Anal. Chem. 1907, 5 9 , 2077-2884
2877
Refractive Index Effects in Cylindrical Detector Cell Designs for Microbore High-Performance Liquid Chromatography Robert E.Synovec Center for Process Analytical Chemistry, Department of Chemistry, University of Washington, Seattle, Washington 98195
Mlnlaturlratlonof the hlgh-performancellquld chromatography (HPLC) technology has placed strlngent requirements on detector flow cell geometry. Preservatlon of flow dynamics and separatlon effldency in microbore and capillary HPLC demand a cyllndrlcal detector flow cell. Refractlve index ( R I ) effects are studied by solving the optical ray tracing dlagram for cylindrical detector flow cell deslgns. R I detection In HPLC Is studied and optlmlzed as a functlon of eluent and caplllary flow cell conditions. The reversed-phase microbore HPLC separation and R I detectlon of aikylbenrenes at a detection ilmlt (3u)of 120 ng are demonstrated. Ramlflcatlons of the R I dependency in cylimlrlcai flow cells for other spectrogcoplc technlques are discussed.
Detectors for high-performance liquid chromatography (HPLC) have been given a significant amount of attention lately (I). The demands of microbore and capillary HPLC technologies on detector performance are well-known ( 2 , 3 ) . Often, the achievement of acceptable performance has necessitated laser-based spectroscopic detection (4). While the emphasis has been primarily on improving analyte detectability and detector selectivity ( 5 , 6 ) ,less emphasis has been placed on optical performance as miniaturization of detector flow cells proceeds. Yet, excellent approaches have been reported, including fluorescence (7,8), light scattering (9, IO), absorbance (11-13), and other optical detector designs (14, 15). One clever approach, first applied by Hershberger et al., employed the sheath-flow cuvette, which has squkre, parallel, light entrance and exit windows (16) The technique has been recently applied in laser-based light scattering measurements of small-volume samples (17-19). It was assumed that laser beam divergence across the sample stream due to refractive index (RI) is negligible (17). This is certainly true if the incident laser beam is not tightly focused and in flow cuvettes in which light rays of the laser beam are incident 90' to the cell window(s), as reported (17). Comparatively, a fairly sensitive RI detector was recently developed, using the sheath-flow technique (20). The changing light ray direction was monitored, induced by the gradient of the eluting solute concentration profile (20). The performance of this RI detector is strong evidence that RI effects in small-volume detection should not be ignored, especially in sample-modulated detection systems (producing solute concentration gradients, i.e., peaks) such as HPLC. Microbore and capillary HPLC methods are often utilized to provide diffusion coefficient information from detected solute retention time and peak width measurements. Diffusion coefficients have often been utilized in conjunction with other experimental techniques to correlate with useful solute properties, such as molecular weight, molar volume, and molecular shape (21,22). Accurate measurements of diffusion coefficients require minimizing flow dynamics aberrations. This places a tremendous burden on detector flow cell geometry. This burden is precipitated by the stringent require0003-2700/87/0359-2877$01.50/0
ments that dead volumes must be minimized and flow profiles must not be perturbed from the point of injection on past the detedor(s). This is particularly true in capillary hydrodynamic LC (23),in sedimentation field flow fractionation (24,25),and in the size exclusion HPLC of macromolecules (21). Thus, it is critical to maintain the flow geometry of the column and tubing as closely as possible to that of the detector flow cell. This is evidently one of the reasons for applying the sheathflow technique in detection (16). Further, it is necessary to reduce dead-volume contributions to improve separation efficiency. Unfortunately, the sheath flow technique is generally employed with a square flow cell, i.e., a geometry that disturbs solute flow profiles and increases dead volume. Further, the ultimate constraints imposed by capillary HPLC technology necessitate spectroscopic detection immediately following the column, i.e., ideally right on the same capillary where the protective cladding has been removed for detection (8). Thus, spectroscopic detection must be made with a capillary, i.e., cylindrical, flow cell geometry. This has been used in the past, with some success, although assumptions were required. For the cylindrical flow cell geometry, it is often assumed that optical path length is a constant over the entire beam profile. This requires an incident beam substantially more narrow than the capillary diameters, i.e., inside and outside. This, in essence, suggests that all incident light rays are normal to the capillary surface, which clearly becomes unrealistic as the capillary dimensions approach that of the incident laser beam diameter, especially if tightly focused. Another implicit assumption is that all light ray trajectories, through the liquid within the capillary, are approximately equal and relatively constant with time as the HPLC effluent passes through. This assumption is not valid, again, as capillary dimensions are miniaturized. Recently, this effect was observed and a method for measuring RI with a cylindrical flow cell design was reported by Bornhop and Dovichi (26). Although the observation was quite insightful, the RI effect on light-ray transmission through a cylindrical flow cell geometry was not def i e d in a quantitative solution. A quantitative solution would allow one to optimize the optical geometry for precise RI measurements in HPLC with a wide variety of eluent compositions and a wide variety of flow cell dimensions and compositions. Further, a quantitative solution allows one to study other important aspects of small-volume spectroscopic detecton, such as path-length dependencies as a function of incident beam position and intensity profile. And also, the quantitative solution provides insight into describing interference effects that occur in cylindrical capillary detection systems as cell dimensions approach incident beam dimensions. Thus, the work reported here is based upon the analytic geometry solution to the ray tracing diagram for a cylindrical capillary flow cell. The optics literature contains many papers with rigorous models for the interaction of light with objects (usually optical fibers) of cylindrical geometry (27-31), based upon a geometrical optics approach (32). Further, the results of an algorithm for calculating reflection and scattering contribu0 1987 American Chemical Society
2878
ANALYTICAL CHEMISTRY, VOL. 59, NO. 24, DECEMBER 15, 1987
By letting ml = tan 012 and bl = (D - r2 cos 8, tan 012), the intersection of eq 4 with the inside capillary interface (point 2) is given by the coordinates
-mlbl x2
=
+ (r12+ m12r12- b12)1/2 1
+ mI2
(5)
and y z = (r12- x22)1/2
Figure 1. Optical ray tracing diagram for a cylindrical flow cell: D , distance offset parallel to the flow cell center axis; n R I outslde the flow mi; n2, RI of the capillary; n3, R I of the Uqukl inside the capillary; r , , inside radius of flow cell; r 2 , outsMe radius of flow cell; Om, final deflected ray angle; 1, 2, 3, and 4, reference points for derivation.
Now, at point 2, a normal is drawn and an angle is calculated, 03, by using eq 3, 5, and 6 and analytic geometry
,,
tions with various cell geometries have been reported (33). The work presented here is concerned primarily with the dominating refraction effects. The study presented here provides tremendous insight for capillary flow cells. Application of the geometric solution for a position-optimized RI detector in microbore HPLC is demonstrated. Such an RI detector has been reported (27),but a quantitative solution for optimizing the detector was not addressed, which is essential for broadening the scope of this technique. Further, RI effects for other miniaturized spectroscopic detectors, especially path-length-dependent measurements, are more readily accounted for by the work presented.
THEORY When light is incident upon a transparent object, the direction and intensity of the transmitted light are dependent upon a variety of experimental factors. For a collimated laser beam, incident upon a cylindrical capillary (e.g., glass, quartz) filled with a solvent, a given ray of incident light can be pictured a distance D parallel from the center optical axis of the capillary, as shown in Figure 1. What one requires is the exact solution for the transmitted (deflected) angle, Om, for this incident ray of light, via four refractions, in terms of readily measured experimental parameters: D (ray displacement distance parallel to the capillary center axis), rl (inside radius of the capillary), r2 (outside radius of the capillary), n, (refractive index of the medium outside the capillary, i.e., air), n2 (refractive index of the capillary), and n3 (refractive index of the liquid within the capillary). Thus a solution to 8, = F(D, rl, r2, nl, n2, n3)is sought by using geometrical optics, as in Figure 1. The derivation will reference the four numbered points where refraction occurs, labeled in Figure 1. Note that the derivation is for what primarily happens to the light ray; i.e., secondary effects such as reflections at each interface, and their subsequent influence, are neglected. This is necessary to maintain simplicity at this point while experimental data validated this approach. At point 1 (Figure 11, the incident light ray produces an angle, 6,) with a normal drawn perpendicular to the capillary outside surface sin 8, = D / r 2 (1) Refraction occurs at point 1and according to Snell’s law (34) nl sin 8, = n2 sin O2 (2) With n2 > nl (with n, the RI of air), the light ray is bent toward the capillary center axis according to (3) o12 = 0, - e2 Now, one must make the transition from point 1to point 2. With Figure 1drawn in conventional x-y coordinates, the line between points 1 and 2 is given by y = x tan OI2 + (D- r2 cos 8, tan 012) (4)
(6)
O3 = sin-’ ( y 2 / r l )- 812
(7)
Application of Snell’s law yields at this interface
n2 sin O3 = n3 sin d4
(8)
Now, it is critical to determine if the light ray, going from point 2 to point 3 (Figure 1))is deflected toward or away from the capillary center axis. Figure 1depicts a ray bent slightly toward the center axis. Consideration of the geometry requires comparing O4 to the sum of O3 and BIZ. In general, for most spectroscopic detection systems, n3 < n2 and O4 > 03. Yet, the sum of O3 and 012was experimentally found to be greater than tI4. Thus, only one case is considered here, namely O3 + B12 > 04. This choice will be discussed in detail later. For this case, between points 2 and 3, the net angle 834 is calculated, relative to a center axis parallel drawn through point 2, by
834 = e3
+ 812 - 84
(9)
Now, the intersection at point 3 may be determined. The line in x-y coordinates between points 2 and 3 is given by y = x tan B,, + ( y z - x 2 tan (10) which depends upon eq 5,6, and 9. With m2 = tan 6, and b2 = (y2- x2 tan 8%)) the intersection of eq 10 with the inside capillary interface (point 3) is given by the coordinates x3
=
-m2b2 - (rI2 + mZ2rl2- b;)ll2 1
+ m?
(11)
and y 3 = (r12-
(12)
The intersection of the light ray at point 3 produces an angle, 05, with respect to a normal. The geometric solution provides, in radians 8, = n / 2
+ e,,
- exl
(13)
where 0,, is determined exactly from tan BXl = - x 3 / y 3
(14)
since in the coordinate system x 3 is negative. At point 3, Snell’s law provides
n3 sin O5 = n2 sin 0, (15) Now, one must compare Os to the difference, O5 - 034. Case A. Os < O5 - 034. Between points 3 and 4 (Figure l ) , the rsultant deflection, BS6, relative to a parallel, relative to the center optical axis, drawn through point 3 is given by 856 = 85 - 634 - 86 (16) The line between points 3 and 4 is given in x-y coordinates as y = -x tan 856 + (y3 + x 3 tan 856) (17) with m3 = -tan OM and b3 = (y3 + x 3 tan fIM). The intersection of eq 17 with the outside capillary interface (point 4) is given by the coordinates
ANALYTICAL CHEMISTRY, VOL. 59, NO. 24, DECEMBER 15, 1987
xq
=
-m3b3 - (rZ2+ m 32r2 - b3 ) '1' 1
+ m3'
W
(18)
C / RI
DETECTION
2879
0
HPLC
and y4 = (r22- x42)1/2
(19)
The intersection of the light ray a t point 4 produces an angle, 07,with respect to a normal at the capillary surface. The geometric solution, in radians, yields 6, = 7?/2 - OX2 - 656 (20) where Ox2 is given by tan
6x2
= -X4/Y4
(21)
n2 sin O7 = nl sin O8 (22) Finally, the deflected angle of interest, Om, is calculated by 8, = 68 - 8, - 656 (23) Case B: O6 > O5 - 634. In contrast to case A (eq 16), 656 is found to be
+ 634
= 66 - O5
(24) and the line between points 3 and 4 in Figure 1 is given by y = x tan OS6
+ (y3 - x 3 tan 6 5 6 )
(25)
in contrast to eq 17. The coordinates x4 and y4 can be calculated as in eq 18 and 19 with m3 = tan OM and b3 = y 3 - x 3 tan 056. In contrast to eq 20, O7 is given by 6, = 7?/2 - Ox,
+ 656
(26)
where OX2 is given by eq 21. Also, eq 22 still applies. Now, for case B, the deflected angle of interest, Om, is calculated by
6, =
68
- 6,
+
(27)
656
Thus, it is possible to calculate the deflected angle, Om, for a transmitted light ray, given an incident light ray, displaced a distance D from a capillary center axis and parallel to that axis. The solution is in closed form, depending upon readily obtained experimental parameters, and more importantly, readily understood in terms of the liquid refractive index, n3, within, and possibly flowing through, the cylindrical capillary. The derivation of 8, (eq 23 and 27) is consistent with the solution provided by applying the formula of Bougner (27,32)
sin-l
(-$
- sin-l
(&)I
(28)
which is a closed-form solution but is not quite as helpful for calculating optical path lengths for the syatem, vide infra. For D/rini
