.42
0.38 0.35 0.33
Cell Volume
P 0.68 0.97 1.37 1.68 1.94 2.16
dispersion demanded by the column. Dispersion effects between column and detector can be extremely important where post column reactors are employed. In such systems a reagent is mixed with the column eluent that reacts with the eluted solute to render it detectable by the detecting system employed.
Under such conditions band
dispersion in the reactor volume between column and detector can be very serious. chapter.
The design of column reactors will be discussed in the next
Turbulence Turbulence in either the detector connecting tube or the cell itself can significantly increase the diffusivity of the solute by convective mixing and reduce band dispersion. However, turbulence in the detector cell itself
Other Factors Affecting Band Dispersion 27 also results in serious noise and thus reduces the detector sensitivity. It fcllows that turbulence in the detector should be avoided. Turbulence can be employed in the detector connecting tube to reduce dispersion by interfering with the regular geometry of the tube.
Providing the lamina nature
of the flow patterns in the tube are broken, this will result in convective mixing and reduce dispersion.
This can be achieved by crushing the tube
almost flat every 2 mm along its length between the column and detector. However, such a system easily becomes blocked, provides significant back pressure on the column and for these reasons an open tube of the correct dimensions is to be preferred. Factors that Indirectly Affect Band Dispersion The band width as drawn by the recorder on the chart paper may be significantly broader than that actually sensed by the detector due to a spreading effect that results from the time constant of the detector amplifier and the recorder itself.
Amplifiers and recorders have an
inherent time constant that arises from their respective circuit design but a further time constant is often purposely introduced to remove high frequency noise.
If this time constant is of commensurate period to the
time standard deviation of the eluted peak then the peak will suffer significant broadening. The effect of amplifier time constant on the shape of detected chromatographic peaks has been elegantly treated by Vandenheuvel ( 3 ) , Schmauch ( 4 ) , and Sternberg (5). For those readers wishing to study the effect of time constants on peak distortion in detectors they are recommended to read the work of Sternberg. Sternberg developed the following equation to describe the pcak shape after being distorted by an amplifier with a time constant of T'.
x
=
Xoe
-t/T'
t/T
- ( t-to) /2Tt2
/~lJe e
where X = the voltage output of the amplifier to the recorder Xo = a constant to = time at the peak maximum Tt = the time standard deviation of the eluted peak t = the elapsed time The explicit solution of the above equation is rather complicated, but the
28
Liquid Chromatography Detectors distortion of a normal Gaussian peak can be determined from the above equation by the use of a computer.
Consider a column 25 cm long and 4.6 mm in
diameter having a dead volume (V
0
)
of 3 ml, an efficiency (n) of 12,000
theoretical plates and operated at a flow rate of 1 ml/min (Q). From the plate theory the standard deviation of the dead volume peak in ml of mobile phase will be
-vO nk
3
-
(12,000)k
=
0.0274 ml
Thus the time standard deviation Tt
=
Q
=
0.0274 x 60
=
1.64 sec.
Thus taking values for the time constant of the amplifier of 0.6 and 1.5 secs and replacing T in the equation by the calculated value of 1.64 sec, t
= 1.5 sec
, T '
5
0.6 sec
Time Figure 1
Curves Demonstrating Peak Distortion Resulting from Significant Amplifier Time Constant
Effect of Amplifier Time Constant the shape ,.f
tiz
resultant peaks can be calculated.
Using a computer the
resultant curves were determined over the range of t = 4 . 9 2 sec to t = 8.2 sec
and the results obtained are shown in figure 1 together with the
original undistorted peak.
It is seen that the effect of a 1.5 second time
constant is to serioilsly distort the peak both by increasing the peak width and reducing the peak height.
This has resulted in a reduction of the
resolving power of the column and the sensitivity of the detector.
It is,
20
c 4J
‘c1
.,+
6
Y
m
a
10
C
.,+ 0)
m LI
U
n dp
0
1
2
Time Constant in sec Column Length 2 5 cm Column Diameter 4.6 nun I. D. Dead Volume 3 ml Figure 2
Curves Relating the Percentage Increase in Peak Width to the Amplifier Time Constant
therefore, important to determine the maximum time constant that can be employed with any given column. Using the same equation and again with the aid of a computer the percentage increase in peak width that will occur for a range of different time constants can be calculated. The results obtained
29
30
Liquid Chromatography htectors for the dead volume peak from colunms of the same dimensions as given previously but with efficiencies of 4,000; 12,000; and 20,000 theoretical plates are given
in
figure
If an increase in hand width of 5% is
2.
tolerated which will result in a reduction in peak height of about 5%, then the maximum permitted amplifier time constant will be 0.48 sec, 0.58 sec and 1 sec for peaks having time standard deviations of 1.26 sec, 1.64 sec and 2.85 sec
respectively.
Thus the maximum time constant that can be
tolerated is seen to be consistently about 35% of the time standard deviation of the peak. IF figure 3 the same calculations have heen carried out to provide similar curves but for a solute eluted at k' = 2. It is seen that the maximum time co!i;tants
that can be tolerated are now 1.32 sec,
1.76 sec and 3.04 sec for peaks having standard deviations of 3.82 sec,
4.93 sec and 8.54 sec respectively. It should also be noted that the maximum amplifier time constant is again consistently 35% of the time standard deviation of the peak. The problem can be approached in an entirely different inanner by using the principle of the summation of variances. The effect of amplifier time constant is to combine two functions, an exponential function and a Gaussian function. As these two functions describe physical phenomena that are not interacting in the sense that they proceed independently of one another, the variance of the combined function is equivalent to the sum of the variances of each individual function. The time variances of an exponential function of the form t 2 e- Tl is T1 and that of a Gaussian function 2
-t 2 T~~
is T~2
It follows that
+
Tt2 = T2
where T2 is the time variance of the resulting pcs4 as described by the recorder.
Now Tt2 is the time variance of the solute band leaving the
column and thus if a band width increase of 5% is considered acceptable (T = 1.05 Tt)
Effect of Amplifier Time Constant then Tt2 + T12 =
(1.05 T )2 = t
thus T12 =
1.103 T
2
t
0.103 T: 2
and T1 = 0.32 Tt Thus the maximum value of T1 that can be tolerated to maintain column resolution will be 32% of the time standard deviation of the eluted peak. The discrepancy between the value of 32% derived in this manner to the value of 35% derived from computer integration results partly from the fact that
in digital integration finite and not infinitesimal steps are used during
a,
m LI
V
Y
Time Constant in sec Column Length 25 cm,Column Diameter 4.6 mm I. D. Dead Volume 3 ml Figure 3
Curve Relating the Percentage Increase in Peak Width to the Amplifier Time Constant
31
32
Liquid Chromatography Detectors
integration procedure and partly due to the assumption that the variance of the distorted peak is at 0.607 of the peak height.
The more correct value
is, therefore, given by the second treatment, namely, 32%.
It follows that
for any column the maximum permitted time constant T' will be given by T'
0.32 VR
-
=
4
n Q Where V and n are the retention volume and efficiency of the peak and Q is R the flow rate through the column. VR should normally be taken for the dead
0
20
I0 Retention Volume ml 1. Column Efficiency
Figure 4
5,000 Theoretical Plates
2.
Column Efficiency
10,000 Theoretical Plates
3.
Column Efficiency
15,000 Theoretical Plates
Curves Relating Retention Volume to Amplifier Time Constant for Columns of Different Efficiencies
33
Recorder Time Constant volume peak as it has the smallest retention volume and the highest efficiency.
The above equation for T is applicable to columns of all
dimensions packed with any stationary phase and employed with any mobile phase.
In figure 4 curves are shown relating the minimum amplifier time
constant that can be permitted to maintain column resolution, to solute retention volume.
The individual curves are for columns of different
efficiency operated at a flow rate of 1 ml/min.
The appropriate time
constant for any column system, within the range given, can be obtained by interpolation. The Time Constant of the Recorder The potentiometric recorder does not have a time constant of the form normally associated with the amplifier which, in general, results from a capacity resistance network intrinsic in the amplifier circuit.
The
response of an amplifier to an instantaneous applied constant voltage is normally an exponential function of time, whereas for a
potentiometric
recorder the response is usually linearly related to time.
The linear
response results from the feed back circuitry incorporated in the sensor system which is necessary for stability. In figure 5 the recorded reading
is plotted against time for an instantaneous applied constant 9 mv signal. The recorder was the Honeywell Electronik 196, 10 mv potentiometric recorder operated with a chart speed of 1 cm/sec and having a specified balancing time of 0 . 5 sec.
It is seen that the response is approximately linear.
Now for a linear function the time variance T’, is given by
where tR is the time taken for the recorder to reach the applied voltage. In the example given T2 =
w2 =
0.035 sec2
and thus the time standard deviation, T = 0.19 sec. It is seen that if a recorder is employed having a balancing time of about 0 . 5 sec the contribution in time variance to the eluted peak is relatively small and by itself would not significantly increase the width of the eluted peak and reduce resolution.
However, it is yet another
34
Liquid Chromatography Detectors contribution to band dispersion and it is the sum of all the dispersive effects that take place subsequent to the column that ultimately limits the chromatoqrapher from realizing the actual efficiency that the column provides.
Figure 5 Response Curve of a Potentiometric Recorder
1
0.5
0.79
1.0
Adsorption Effects in the Detector When using a mobile phase that contains a small percentage of a polar solvent contained in a nonpolar solvent, for example ether or ethanol in heptane, a layer of the more polar solvent is often adsorbed on the surface of the quartz or glass detector cell.
If a solute is eluted that is more
polar than the ether or ethanol then the polar solvent is displaced from the
Summary of Dispersion Effects cell walls and is replaced by the solute.
This displacement results in a
spurious peak or, more often, a distorted peak shape.
The effect of
adsorption on the walls of the detecting cell cannot be treated quantitatively and when it occurs it should be eliminated by choosing an alternative mobile phase. Summary There are a number of band dispersing processes that occur in various parts of the detector and its associated electronics.
By careful design,
any one of these processes can usually be reduced to a level where it does not significantly affect column performance.
However the total dispersion
that takes place is a combination of all these effects and thus the T will be given by resulting time variance of an eluted peak '
2
T
Tc
=
2
+ T12 + T22 + T32 +
T4
2
where 2 . is the time variance of the peak leaving the column. Tc T12 is the time variance resulting from dispersion in the detector con-
necting tubes. T22 is the time variance resulting from dispersion in the detector cell. Tf
is the time variance resulting from the time constant of the amplifier.
2 is . the time variance resulting from the time constant of the recorder. T4
If a 5% increase in band width is tolerated then T2 = T12
+
(T12
+ T22 + T3
T22
+
T32
and
(1.05 C)t
+ T42
+
=
0.103
T42)t =
TC2
0.32 Tc.
If it is assumed that in the future a 25 cm long column 4.6 mn I.D. with a
35
36
Liquid Chromatography Detectors dead volume of 3 ml and operated at a flow rate of 1 ml/min will provide 20,000 theoretical plates then the time standard duration of the peak, Tc, will be about 1.27 sec. It follows that 2 (T1
+
2 T2
+
T32
+
=
T42)”
0.41 sec.
It is seen from the above equation that the detecting system has to be very carefully designed if the efficiency of 20,000 theoretical plates is to be realized.
It is an unfortunate fact that with many present day detecting
systems for liquid chromatography, an efficiency of even 10,000 theoretical plates from such a column would never be seen.
If 10,000 theoretical plates
were indeed achieved then the column would, in fact, be providing very much more than the 10,000 theoretical plates measured.
It is hoped that the
detector manufacturers will, in the future, design detectors having specifications that are appropriate to the column efficiencies presently obtainable and also be suitable for even higher efficiencies that will result from future developments in column technology.
REFERENCES 1.
A.
Klinkenberg, Gas Chromatography 1960, (Ed. R. P. W. Scott), Butter-
worth, London, 1960, p. 182. 2.
R. P. W. Scott and P. Kucera, J. Chromatogr. Sci., 9 (1971) 641.
3.
F. A. Vandenheuvel, Anal. Chem.,
35 (1963) 1193.
4.
L.
5.
J. C. Sternberg, Advances in Chromatography, (Ed. J. C. Giddings
S.
Schmauch, Anal. Chem.,
31 (1959) 225.
and R. Keller), Marcel Dekker, New York, Vol. 11, 1966, p. 206.