Anal. Chem. 1998, 70, 51-57
Effects of Sample Dimension and Dye Distribution Characteristics in Absorption Microspectroscopy Haeng-Boo Kim, Shuri Yoshida, and Noboru Kitamura*
Division of Chemistry, Graduate School of Science, Hokkaido University, Sapporo 060, Japan
Models and the relevant equations giving optical absorbance of an analyte (i.e., dye) confined in a small volume with different shapes and sizes (sample dimension, D) are given, and factors governing absorbance are discussed. Dye absorbance is shown to be dependent on the threedimensional sample structure: films (1-D), tubes (2-D), and spheres (3-D). The magnitude of the dimension effect on absorbance is determined by the size of both the sample and the probe beam. Furthermore, dye distribution characteristics in 2-D and 3-D systems also affect absorbance; a dye distributed exclusively to the surface layer of a 2-D or 3-D sample gave an absorbance 0.50or 0.33-fold of that homogeneously distributed, respectively. If a dye is distributed in the inner volume of the sample alone, absorbance increases over that for homogeneous distribution. Effects of sample dimension, dye distribution characteristics, and the size of the probe beam in absorption microspectroscopy are discussed on the basis of the proposed models, and an experimental check of the models is presented. Spatially resolved analyses are of primary importance for studying chemical and physical characteristics of a wide range of heterogeneous materials, including biological tissues, polymers, and semiconductors. For this purpose, the use of an optical microscope is very convenient, and so far various microspectroscopies have been developed: absorption, fluorescence, and IR spectroscopies and Raman scattering.1,2 In particular, fluorescence microspectroscopy is a powerful means to detect an ultratrace amount of an analyte confined in a minute volume, and it has been applied to a detection method for separated components in capillary electrophoresis3 and in micrototal analysis systems.4 Single-molecule detection in various systems has been also achieved by fluorescence spectroscopy.5 Although fluorescence spectroscopy certainly has a high sensitivity, quantitation of an analyte concentration requires a knowledge of the relation(1) Rochow, T. G.; Tucker, P. A. Introduction to Microscopy by Means of Light, Electrons, X Rays, or Acoustics; Plenum Press: New York, 1994. (2) (a) Masuhara, H., De Schryver, F. C., Kitamura, N., Tamai, N., Eds. MicrochemistrysSpectroscopy and Chemistry in Small Domains; NorthHolland: Amsterdam, 1994. (b) Masuhara, H.; Sasaki, K. Anal. Chim. Acta 1995, 299, 309-318. (3) Baker, D. R. Capillary Electrophoresis (Techniques in Analytical Chemistry); Wiley-Interscience: New York, 1995. (4) For example, see: van den Berg, A., Bergveld, P., Eds. Micro Total Analysis Systems; Kluwer Academic Publishers: Dordrecht, 1995. (5) For example, see: Chen, D. Y.; Dovichi, N. J. Anal. Chem. 1996, 68, 690696. S0003-2700(97)00572-6 CCC: $14.00 Published on Web 01/01/1998
© 1997 American Chemical Society
ship between fluorescence intensity and the analyte concentration. On the other hand, absorption spectroscopy gives directly the analyte concentration if the molar absorptivity data are available. Nonetheless, reports on absorption measurements under a microscope are limited, one reason being the requirements for microscope-optics alignments to monitor transmitted light intensity through a minute sample, by which the quality of the data is determined. Previously, we developed a laser trapping-microspectroscopy system and demonstrated that absorption spectroscopy could be done for individual microparticles in solution.6-10 Spectroscopic characterization of sample solutions confined in microcapillaries or microchannels fabricated on substrates has also been explored. In order to apply absorption microspectroscopy to various samples and systems, factors governing spectroscopic data should be discussed in detail. The main aim of the present study is to discuss the sample shape effect on optical absorbance determined under a microscope. Analytical chemists deal with various samples, such as films (one-dimension, 1-D), tubes (2-D), and spheres (3-D), so that effects of a sample dimension (shape and size) on absorbance should be clarified. For absorption measurements, a probe beam possessing a finite beam diameter irradiates a sample with various shapes and sizes, so that a sample dimension would influence absorbance. Thus, discussion of the relationship between probe beam size and sample dimension is important to evaluate observed data. Furthermore, the three-dimensional distribution characteristics of an analyte in a sample are generally unknown, and this might also influence spectroscopic data. To the best of our knowledge, however, the effects of these factors on absorbance have been never reported. Therefore, we studied the effects of sample dimension and distribution characteristics of an analyte in a sample on optical absorbance. In this paper, we report models and mathematical equations giving absorbance in 1-D, 2-D, and 3-D systems and demonstrate that Lambert-Beer’s law is not necessarily warranted for microsamples. An experimental check of the present models and derived equations has been also explored. (6) Funakura, S.; Nakatani, K.; Misawa, H.; Kitamura, N.; Masuhara, H. J. Phys. Chem. 1994, 98, 3073-3075. (7) Kitamura, N.; Nakatani, K.; Kim, H.-B. Pure Appl. Chem. 1995, 67, 79-86. (8) Kitamura, N.; Hayashi, M.; Kim, H.-B.; Nakatani, K. Anal. Sci. 1996, 12, 49-54. (9) Kim, H.-B.; Yoshida, S.; Miura, A.; Kitamura, N. Chem. Lett. 1996, 923924. (10) Kim, H.-B.; Hayashi, M.; Nakatani, K.; Kitamura, N.; Sasaki, K.; Hotta, J.; Masuhara, H. Anal. Chem. 1996, 68, 409-414.
Analytical Chemistry, Vol. 70, No. 1, January 1, 1998 51
Chart 1. Schematic Illustrations of 1-D, 2-D, and 3-D Samples for Homogeneous Dye Distribution Model
Figure 1. Schematic illustration of a laser trapping-absorption microspectroscopy system.
MODELS AND THEORETICAL EQUATIONS General Assumptions. We assume that absorption measurements are performed by a system illustrated in Figure 1. A probe beam (typically, a Xe lamp) is introduced to an optical microscope and irradiates a microsample (i.e., dye-doped thin films, microtubes containing a dye solution, or dye-doped microspheres) through an objective. The transmitted light beam from the sample is passed through a pinhole, reflected by a half mirror set under the microscope stage, and led to a polychromator-photodetector set to determine its intensity (I). The incident light intensity (I0) is determined for the relevant dye-free sample. In the case of experiments on single microparticles dispersed in solution, a laser trapping technique is applied to choose and suppress the Brownian motion of the particle by introducing a 1064-nm laser beam (CW Nd:YAG laser) to the microscope coaxially with the probe beam.7,8 It is very important to irradiate the probe beam at the center of a microparticle, as described later. The I0 value is determined by the method mentioned above or by passing the probe beam in the solution phase near the particle under the same conditions. For precise and reliable absorption measurements under a microscope, the following points must be considered: (i) the size and shape of both the probe beam and the sample, (ii) the position of the probe beam irradiating a sample, (iii) refraction and/or reflection of the probe beam by the sample, and (iv) distribution characteristics of the dye in the sample. These factors become more and more important with decreasing size of the sample, and all these factors must be considered in discussing microspectroscopy data. As the first approximation, however, we make in the following discussion the assumptions that a parallel beam is employed for monitoring absorption and refraction/reflection of the probe beam by the sample is ignored. These assumptions are satisfied experimentally when a paraxial ray of a microscope objective is used as a probe beam.8 The size of a pinhole set under a microscope stage is also important to fulfill this condition. In the actual experiments, the effects of light refraction/reflection on an absorption spectrum become serious for samples having a nonflat surface. Therefore, this is discussed in a separate section of the paper. In the following, we describe models and theoretical 52
Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
equations for probe beam size, sample dimension (size and shape), and dye distribution effects on optical absorbance in microspectroscopy. Probe Beam Size Effects on Absorbance. We consider absorption measurements of a film (1-D), tubular (2-D), or spherical sample (3-D), in which a dye is distributed or incorporated homogeneously as illustrated in Chart 1. When a parallel probe beam with the radius of h is introduced to the center of the sample, the observed dye absorbance (Abs) is given as in eq 1,
Abs ) 2rC0ζ(h,r)
(1)
where and C0 are the molar absorptivity and the concentration of the dye, respectively. r represents the half-thickness of a film or the radius of a tube/sphere. ζ(h,r) is a correction term for an effective optical path length (lef), since 2r does not necessarily correspond to lef, depending on the sample shape, h, and r. In the present discussion, we define lef as
lef ) 2rζ(h,r) )
V πh2
(2)
where V is the volume of the sample irradiated by the probe beam. For a film sample, V is equal to the cylindrical volume produced by the probe beam (Vcyl ) 2πrh2) in the film, so that lef is essentially identical to 2r (ζ ) 1) as long as the probe beam is included in the X-Y plane of the sample,
lef(1-D) ) 2r
(2a)
For 2-D and 3-D systems, on the other hand, 2r does not
Chart 2. Schematic Illustrations of 1-D, 2-D, and 3-D Samples for Inhomogeneous Dye Distribution Model
Chart 3. Schematic Illustrations of Two-Phase Model (a) and Linear Gradient Model (b)
correspond to lef, since V is not identical to Vcyl even at h e r. For a given h, the surface curvature of a sample becomes important with decreasing r, and lef should depend on the h/r ratio. In such a case, the contribution of the ζ(h,r) term to Abs is of primary importance. V is dependent on the sample dimension and the relationship between h and r (h e r or h > r). Our calculations gave the following equations:
her
(
)
xr2 - h2 1 2 r π + 2hxr2 - h2 - 2r2 sin-1 lef(2-D) ) 2h r 3 2 2 3/2 4 r - (r - h ) lef(3-D) ) 3 h2
(2b)
Abs ) 2{(r - θ)Cin + θCs}
(3)
(2c) We define the partitioning ratio of a dye between two phases, E. For a 3-D system, as an example, E is given as
h>r lef(2-D) )
to r and the probe beam passes the center of the sample, Abs is given as in eq 3.
2 1 r2 πr + 2 r2xh2 - r2 - 2 xh2 - r2 2 h h
xh2 - r2
r sin-1 4 r3 lef(3-D) ) 3 h2
/3π{r3 - (r - θ)3}Cs
4
E)
h
4
3
/3πr C0
Cs ) {1 - (1 - R)3} C0
(4)
(2d)
(2e)
where R ) θ/r and C0 represents the total concentration of the dye in the sample. All of the dye molecules are included in the sample volume,
/3πr3C0 ) 4/3π(r - θ)3Cin + 4/3π{r3 - (r - θ)3}Cs
4
Clearly, the observed dye absorbance depends on h and r as well as on the sample dimension. Effects of Inhomogeneous Dye Distribution on Absorbance. When a dye distributes inhomogeneously in a sample, other effects of sample dimension on Abs are expected. Here, we consider a simple but important case, where a dye is distributed to both the surface layer (thickness of θ) and the inner phase of the sample, as illustrated in Chart 2. A typical example for this is an adsorption layer on a tube/particle surface, a polymeric resin wall of a capsule, or a liquid phase on a spherical chromatographic resin bead. Two-Phase Model. First, we assume a two-phase model, in which the boundary between the surface layer and the inner phase is very sharp, and the concentrations of a dye distributed to the surface layer and the inner phase are constant at Cs and Cin, respectively (Chart 3a). When h is sufficiently small compared
so that we obtain
C0 ) (1 - R)3Cin + {1 - (1 - R)3}Cs
(5)
Similarly, the equations giving E and C0 in an n-dimensional system are derived as in eqs 6 and 7, respectively.
Cs E ) {1 - (1 - R)n} C0
(6)
C0 ) (1 - R)nCin + {1 - (1 - R)n}Cs
(7)
These relations afford Cin and Cs in n-dimension as in eqs 8 and Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
53
9, respectively.
system are given as in eqs 14a and 14b, respectively.
1-E C0 (1 - R)n
(8)
E C0 1 - (1 - R)n
(9)
Cin ) Cs )
Cin )
Cs )
[
n+1 E+ R{(2 - R)n-1 + n - 1}
{
}
RE 1-E + C0 ) 2rξC0 n-1 (1 - R) 1 - (1 - R)n
(10)
E 1-E + 1-R 2-R
(11b)
ξ(3-D) )
1-E E + 2 (1 - R) 3 - 3R + R2
(11c)
(0 e |t| e |r - θ|)
(
)
Cs - Cin Cs - Cin |t| + Cs Rr R (|r - θ| < |t| e |r|) (12)
Therefore, Abs can be calculated from eq 13.
∫ C(t) dt
Abs ) 2
{
r
0
}
1 ) 2 r(1 - R)Cin + θ(Cs + Cin) 2
(13)
As in the case of the calculations of Cin and Cs in the two-phase model, Cin and Cs in the linear gradient model for an n-dimensional 54
Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
(15)
where ξ′ is a correction term for Abs caused by the linear gradient dye distribution in a sample:
ξ′ )
(n + 1) 1-E 1 E+ + (1 - R)n-1 2 (2 - R)n-1 + n - 1 (n - 1)R2(2 - R)n-2 1 (1 - E) (16) 2 {(2 - R)n-1 + n - 1}(1 - R)n
The ξ′ value for a 1-D, 2-D, or 3-D system is as follows:
ξ′(1-D) ) 1
Linear Gradient Model. The two-phase model mentioned above might be too simple and not realistic. Thus, we consider here a linear gradient model, where a dye distribution is assumed to vary linearly from the sample surface to the inner phase, as illustrated in Chart 3b. Although this is also oversimplified compared to actual systems, the present model is sufficient to demonstrate sample dimension effects on Abs. In Chart 3b, Cin has the same meaning as in the two-phase model, while Cs represents the dye concentration at the surface of the sample. The dye concentration at a given position from the center of the sample (C(t)) is then given as
C(t) )
Thus, substitution of eqs 14a and 14b into eq 13 gives
(11a)
ξ(2-D) )
C(t) ) Cin
]
Abs ) 2rξ′C0
where ξ is a correction term for Abs caused by inhomogeneous dye distribution in a sample. Equation 10 indicates that the ξ value is a function of E, R, and sample dimension (n). Therefore, even in the case that the size of the sample (r) and the total amount of dye (C0) are the same, Abs depends on distribution characteristics of the dye as well as on sample dimension (n):
ξ(1-D) ) 1
(14a)
(2 - R)n-2(nR - 2) - (n - 1) (1 - E) C0 (14b) {(2 - R)n-1 + n - 1}(1 - R)n
For the two-phase model, consequently, we derived a general formula for Abs as
Abs ) 2r
1-E C0 (1 - R)n
(17a)
ξ′(2-D) )
R2(1 - E) 3E 1-E + + 1 - R 2(3 - R) 2(3 - R)(1 - R)2
ξ′(3-D) )
2E 1-E + + (1 - R)2 (2 - R)2 + 2 R2(2 - R)(1 - E) {(2 - R)2 + 2}(1 - R)3
(17b)
(17c)
For the 1-D system, the ξ′ value is unity, as in the case for the two-phase model, so an inhomogeneous dye distribution in a film does not influence absorbance. For the 2-D and 3-D systems, however, Abs is shown to be dependent on the three-dimensional distribution characteristics of the dye in the sample. RESULTS AND DISCUSSION We derived the theoretical equations giving dye absorbance for 1-D, 2-D, and 3-D samples with different dye distribution characteristics. In the following, we discuss general features of the effects of sample dimension and dye distribution characteristics on absorbance. Probe Beam Size Effects on Absorbance. The relationship between ζ and h/r for a 2-D or 3-D system, calculated on the basis of eqs 2b-e (homogeneous dye distribution model), is shown in Figure 2. As described in the preceding section, the ζ value for a 1-D system is unity as long as the probe beam is included in the X-Y plane of the sample, so that the Lambert-Beer’s law always holds for film samples. For 2-D and 3-D systems, ζ is almost unity when the probe beam size is small enough compared to r (h/r ≈ 0), and thus correct absorbance can be obtained under such conditions. As
Figure 2. Sample dimension effect on dye absorbance (homogeneous dye distribution model): 2-D, broken curve, and 3-D, solid curve.
demonstrated clearly in Figure 2, however, the ζ value decreases with increasing h/r. Furthermore, the h/r dependence of ζ is different between 2-D and 3-D systems: this is the sample dimension effect. At h ) r, as an example, the ζ value for the 2-D or 3-D system is π/4 (0.785)- or 2/3 (0.667)-fold of that at h/r ) 1, respectively. In such a case, therefore, Abs does not represent the correct amount of dye in the sample, and a correction of Abs by ζ is absolutely necessary for quantitation. If a 1% error in determining correct Abs is accepted, the h/r value should be smaller than 0.25 or 0.20 for the 2-D or 3-D sample, respectively. These results indicate that the relationship between probe beam size and sample dimension is quite important when discussing absorbance determined under a microscope. Absorption measurements of microtubular samples are very important as the detection method for separate components in high-performance liquid chromatography and in electrophoresis, and those of microspheres are applied to solid-phase concentration/quantitation of an analyte. The probe beam size effects on observed Abs should thus be considered carefully in such studies. Effects of Inhomogeneous Dye Distributions on Absorbance. Distribution characteristics of a dye in a sample also strongly influence Abs. In Figure 3, the ξ values calculated from eq 11 (two-phase model) were plotted against R at several E values under the assumption of h/r ≈ 0. The linear gradient model (eq 17) gave results that were almost identical with those in Figure 3, so that the data are not shown here. For a 1-D system, both ξ and ξ′ are unity, regardless of E and R (eqs 11a and 17a). An inhomogeneous dye distribution in film samples, therefore, does not affect Abs. This is readily understood, since Abs is determined by the number of dye molecules in the optical path, and thus Abs observed in a 1-D system is essentially independent of such inhomogeneities along the probe beam axis (Z-axis). To the contrary, ξ values in the 2-D and 3-D systems depend highly on both E and R, as can be clearly seen in Figure 3. When a dye is distributed exclusively in the inner phase (E ) 0) and the thickness of the surface layer (θ) is sufficiently thin compared to r (R ) θ/r 0.1). In such cases, Abs no longer represents the correct dye concentration, and a correction of Abs by ξ is necessary for quantitation. For a given r, an increase in θ renders a decrease in the radius of the inner
Figure 3. Sample dimension effect on dye absorbance at several E values (two-phase dye distribution model): 1-D, dotted line; 2-D, broken curve; and 3-D, solid curve.
phase (volume reduction of the inner phase). Since the dye is not distributed to the surface layer, the dye is concentrated to the inner phase, with the surface layer acting as a cell spacer for conventional absorption measurements: this is the concentration effect. This leads to an apparent increase in Abs. In the 3-D system, the inner phase is surrounded by a spherical spacer layer, so that the concentration effect is more pronounced compared with that in the 2-D system. Actually, the R dependence of ξ in the 3-D system is steeper than that in the 2-D system, as can be clearly seen in Figure 3a: this is the sample dimension effect. When a dye is distributed in the surface layer alone (Figure 3c, E ) 1), on the other hand, ξ in the 2-D or 3-D system is determined by the second term of the right-hand side of eq 11b or 11c, respectively. Similar to the case at E ) 0, ξ is independent of R at < 10-2. It is noteworthy, however, that the absolute ξ value is not equal to unity, and, with decreasing R, ξ(2-D) and ξ(3-D) reach 0.50 and 0.33, respectively. For the linear gradient model, analogous calculations indicate that the ξ′ values also converge to ξ′(2-D) ) 0.50 and ξ′(3-D) ) 0.33 at R f 0. In the case of the 2-D and 3-D systems, therefore, an inhomogeneous dye distribution renders an apparent decrease in Abs, in marked contrast to the results at E ) 0. This is because the dye, distributed outside the probe beam, does not contribute to optical absorption. Therefore, the number of dye molecules within the optical path decreases either with a decrease in θ or an increase in r (i.e., a decrease in R), which brings about a decrease in Abs. With increasing R, the mole number of the dye contributing to optical absorption increases, so that the ξ value increases and reaches unity at R ) 1.0. These discussions indicate that the number of dye molecules contributing to an absorption event depends on the sample structure. The effect is much larger for the 3-D system compared to that for the 2-D system: this is the sample dimension effect. At 0 < E < 1.0, ξ varies from 0.33 to ∼102, depending on E and the sample dimension, as shown in Figure 3b. Relationships Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
55
Figure 5. Capsule diameter dependence of absorbance (427 nm) for DO13/toluene microcapsules dispersed in water.
Figure 4. E dependencies of dye absorbance at several R values (two-phase dye distribution model): 1-D, dotted line; 2-D, broken line; and 3-D, solid line.
between E and ξ at several R values are also summarized in Figure 4. When θ is thin enough (R ) 10-3), the ξ value decreases linearly from 1.0 at E ) 0 to 0.50 or 0.33 at E ) 1 for the 2-D or 3-D system, respectively (Figure 4a). At E ) 0, an increase in R brings about concentration of the dye in the inner phase, so that ξ becomes larger than unity, as described above. However, ξ decreases linearly with an increase in E, irrespective of the R value for both the 2-D and 3-D systems (Figure 4). Thus, the variation of ξ with E is pronounced for a larger R ()θ/r) value. These results also demonstrate that Abs observed under a microscope is dependent strongly on both sample dimension and dye distribution characteristics. Diameter Dependence of Absorbance for Individual Microparticles. The above discussions indicate that dye distribution characteristics in a sample should be known for quantitative analyses of microspectroscopic data. However, this is generally very difficult for microsamples. Although confocal fluorescence microspectroscopy can determine a three-dimensional spatial distribution of a fluorescent dye in materials, the method is still uncommon and cannot be applied to nonfluorescent samples. For dye-doped microparticles, as an example, a possible approach for this is to study a diameter (d ) 2r) dependence of dye absorbance. An Abs vs d plot should be linear if the dye is distributed homogeneously in the particle (Cin ) Cs ) C0 in eq 5 or 12), and the slope of the plot gives C0 (ξ ) 1). According to the present results, the slope of the plot is dependent on E if the two-phase or linear gradient model is applied. When the surface layer is thin enough (R < 10-2, Figure 3), R in eq 11c can be ignored, so that we obtain
2 Abs(3-D) ) 1 - E dC0 3
(
)
(18)
If E is independent of the size of a sample, Abs increases linearly with an increase in d for a given C0, so the E value can be 56 Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
Figure 6. Capsule diameter dependence of absorbance (424 nm) for ZnTPP/toluene microcapsules dispersed in water (data were taken from ref 9).
estimated from the slope of the plot. As discussed in the preceding section, ξ becomes unity when the dye distributes exclusively in the inner phase (E ) 0), while it should be 0.33 at E ) 1.0. When a d dependence of Abs does not obey eq 18, E should be dependent on the particle diameter. In order to test this experimentally, we studied the size dependence of Abs for individual microparticles on the basis of a laser trappingmicrospectroscopy technique. Figures 5 and 6 show typical examples of the diameter dependence of dye absorbance obtained for single melamine resin wall microcapsules containing a toluene solution of Disperse Orange 13 (DO13, azo dye) and zinc tetraphenylporphyrin (ZnTPP), respectively. The sample was dispersed in water, and single capsules were laser-trapped for absorption measurements. A paraxial ray of the microscope objective (×100) was used as a probe beam, and its diameter was set at ∼1 µm (h ) 0.5 µm, diameter of the entrance pinhole ) 100 µm).11 In the case of DO13, Abs at 427 nm increased linearly with d of the capsule. The value of DO13, calculated from C0 (0.01 M) and the slope of the plot ((427) ) 2.7 × 104 M-1 cm-1), agreed with that determined in a homogeneous toluene solution by a separate experiment (2.1 × 104).12 Therefore, we concluded that DO13 is distributed homogeneously in the inner toluene solution of the capsule. (11) For details of preparation and absorption measurements of the microcapsules, see ref 9. The dye was not distributed to the water phase, as confirmed by separate experiments. (12) The value determined for the capsules was slightly larger (∼30%) than that in a homogeneous toluene solution. This is not characteristic of the DO13 microcapsules, and analogous result have been observed for pyrene/ melamine resin capsules (Koshioka, M.; Misawa, H.; Sasaki, K.; Kitamura, N.; Masuhara, H. J. Phys. Chem. 1992, 96, 2909-2914). We suppose that the dye/toluene solution is concentrated during synthetic procedures (heating at 65 °C for 2 h).13
For the ZnTPP microcapsules, on the other hand, although Abs increased with d, that extrapolated to d ) 0 did not cross the original point of the plot.9,13 In Figure 6, the broken and solid lines represent the slopes predicted by eq 18 at E ) 0 and 1, respectively (C0 ) 1.9 × 10-3 M, (424) ) 5.4 × 105 M-1 cm-1). Clearly, the observed data agreed with neither those expected from E ) 0 nor from E ) 1. Furthermore, a variation of E between 0 and 1 could not reproduce the results in Figure 6. An inspection of the data indicates that Abs varies from that expected at E )0 (broken line) to the value at E ) 1 (solid line) with increasing capsule diameter. The results imply that the dye is likely to distribute to the inner toluene solution for smaller capsules, while it tends to distribute to the capsule wall with increasing d. One possible explanation for this assumes a d dependence of E. We suppose that the partitioning ratio of ZnTPP between the resin and toluene phases of the capsule is determined during polymerization of the melamine resin wall around a dye/toluene droplet.9 Since polymerization is expected to proceed relatively faster for smaller capsules (droplets) compared to that for larger capsules (droplets), ZnTPP will be likely to distribute to the resin wall for larger capsules. If this is the case, the d dependence of Abs should show a sigmoidal curve. Actually, we succeeded in analyzing the results in Figure 6 by using such an assumption. Further detailed discussion on this will be reported in a separate publication.13 The present results demonstrate that a discussion on a d dependence of Abs based on the model equations described in the preceding sections is quite important, and this can provide information on dye distribution characteristics in microparticles. At the present stage of investigations, we do not have any data on the d dependence of Abs in a 2-D system. However, discussions analogous with those mentioned above are possible, so that detailed information on dye distribution characteristics in a 2-D system will be also obtained through such an approach. Effects of Refraction/Reflection of a Probe Beam in Microspectroscopy. Refraction and/or reflection of a probe beam by a small tube/spherical sample becomes important when d ()2r) decreases to the order of micrometers. However, such effects on an observed spectrum and/or Abs have been ignored in the above models and discussions. Since a theoretical discussion on the effects is far beyond the scope of the present study, we discuss qualitatively the light refraction/reflection effects in microspectroscopy. When the h/r value is sufficiently small, the sample surface irradiated by a probe beam is considered to be flat, even for 2-D and 3-D samples. Therefore, refraction/reflection of the beam by the sample can be minimized or ignored as in the case of film samples. In laser trapping-absorption microspectroscopy of single dye-doped microparticles, actually, a correct absorption spectrum of the particle with r ) 35-70 µm can be obtained when h is set 0.5 µm (h/r ≈ 0.007-0.014), even by the determination of I0 with the probe beam being passed in the solution phase near the particle. However, the spectral band shape observed for the particles with r < 4 µm (h/r > 0.1) under the same experimental conditions was distorted considerably. This is because light refraction/reflection cannot be corrected by the I0 value determined by the above method. A correct spectrum can be obtained
only when I0 is determined for a dye-free particle with a size identical to that of the sample.8 Under the condition of h/r > 0.1, refraction/reflection of the probe beam by a sample particle cannot be ignored, and this effect on the spectrum should be compensated by the determination method of I0. Light refraction/reflection is also caused when a probe beam is not passed at the center of a tube or spherical sample. For the dye-doped microparticles, a displacement of a probe beam from the center of the particle led to considerable distortion of the spectrum as well as to high background signals.8 For precise and reproducible measurements, an optical trapping technique, capable of suppressing the Brownian motion of a particle in solution, is indispensable. Furthermore, since a probe beam can always pass the center of a particle if the probe beam is introduced to the microscope coaxially with a trapping laser beam, laser trapping has a high potential to conduct absorption measurements of microparticles under a microscope. By optimization of these factors, absorption microspectroscopy of single particles as small as r ) 4 µm has been achieved.7-9,13 CONCLUSIONS In this paper, we demonstrated that sample dimension (size and shape) and solute distribution characteristics in a sample strongly affect the optical absorbance. Usually, sample dimension is not considered in absorption spectroscopy. However, the present results indicate that this is very important in order to evaluate C0 by absorption microspectroscopy. In particular, microtubes and microspheres are widely used in analytical sciences, so that special care should be taken in quantitative discussion of optical absorbance for these samples determined under a microscope. Furthermore, dye distribution characteristics in a sample (2-D and 3-D) are shown to be estimated by a d dependence of Abs. Generally, a determination of the spatial distribution of a solute in a microsample is difficult, so that the present approach is very useful in various systems. The present models and assumptions are quite simple. In particular, the whole discussion is made on the basis of the assumption of the use of a parallel probe beam. Clearly, this is oversimplified since a probe beam under a microscope is more or less conical, even with the use of a paraxial beam. In order to reveal such an effect on Abs, further discussion based on detailed ray optics will be required. However, since the aim of the present study is to demonstrate the importance of sample dimension and dye distribution characteristics in determining Abs, we consider that the present results deliver a very useful means to discuss microspectroscopy data. Experimental applications of the present models and equations to various samples are now in progress in this laboratory and will be reported in separate publications.13 ACKNOWLEDGMENT N.K. is grateful for a Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan (08404051) for partial support of the research. Received for review June 3, 1997. Accepted October 14, 1997.X AC970572A
(13) Kim, H.-B.; Yoshida, S.; Miura, A.; Kitamura, N. Anal. Chem. 1998, 70, 111-116 (in this issue).
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Abstract published in Advance ACS Abstracts, December 1, 1997.
Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
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