Langmuir 1993,9, 282Ei-2831
2825
Picosecond Fluorescence Studies of Energy Transfer on the Surface of Poly(buty1 methacrylate) Latex Particles Kenichi Nakashima,l Yuan Sheng Liu, Ping Zhang, Jean Duhamel, Jianrong Feng, and Mitchell A. Winnik’ Department of Chemistry and Erindale College, University of Toronto, 80 St. George Street, Toronto, Ontario, Canada M5S 1Al Received February 18,1993. In Final Form: July 9, 199P
Direct energy transfer experiments were carried out with rhodamine 6G and malachite green adsorbed on the surface of poly(buty1methacrylate) latex spheres. A carefulanalysis of the fluorescence decay data using the Klafter-Blumen model showsthat the latex surface is smooth on a length scale of 8 nm. Problems encounteredinthe fractal dimensionanalysis,especiallythe uncertaintyof the recovered apparent dimension, and the dependence of the recovered parameters on the selection of models, are discussed in detail. Fluorescence quenching experiments can provide important information about polymer and colloidal systems. AB our knowledge of these processes grows, new types of applicationsemerge. One clam of applicationscomes from analysis of fluorescence quenching kinetics in systems in which no mass diffusion occurs during the excited-state lifetime. In these systems, if the rate of reaction between a pair of reactants [w(r)l depends upon the distance r between them, then the decay rate of the excited fluorophores will depend upon the spatial distribution of the reactant pairs in the system. We are specifically interested in direct nonradiative energy transfer (DET) by the dipole coupling (Farstel) mechanism? for which w(r) Y (Ro/r)e, and Ro is a characteristic distance dependent only on the spectroscopic properties of the reactants. A very special situation occurs if the reactants are confiied to a restricted The term “restricted geometry” refers to a confining space characterized by at least one dimension not much larger than Ro. The confining space can be a surface, a thin cylinder, a small sphere, or an irregular geometry with a thickness comparable to Ro. Under these circumstances, the majority of reaction pairs are confined in this space. They have a different pair distribution than pairs in bulk. As a consequence, the reaction kinetics become sensitiveto the size and shape of the confining space. DET between adsorbed or incorporated dyes has been used to probe the microsctructure of various materials. There have been extensive investigations of microdomains in porous glass,6J and the pore structure in silica.a11 The work of Drake at Abstract published in Advance ACS Abstracts, October 1,1993. (1) On leave from the Laboratory of Chemistry, College of Liberal A r b , Univeristy of Saga, 1 Honjo, Saga 842, Japan. (2) (a) FWter, Th.Z . Naturforsch., A 1949,4,321. (b) Birks, J. B.; Photophysics of Aromatic Molecules; Wiley: New York, 1970. (3) Blumen, A.; Klafter, J.; Zumhofen, G. J. Chem. Phys. 1986, 84, 1387. (4) Yang, C. L.; Evesque,P.; El-Sayed,M. A. J. Chem. Phys. 1986,82, 3442. (5) Klafter, J.; Drake, J. M.; Reaction Dynamics in Restricted Geometries; Wiley: New York, 1990. (6) Even, U.; Rademan, K.; Jortner, J.; Manor, N.; Fteisfeld, R. Phys. Reu. Lett. 1984,52,2164. (7) Dozier, W. D.; Drake, J. M.; Klafter, J. Phys. Rev. Lett. 1986,56, 197. (8)Levitz, P.; Drake, J. M.; Klafter, J. J. Chem. Phys. 1988,89,5224. (9) Levitz, P.; Drake, J. M. Phys. Rev. Lett. 1987,58, 686. (10) Turro, N. J.; Zimmt, M. B.; Gould, I. R. J. Am. Chem. SOC.1986, 107,5826. (11) (a) Rojamki, D.; Huppert, D.; Bale, H. D.; Xie, D.; Schmit, P. W.; Farin, D.; Seri-Levy,A.; Avnir, D. Phys. Rev. Lett. 1986,56, 2505. (b) Pines-Rojansky,D.; Huppert, D.; Avnir, D. Chem. Phys. Lett. 1987,139, 109.
Exxon, with LevitzeJ2and in collaboration with Klafter,6I8 deserves special mention. Other applications include studies of the internal morphology of submicrometer nonaqueous dispersion particle^,'^ and the internal structure formed spontaneously when one casts a film from a solution containing both a homopolymer and a graft copolymer.l4 A classic example analyzed theoretically by Levitz et al.8 is the case of energy transfer between dye pairs adsorbed to the surface of a sphere. If the radius of the sphere is much larger than Ro, then the surface of the sphere appears flat to the adsorbed dyes. One anticipates energy transfer kinetics characteristic of pairs distributed in two dimensions. If the radius of curvature of the sphere becomes comparable to, or smaller in size than, Ro, then the reactants experience the finite size of the sphere. The reaction rate changes, and a crossover is said to occur. Experiments of this type were carried out by Yamazaki and co-workers16 for dyes adsorbed to the surface of unilamellar phospholipid vesicles. The vesicle radii were much larger than the ROvalues of their dye pairs. Their results indicate that at low levels of dye adsorption the surfaceswere smooth. At higher dye coverage, changes in the fluorescence decay parameters were interpreted in terms of a nonuniform (fractal) structure to the dye adsorption pattern on the surface. A very similar application of DET that comes to mind is the direct examination of the surface of latex particles dispersed in water. Electron microscopy (EM) experiments show that the surface of some latex particles is smooth16 and that other particles have a raspberry-like texture.17 The resolution of EM experimentson polymeric materials is limited. In addition, one never knows how sample preparation affects the sample before the observation is made. It would be very attractive if one could (12) (a) Drake, J. M.; Klafter, J.; Levitz, P. Science 1991,261, 1574. (b) Drake, J. M.; Klafter, J. Phys. Today 1990,43 (May), 46. (c) Levitz, P.; Drake, J. M.; Klafter, J. Chem. Phys. Lett. 1988,148,557. (d) Drake, J. M.; Levitz, P.; Sinha, S. K.; Klafter, J. Chem. Phys. 1988,128, 199. (13) Pekcan,0.;Winnik, M. A,;Croucher,M. D. Phys. Rev. Lett. 1988, 61,641. (14) Pekcan, 0.;%an, L. S.; Winnik, M. A.; Croucher, M. D. Macromolecules 1990,23, 2210. (15) (a)Tamai,N.;Yamd,T.;Yamazaki,I.;Mataga,N.In Ultrataut Phenomena V; Fleming, G. R., Siegman, A. E., E&.; Springer-Verlag: Berlin, 1986; pp 449-453. (b) Yamazaki, I.; Tamai, N.; Yamazaki, T. J. Phys. Chem. 1990,94,516. (16) Winnik, M. A.; Zhao, C.-L.; Shaffer, 0.;Shivers, R. 5. To be published. (17) Bassett, D. R. In Science and Technology of Polymer Colloids; Poehlein, G. W., Ottewill,R. H., Goodwin,J. W., Eds.; NATO AS1 Seriee; Martinus Nijhoffi Boston, 1983; Vol. 1, p 220.
0143-1463/93/2409-2825$04.00/0Q 1993 American Chemical Society
Nakashima et al.
2826 Langmuir, Vol. 9,No. 11,1993
Table I. Recipe for Latex Preparation first stage second stage
MALACHITE GREEN
Figure 1. Structural formulas for the dyes R6G and MG.
simply bind a pair of dyes to latex particles in water, and interpret the energy transfer kinetics in terms of the roughness or smoothness of the particle surface. For typical values of Ro, this kind of experiment would be sensitive to roughness on the scale of 2-15 nm. In latex systems,particularly with cationic dyes and anionic latex, dye binding might also be sensitive to the distribution of charges on the latex surface. Here we report our first results on the use of DET to study surfacestructure in latex particles. We employ poly(butyl methacrylate) (PBMA) microspheres,prepared by emulsion polymerization with an anionic initiator (potassium persulfate, KPS). FollowingYamazaki,15we have tried to use either rhodamine 6G (R6G) or rhodamine B (RBI as the energy donor. For reasons which will be discussed below, our apparent dimension analysis was carried out using the resulta of R6G experiments only. We use malachite green (MG) as the energy acceptor, and follow the energy transfer kinetics through careful monitoringof the donor fluorescencedecay profile as a function of the amount of adsorbed acceptor. All three dyes contain a single positive charge and are otherwise reasonably hydrophobic in composition. The structures of R6G and MG are given in Figure 1.
Theory DET describes the transfer of energy from a donor (D) to an acceptor (A) without intervening donor-donor transfer. The rate of this transfer depends upon the separation distance r between D and A, W ( r ) = (3/2)K2T~-'(Rdr)s (1) with s = 6 for dipole-dipole interaction. Here TD is the unquenched donor lifetime, and K~ is an orientation factor containing the angular dependence of the dipole-dipole interaction. ROis the critical transfer distance,
where N Ais Avogadro's number, n the refractive index of the medium, @D the donor fluorescence quantum yield, FD(u)the fluorescence intensity of the donor at frequency v, and ~ A ( v the ) molar decadic extinction coefficient of A. A general model of isotropicDET was derived by Klafter and Blumen (KB).18 For a small fractionp of sites occupied by acceptors, the survival probability @ ( t )of the excited donors is given by
where pdr) describes the spatial distribution of acceptors around a donor assumed to be located at the origin. When p o ( r ) follows a fractal model, evaluation of the integral leads to (18) (a) Klafter, J.; Blumen, A. J.Phys. Chem. 1984,80, 875. (b) J. Lumrn 1985, 34, 77.
water BMA"
90mL 2 mL
KPS
0.03 g
ammonia(29%) temperature time
0.007g 80 OC
a
l h
water KPSb ammonia (29%) BMA temperature time
10 mL 0.02 g 0.007 g 10 mL 8OOC
5h
Butyl methacrylate. Potassium persulfate
where a is the size of a site, F is a geometric factor, r ( n ) is the complete gamma function, and g' is an isotropic angular factor, g1 = ( ( 3 / 2 ) ( ~ ~ which ) ) ~ / depends ~, upon how the dipoles are averaged. For isotropic DET, two situations one commonlyencounters involve either rapidly reorientingdipoles ( ( K ~ =) 2/31 or random pairs of immobile dipoles ( ( K ~ )= 0.475). Equation 4 constitutes a generalization of the Fijrster equation. The prefactor preceding the g T ( x ) term in eq 4 is equal to the number of acceptors within a critical radius Ro. For a homogeneous Euclidean medium (51 = d), one finds8JgF = ?r for 2 = d = 2, and F = 4%/3for a = d = 3. The full expression for energy transfer in two dimensions,which will be of interest to us in interpreting experimental data, is given by the expression @ ( t )= eXp[-t/rD
- (P/a2)?rR02gIr(2/3)(t/r~)1'3] (5)
In the case of donors and acceptorsadsorbed to the surface of a sphere,one has to take into account the finite geometry of the system in evaluating the integral in eq 3. For a sphere of radius R , Levitz et al. have shown that one recovers eq 5 when R >> Ro. This will normally be the case for latex particles with smooth surfaces. When R = Ro, one has a crossover at long times in the decay, due to the finite size of the sphere. When consideringthe decay profile for dye adsorption onto a rough sphere, other features of the geometry come into play. For large spheres with surface undulations having a local radius of curvature R' >> Ro, one recovers the expression for energy transfer in two dimensions. If, however, roughness is important on a scale of R C &,then the decay kinetics become sensitive to the roughness. The exact shape of the decay profile for @ ( t )will depend upon the local geometricaldetails, but we can speculate that at short times not too close to zero, the energy transfer rate will be sensitive to the third dimension created by the roughness. In this time domain, the decay will exhibit three-dimensional DET, crossing over to d = 2 at later times. Experimental Section Poly(buty1methacrylate) (PBMA)microsphereswere prepared by surfactant-free, two-stage emulsion polymerization. The recipe is given in Table I. The latex particle size distribution was determined by dynamic light scattering using a Brookhaven Model BI-90 particle sizer. The particles were narrowly distributed in size with a mean diameterof 311 nm. Stock solutions of the donors and the acceptor were prepared in purified ultrafiltered water, to give a R6G concentrationof 8.8 X loJ M, and a MG concentrationof 17.6 X 106 M. Aliquots of the PBMA dispersion and the two stock dye solutions were mixed in a 10mL volumetric flask, to obtain the requiredsample concentrations (19) Blumen, A. Nuouo Cimento B 1981,63, 50.
Langmuir, Vol. 9, No. 11, 1993 2827
Energy Transfer on the Surface of PBMA Particles
of R6G, MG, and latex. By this procedure the donor molecules were first adsorbed onto the latex, followed by addition of the acceptor. Steady-statefluorescencemeasurementswerecarried out with either a Spex Fluorolog 2 spectrometer or a Hitachi F-4000 spectrometer. Excitation and fluorescencespectrawere corrected by a conventional rhodamine B method and by the use of a standardtungsten lamp with known color temperature, respectively. A sample cell with a 2-mm optical path length was used in the front-face configuration to minimize reabsorption of sample fluorescence. Due to the smallStokesshiftsforRB andR6G,% this effect can be significant. In order to fiid out if there is any interference with our measurements of R6G, we have carefully examined the latex sample used in this work, with and without MG but in the absence of R6G, and found that at the excitation and emission wavelengths employed,their emission is negligible. Binding isothermswere measured as follows. Dilute solutions of MG or R6G in water were mixed in a 5 1 volume ratio with a PBMA latex dispersion (5.28 mg/mL). The dispersions were allowed to equilibrate (minutes) and then were centrifuged at 12000 rpm at 20 O C for 1 h. The supernatant solution was removed and examined by UV-vis spectroscopy to determine the dye concentration: 545 nm for R6G and 618 nm for MG. The amount of dye adsorbed on the latex sphere was calculated by using the equation N . = Vm~([Clo[Cl)lm
where m is the mass of the latex (g), [C]Oand [C] are the initial and equilibrium (supernatant)concentrations of the dye (pmoll L), respectively, and Vml is the volume of the solution (L). Picosecondfluorescencedecaycurveswere measuredby a timecorrelated single photon counting system equipped with a NdYAG laser (Coherent Antares 76-8) synchronously pumping a cavity-dumped dye laser (CoherentModel 700) as an excitation source. A microchannel plate photomultiplier (Hammamatau R15640-1)was used as a detector.21 The output from the dye laser was freqeuncy-doubled to get a 300-nm beam, which was used to excite the donor dye R6G. The fluorescence emission was observed at 555 nm. In order to minimize the interference of reflected excitation light on the fluorescence detection, the angle of incident light to the cell surface was carefully adjusted to 60°. In addition, sharp cutoff glass filters were set on the emission side of the sample chamber. A Glan-Taylor prism polarizer was also put on the emission side at the magic angle (54.75O)to the polarized plane of excitation light so as to remove the effect of photoselection.22 A Ludox dispersion was used to obtain the excitation profile of the system, indicating an instrumental response time (fwhm)of 100 ps. Results and Discussion In an apparent dimension analysis, as pointed out earlier by Levitz, Drake, and Klafter8, it is essential to select the experimental conditions properly so that the donor molecules have a well-defined lifetime TD. In this regard, RB is expected to be more problematic than R6G, because it may exist in three different forms and each form of the molecule has a different lifetime.20 In order to find the best conditions to carry out the decay measurement, we observed the apparent fluorescencequantumyield of these dyes, and the binding isotherms of R6G and MG in the latex. R6G and MG adsorb to the surface of PBMA latex particles in water. When we began these experiments, we were concerned that equilibrium might require long time periods, and the samples were sonicated and allowed to stand several hours before measurements were carried out. ~~~
~
(20) (a)Chang,T.-L.;Cheung, H. C. J.Phys. Chem. 1992,96,4874. (b) Arbeloa, I. L.; Rohatgi-Mukherjee, K. K. Chem. Phys. Lett. 1986,128, 474. (c) Sadkowski, P. J.;Fleming,G. R. Chem.Phys.Lett. 1978,57,526. (21) Yamazaki, I.; Tamai,N.; Kume, H.; Tsuchiya, H.; Oba, K. Reu. Sci. Instrum. 1985, 56, 1187. (22) OConnor, D. V.; Phillips, D. Time-Correlated Single Photon Counting; Academic Press: London, 1984.
71
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glL
Figure 2. Dependenceof the relative fluorescence intensityof (a)R6G ([R6Gl= 1.04 X 10-8M) and (b)RB ([RBI = 1.62 x 10-8
M)on various PBMA latex concentrations. The relative intensity
was measured at the band maximum at 558 nm for R6G and 586 nm for RB. ZOis the maximum band intensityof (a) 1.04 x 10-8 M R6G and (b) 1.62 X 10-8 M RB bulk solution. Increasing sample turbidity with increasing PBMA particle concentration makes it impossibleto determine the true fluorescencequantum yields.
We now know that equilibration of R6G with the latex surface, a t the low bulk concentrations employed here (10-5-10-6 M), occurs on a time scale of minutes or faster.23 Binding of the Dyes t o the Latex. It is well-known that RB and R6G form nonfluorescent dimers in solution, and it is thought that dimer formation is negligiblein very dilute solution (ClW M).24 In the presence of colloidal particles, however, if the dyes become adsorbed and therefore concentrated on the surface of the particles, dimer formation will be enhanced.16 Figure 2 shows the dependenceof the fluorescenceintensity of a R6G solution as a function of the amount of PBMA latex added. With the increase of the PBMA concentration, a t first the intensity decreases as a result of adsorption of the dye on the surface and formation of surface dimer, eventually passing through a minimum value. At higher PBMA concentrations, the fluorescence intensity increases due to dissociation of surface dimers when the surface area becomes sufficient, eventually exceeding that of the dye solution. At very high latex concentrations, the measured fluorescence intensity again decreases. This is probably a consequence of the high turbidity of the solutions rather than some more fundamental spectroscopic property of the system. In the case of RB, a similar trend is seen, although a minimum in the fluorescence intensity is lacking. The data in Figure 2 establish that R6G does indeed bind to the surface of the latex, and we infer from the data that at high PBMA/R6G ratios the amount of dimer formation on the surface is negligible. Binding isotherms for MG and R6G in the presence of our PBMA latex dispersion are shown in Figure 3. Both isotherms are similar in shape, with a sharp break point (23) Nakashima, K.; Duhamel, J.; Winnik, M. A. J. Phys. Chem., in press. (24) Selwyn, J. E.; Steinfeld, J. I. J. Phys. Chem. 1972, 76, 762.
Nakaehima et al.
2828 Langmuir, Vol. 9, No. 11, 1993 r’
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U T i m e (ns) Figum 4. Fluorescencedecay curve of R6G adsorbed on a PBMA latex dispersion. W G I = 1.00 X 1o-S M, and [PBMA] = 53.6 g L-I. The decay curve was collected in 512 channels to 20K CPC, at bx = 300 nm and bm = 565 nm, fitted to single exponential.
P
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1
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i
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60.0 80.0 100.0 120.0 Supernatant Concentration of R60 Dyr, p o i l L
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Figure 3. Binding isotherms of (top)MG and (bottom)R6G in the latex aqueous suspensions. The 3c axis is the supernatant concentration[C](pmol/L). They axis is the micromoles of MG or R6G adsorbed per gram of latex N,. at 7 pmol of dye/g of latex. At higher dye concentrations, the surface coverage does not saturate, but increases linearlywith a much smallerslope. Similar type isotherms were observed by Levitz et ale8for binding of these dyes to silica gel particles. They attributed the high concentration binding to dimer formation by the dyes on the particle surface. Our binding isotherms were used primarily as a guide for selection of the proper R6G/latex and MG/latex ratios for meaningful experiments. It is important for us to avoid as much as possible the presence of dimers on the latex surface. Energy transfer from monomeric donor to nonfluorescent dimer will competewith energytransfer to MG and distort our results. If energy transfer from the donor dye to dimer pairs is significant, it will participate in quenching the donor fluorescence, making the donor decay profiie nonexponential in the absence of MG. In order to minimize the presence of dimers, we employed 1.10 X 1O-e M donor dye and 53.5 mg/mL latex (cf. Figure 2). From the binding isotherm of R6G and MG on the PBMA surface, we calculate that for 1.00 X 1O-e mol/L R6G and 6.58 X 1W to 6.58 X lV moVL MG (for 5.35 w t % latex suspensions), 98.2% of R6G and 98.0-95.6% of MG dyes are on the latex surface. The latex system differs from the silica gel system in that these latex particles have a fixed number of anionic (sulfate) groups on the surface. We have some indication that the breakpoint in the binding isotherm occurs in the vicinity of one cationic dye bound per anionic surface charge. This result needs to be confirmed through careful titration of surface charge on the latex and examination of particles with different surface charge densities. If this turns out to be correct, then it may also happen that the distribution of MG and R6G on the particle surface may reflect the distribution of surface charges, and that DET measurements could provide information about this
distribution. Thus, there is good reason to pursue these types of experiments in more detail in the future. DET Experiments. The fluorescencedecay profile for R6G adsorbed to PBMA is shown in Figure 4. This decay curve was collected to 20K counts in the peak channel (CPC) and over 512 channels. The profile is exponential over two decades of the intensity decay, exhibiting a lifetime of 3.93 ns. Attempts to increase the CPC further made apparent small deviations from exponentiality of the decay curves. For this reason we are restricted to discuss the subject at a datum precision level of about 20K CPC. The situation with RB was noticeably worse: the decay profile deviated from a single exponential decay even below 20K CPC. Therefore, RB was considered not the best candidate as a donor in this study. We note that these deviations from exponential decay are consistent with a tiny fraction of the dye in the aqueous medium. We comment further on this possibility in the next section. Addition of even small amounts of MG to the system leads to fluorescence quenching. The donor fluorescence decay profiles [I&)] become nonexponential, and the deviations from exponentiality become more pronounced as the MG concentration is increased. In Figure 5 we present the decay curves for the R6G/PBMA sample in the presence of various bulk concentrations (0-6 X 1 V M) of MG. Data Analysis Technique and Results. In a direct energy transfer experiment, the intensity decay of the donor fluorescence is affected by the spatial distribution of acceptors as described by eq 3. Fitting the data to eq 3, however, is normally difficult without a model incorporating detailed knowledge of the geometryof the system. As Levitz et al.8 point out, a useful strategy for fitting experimental data is to use a two-parameter expression of the form ID(t) = A eXp[-t/T, - p(t/TD)a’s]
P = pF(Rda)ag’l”(l- a/6)
(6) (7)
This fitting function is based on eq 4, originally derived for the fractal model. For direct energy transfer in a restricted geometry, fluorescence decay data fitted to eq 6 often yield noninteger values for Under these circumstances, the value of has no intrinsic meaning as a fractal dimension, but rather suggests the importance of a crossover between terms describingthe decay kinetics. It is therefore referred to as the “apparent dimension” of the geometry.
a
a.
Energy nansfer on the Surface of PBMA Particles I
I
Langmuir, Vol. 9, No.11, 1993 2829 Table 11. Results of Fluorescence Decay Data Fitted Without the A2 Term
4
I
io4
A
cr .rl
lo3
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al
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5.5
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1 2 3 4 5 6 7 8 9 10
[MG]X le (moUL)
A
P
a
reducedxz
0.658 1.316 1.974 2.632 3.290 3.948 4.606 5.264 5.922 6.580
1.OOO 1.OOO 1.OOO 1.OOO 1.OOO 1.OOO 1.OOO 1.OOO 1.OOO 1.OOO
0.576 1.180 1.755 2.610 3.192 3.629 4.387 5.255 6.746 7.304
1.91 1.95 1.85 1.80 1.71 1.60 1.47 1.28 1.04 0.96
1.2289 1.0212 1.2416 1.3281 1.4902 1.7661 2.2663 3.0871 4.2263 5.1441
Table 111. Results of Fluorescence Decay Data Fitted with the A2 Term Included
al
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5.5
11
16
22
T i m e (ns) Figure 5. Fitted fluorescence decay curves (top) and residuals (bottom) for samples of R6G adsorbed on the PBMA latex spheres, at various MG concentrations. [R6G] = 1.10 X 1 V M, and [PBMA] = 53.5 g L-l; from top to bottom, [MG] X 106 = 0,0.658,1.316,1.974,2.632,3.290,4.606,5.922 M (bulk concentration). Lx= 300 nm,and Lm= 555 nm. Under some circumstances, it is necessary to introduce an additional term, A2 exp(-t/m) (A2 term), which may represent the donor molecules that are trapped in an environment which is free of acceptor molecules. The fitting function then becomes
Extensive computer simulations carried out in this laboratory25show that the existence of an unquenched component (A2 term) affeds sensitively the quality of information about the apparent dimension contained in DET data, especially when the P parameter is small. With the increase of the relative weight of the A2 term or the AdA1 ratio, the accuracy and precision of the recovered apparent dimension rapidly decline, indicated by a more and more flat x2 surface with respect to the dimension. Consequently, the dimension parameter is more difficult to recover accurately. Recently we have developed a programz5which optimizes the parameter a using a grid searching method,” for each a tested, the program then optimizes the parameters AI, Az, and P using the Marquardt algorithm.26 By this approach we can ensure that the program does not terminate at any local minimum of the x2 surface. A byproduct of this approach is that we (26)Liu, Y.S.;Winnik, M. A. Unpubliahed work.
(26) Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill Book Co.: New York, 1969.
1 2 3 4 5 6 7 8 9 10
[MG]x106(mol/L)
A1
Az
P
d
reducedxz
0.658 1.316 1.974 2.632 3.290 3.948 4.606 5.264 5.922 6.580
0.710 0.946 0.979 0.988 0.991 0.992 0.995 0.996 0.997 0.997
0.290 0.054 0.021 0.012 0.009 0.008 0.005 0.004 0.003 0.003
0.752 1.214 1.697 2.457 2.964 3.287 3.936 4.469 5.214 5.641
2.80 2.28 2.16 2.22 2.19 2.21 2.07 2.04 1.99 1.85
1.2155 1.0075 1.2113 1.1520 1.1338 1.0989 1.3284 1.4179 1.6556 1.6621
can draw the x2 surface for each set of data and estimate the uncertainty of the recovered dimension through this surface. Both eqs 6 and 8 have been employed to fit the decay data derived from the experiments. In Tables I1 and I11 are listed the results of fitting a set of experimental data of the RGG/MG/PBMA latex system to these two equations. A comparison of these two tables shows that the dimension obtained by fitting the data without the A2 term are systematically smaller than that obtained with the A2 term included, and for samples with higher acceptor concentrations, they correspond to higher or even unacceptable x2 values. It is interesting to note that when the P parameter is relatively small, both equations can fit the data properly, although they give greatly different AdA1 ratios. This suggests that when P i s small, the recovered AdA1 ratio may not be reliable. On the other hand, if the P parameter is large, omission of the A2 term of a magnitude only 0.3-1 % of that of the A1 term may came a remarkably increased x2 value and great changes to the recovered P and a parameters. These results are consistent with those obtained from computer simulations: a correctselectionof a fitting model becomes more and more critical with an increase of the P value. For example, when a simulated decay is generated at a P value of 0.5 without an A2 term, and is fitted to eq 8, a false A2 term may occur with a perfect fit. At a P valueof2andanA2/A1ratioof0.1, ifthedecayisanalyzed without the A2 term, a negative error of about 50% in the recovered dimension with an unacceptable x2 may be obtained. When a simulated decay is generated at P = 2 and without the A2 term, and it is fitted with the A2 term included, the correct dimension and A2 = 0 can be recovered. We therefore consider that at low acceptor concentrations (sample nos. 1-31, the information about P and a derived from eq 6 is more likely to be “true”, although both models may give excellent fits to the data. At higher acceptor concentrations, however, only resulta derived from eq 8 are correct. Thus, these results suggest that the true dimension is about 2, and the true weight of A2 is about 0.6%. At present, we are not certain about the physical interpretation about this small “unquenched component”. For a heterogeneous system like a latex
Nakashima et al.
2830 Langmuir, Vol. 9, No. 11,1993
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C x 100 / nm-2
apparent dimension
Figure 7. P v8 C obtained for RGG/MG/latex by fitting fluorescencedecay curves to eq 8, solid circles, and to eq 6, open squares. The horizontal axie is the surface concentration (molecules/nm2)of MG assuming complete adsorption.
1.8 1.7 1.6
Table IV. Ib Valuee Calculated for R6G in Various Media
1.5
medium water PS latex
1.4 1.3 12
Ro (nm) 6.30 6.22
medium PBMA latex
Ro (Inn) 6.66
latex spheres. Thus,pla2is the number density of acceptor molecules on the surface, hereafter designated as C. Equation 9 can then be written as
1.1 1 .o 1 0
1 5
2 0
2 5
3 0
apparent dimension Figure 6. Examplesof the recovered x2 surfaces from the decay curvesshown in Figure 5. (a) From bottom to top, the x2 surfaces correspond to data nos. 5,6,7,8,9, and 10 in Table 111. (b) From bottom to top, the x2 surfaces correspond to data nos. 1,2,3,4, 5,and 6, in Table 11. The apparent dimensions are indicated by the minimums of these curves. dispersion, one can imagine many possible sources for contributors to this term. Although unlikely here, one would expect this kind of contributionfrom a s m a l lfraction of donor molecules trapped deep inside the latex. In the current experiments, the A2 term most likely originates froma small fraction of donor moleculesin the water phase. This would be consistent with the binding isotherm studies. Parta a and b of Figure 6 show some recovered x2 surfaces when the decay data were fitted to eqs 8 and 6, respectively. The uncertainty of the recovered dimensions can be estimated by the variation of a which causes an increase of the reduced x2 by l l m , where m is approximately the number of channels being fitted.26 In our case l l m is about 0.0021. From an individual curve the uncertainty of the recovered apparent dimension is estimated to be about f0.05. Direct calculation for the whole set of experiments using the recovered dimensionsin Table 111,disregarding the d obtained at the lowest MG concentration, gives an average dimension of 2.11 f 0.14. These results thus imply a flat surface on the length scale probed. Accepting = 2 and taking F = A, g' = 1.0 (assuming ( K ~ )= 2/3), and I'(213) = 1.355, we can rewrite eq 7 as
P = 1.355u@/a2)R,2 (9) By definition, the probability p of an acceptor occupying a site on the surface of a latex sphere can be calculated bY p = Na2/4uR2 (10) where N is the number of acceptors adsorbed per latex sphere, a2 is the area of a site, and R is the radius of the
P = 1.355uCRt
(11)
In Figure 7 we plot values of P obtained from decay data analyses as a function of C, the molecular number densityof MG on the surface (molecules/nm2). The surface number densities were calculated from the quantity of MG added, together with the mean particle diameter (311 nm) and the density (1.07 g cm3) of PBMA under the assumption that essentially all the MG molecules are adsorbed on the latex, and that the latex particle is a sphere with a smooth surface. In the plot of P vs C, a reasonable straight line passing through the origin can be drawn through the data, especiallythose derived from fitting the decays to eq 8. This plot suggests again that the results derived from fitting the decaysto eq 8 are better, compared with those derived from eq 6, in a broad range of MG surface concentration. The finding that d = 2 in these experiments provides support for the idea that the latex has a smooth spherical surface. Interpretation of the P values is more delicate. First, it requires knowledge of the Ro values for energy transfer for the surface-bound dye pairs. On the basis of the solution properties of the chromophores in water, one calculates values of 6.0 nm for RGG/MG. These values should be modified somewhat through adsorption of the dyes to the surface. In addition, one needs a more precise value for the orientation factor SI. We have measured the absorption spectra of malachite green and the emission spectra of R6G in water, and in the presence of polystyrene (PSI latex and PBMA latex usingintegratingsphere optics. The Ro values calculated from these spectra are listed in Table IV. The donor quantum yields (@D) used for these calculations are those in water, correctad for the change in donor lifetime and change in refractive index of the environment. For details, see ref 23. We calculate @D = 0.59,0.64, and 0.63 for R6G in water, in PS latex, and in PBMA latex, respectively. The slope of the plot of P vs C for the R6G/MG system in Figure 7 is about 135. Substituting PIC = 135 into eq
Energy Transfer on the Surface of PBMA Particles 11,we would calculate a value of Ro of 5.63 nm, which is reasonably consistent with the literature data (6.23 for the R6G/MG pair in vesicle systems and our own calculation based on the spectroscopic data for this pair in PBMA latex (6.66 nm). We suspect that the small discrepancy is caused by some assumptions used in this calculation which are not appropriate. For instance, ifwe assumerandomly oriented immobile dipoles and take ( it2) = 0.475, the calculated Ro value would be 6.67 nm which is in excellent accord with our measured data. This seems to indicate that the rotation of the dye molecules on the surface is restricted on the nanosecond time scale. We will defer for future experiments a more quantitative interpretation of P values in terms of a detailed model for energy transfer on spherical surfaces.28 The most important result of the experimental data presented above is that the observed energy transfer kinetics are those characteristic of DET in twodimensions. This implies that over the distance scale probed by R6G/ MG, the surface appears smooth. The apparent flatness of the surfaceis of c o m e related to the muchlargerparticle radius than the RO values characterizing energy transfer for the dye pairs. To estimate the scale over which the surface appears flat and smooth, we will use the Ro values cited above, with the assumption that ( 3 / 2 ) ( ~ = ~ )1, to calculate the maximum distance, Rm,, probed in these experiments. Rm, = Ro(tmar/TD) 1/6 (12)
where t,,, is the maximum time on the time axis for (27) Tamai, N.; Yamazaki, T.; Yamazaki, I.; Mizuma, A.; Mataga, N. J. Phys. Chem. 1987,91, 3603. (28)Nota that we are unable to take rigorous account of the difference in the index of refraction between PBMA (n= 1.46) and water (n= 1.33).
Langmuir, Vol. 9, No. 11,1993 2831
fluorescence decay measurements for which useful information is still obtained. In our experiments, a reasonable value for t- is 20 ns (cf. Figure 4 and 5 ) for which we calculate R m , = 8 nm for R6G/MG. Summary The experiments reported here indicate that the poly(butyl methacrylate) latex microspheres which we prepared by emulsion polymerization have surfaces which are locally smooth on a length scale of 8 nm. This study shows that a fractal dimension analysis is an extremely delicate task, requiring not only carefully designed experiments but also an appropriate method of data analysis. A few additional comments are in order about the system. Since this information is obtained by adsorbing cationic dye pairs to the surface of anionic particles, it is interesting to contemplate how the binding pattern is affectad by the distribution of charges on the surface. There are no known methods for determining the pattern of charge distribtion, and our finding of a = 2 suggests that this distribution is relatively ~ n i f 0 r m . lThis ~ may be a consequence of the fact that the particles are synthesized at a temperature (80OC) significantlyhigher than the glass transition temperature (Tg= 30 OC) of PBMA. At this temperature, the latex particle can be thought of asa liquid droplet, and the sulfate groups which bear the charge are able to redistribute themselves on the surface. We expect that it will be very interesting to extend this kind of experiment to other latex systems, particularly those in which TEM measurements give indications of surface roughness. Acknowledgment. The authors thank NSERC Canada and the Ontario Centre for Materials Research for their financial support.
