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A new and facile method for measurement of apparent density of monodisperse polymer beads
Talanta 80 (2010) 1681–1685
Contents lists available at ScienceDirect
Talanta journal homepage: www.elsevier.com/locate/talanta
A new and facile method for measurement of apparent density of monodisperse polymer beads Qi Zhang a,b , Balasubramanian Srinivasan c , Yuanpeng Li a , Ying Jing a , Chengguo Xing c , Jin Chang b,∗ , Jian-Ping Wang a,∗ a b c
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA School of Materials Science & Engineering, Tianjin University, Tianjin 300072, PR China Department of Medicinal Chemistry, College of Pharmacy, University of Minnesota, Minneapolis, MN 55455, USA
a r t i c l e
i n f o
Article history: Received 28 July 2009 Received in revised form 2 October 2009 Accepted 5 October 2009 Available online 12 October 2009 Keywords: Apparent density Viscosity Hybrid beads Polymer beads Micro-total analysis systems Suspension stability
a b s t r a c t The apparent density, an intrinsic physical property of polymer beads, plays an important role in the application of beads in micro-total analysis systems and separation. Here we have developed a new, facile and milligram-scale method to describe the motion of beads in aqueous solution and further detect the apparent density of beads. The motion of beads in solutions is determined by the viscosity of solutions and the density difference between beads and solutions. In this study, using various glycerol aqueous solutions with certain viscosities and densities, the motion time (i.e. floating or sedimentation time) of hybrid polymer beads was experimentally measured and theoretically deduced, and consequently, the apparent density of monodisperse beads can be quickly and easily calculated. The results indicated that the present method provided a more precise way to predict the movement of hybrid beads in aqueous solution compared with the approach for commercial use. This new method can be potentially employed in flow cytometry, suspension stability, and particle analysis systems. © 2009 Elsevier B.V. All rights reserved.
1. Introduction In micro-total analysis systems (TAS), monodisperse hybrid polymer beads have been widely used in many fields of biological analysis, high-throughput screening, and immunoassay and among others [1–3]. However, due to the unmatched densities of hybrid polymer beads and aqueous solutions [4], a well suspension of polymer beads in aqueous solutions cannot be achieved, resulting in a higher measurement error and lower signal-to-noise ratio [5]. Obviously, a feasible solution is adjusting either the densities of polymer beads [4] or those of aqueous solutions (e.g. an addition of glycerol or sucrose [6]). Although, the densities of polymer beads can be measured by a density meter (ASTM D4052) [7], the measurement of apparent densities of hybrid beads is still a grand challenge due to hybridization of nanoparticles, surface hydrophilic liquid layer [8] and inner nano-scale pores [9]. Here a simple and robust method that can precisely describe the movement of hybrid polymer beads in glycerol aqueous solution, was demonstrated, and further applied to the measurement of apparent density. And it also has potential use in flow cytome-
∗ Corresponding authors. E-mail addresses: [email protected] (J. Chang), [email protected] (J.-P. Wang). 0039-9140/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.talanta.2009.10.004
try, microfluidics chips, bead suspensions, and particle analysis systems. 2. Materials and methods 2.1. Materials Monodisperse polymer beads (PC06N/6182, with carboxyl group, size 9.77 m) were purchased from Bangs Laboratories, Inc. Uniform magnetic beads were obtained from Micromod (micromod Partikeltechnologie GmbH, Micromer® -M, Prod.-No. 08-01-104, with amine group, size 10 m). The densities of polymer beads and magnetic beads provided by vendor are 1.062 and 1.1 g/cm3 , respectively. Sodium dodecyl sulfate (SDS) and glycerol (≥99% (GC)) was procured from Sigma. The water used in all experiments was de-ionized water. 2.2. Motion of beads in glycerol–water mixtures 0.5 mg of polymer beads or commercial magnetic beads was dried under air at room temperature and re-dispersed in a 1.5 mL vial containing 0.5 mL of glycerol–water mixtures. Owing to the lower or higher densities of beads compared with those of glycerol solutions, the beads were buoyed up or sunk down in the
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Fig. 1. Schematic description of bead motion in various glycerol–water mixtures (the darker color means higher mass fraction of glycerol in mixture); (a) 50 wt% glycerol solution, the digital pictures of a1 and a2 were taken at 1.2 and 2.7 h, respectively. (b) 23 wt% glycerol solution, b1 and b2 at 3.5 and 7.5 h, respectively. (c) 10 wt% glycerol solution, c1 and c2 at 0.7 and 1.4 h, respectively. The beads were obtained from Bangs Laboratories, Inc.
solution. The total time taken for entire-bead floating to the surface or sedimentation to the bottom of glycerol solutions was defined as floating or sedimentation time of beads. The accuracy of the measurements was evaluated from 5 repetitive determinations (repeatability) of the motion time of polymer beads in the same glycerol aqueous solution, and the relative standard deviations (RSD) were also calculated. The schematic process and digital photos are shown in Fig. 1. We have included more digital photos taken at different time in supporting information (see Fig. S1). 2.3. Characterization The images of beads were taken using a microscope (Olympus, IX 51). Flow cytometry (Coulter, Epics XL) was utilized to measure the bead’s relative size (forward light scatter, FS signal), relative granularity or internal complexity (side light scatter, SS signal). Flow cytometry was set up according to the standard operating procedure. During detection, a total of 20000 events were recorded for each time point, and the beads dispersed in water or glycerol solutions were detected at the low flow rate with a concentration of 1 mg/mL. The populations of single beads and doublet beads were
selected based on forward and side scatter. The mean and RSD of these populations were also measured. All flow cytometric data were performed using FlowJo 5.7 software (Tree Star, San Carlos, CA, USA). 3. Results and discussions 3.1. Densities and viscosities of glycerol–water mixtures The densities of glycerol–water mixtures were measured by comparing their mass to their volume. The viscosities were calculated using the following empirical equation [10]. ln m − ln w 2 = Cm [1 + (1 − Cm ){a + bCm + cCm + · · ·}] ln g − ln w
(1)
where is kinematic viscosity ( = /), is dynamic viscosity, is the density, Cm is the mass fraction of glycerol in the mixture, and subscripts m, g and w denote the glycerol–water mixtures, glycerol and water, respectively. The constants a, b and c at 20 ◦ C are assigned to −0.76728, 0.12153 and −1.41519, respectively. As shown in Table 1, the detected densities are close to those provided by the vendor, and the interpolated viscosities are in accordance
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with those published [10] (the comparison between the densities and viscosities in the present study and those published were shown in Tables S1 and S2 in supporting information, respectively). 3.2. Theory of bead movement in aqueous solution The net force, Fb , of a bead in aqueous solution is composed of following two contributions, Fb = Fb,1 − Fb,2
(2)
where Fb,1 and Fb,2 are the sum of the buoyant force and the bead’s weight, and the force of movement resistance (Stokes’ drag force), respectively. According to Archimedes’ principle, the Fb,1 is given by, Fb,1 = Vb × (m − b ) × g Fig. 2. The floating and sedimentation time of polymer beads in various glycerol–water mixtures, the positive and negative values indicate that the beads are floating and sedimentating, respectively.
Table 1 The density and viscosity of glycerol–water mixtures. Mass fraction [g/g]
Density [g/mL]
viscosity [mPa s]
0.0000 0.0996 0.1510 0.1758 0.2002 0.2084 0.2202 0.2301 0.2402 0.2502 0.2601 0.3006 0.3802 0.5000
0.9984 1.0218 1.0323 1.0392 1.0449 1.0512 1.0537 1.0558 1.0594 1.0606 1.0634 1.0727 1.0925 1.1251
1.0085 1.2804 1.4920 1.6178 1.7546 1.8123 1.8868 1.9527 2.0260 2.0974 2.1744 2.5252 3.4571 5.9486
(3)
where Vb is the volume of a bead, m and b are the density of the aqueous solution and hybrid polymer beads (or magnetic beads), respectively, and g is the gravitational acceleration. The force of the movement resistance (Stokes’ drag force [11]) Fb,2 is obtained by the following equation. Fb,2 = 6Rv
(4)
where is the viscosity of aqueous solution, R is the radius of a bead, and v is the velocity. Hence, the Fb is deduced by, Fb = Fb,1 − Fb,2 = Vb × (m − b ) × g − 6Rv =
4 3 R × (m − b ) × g − 6Rv 3
(5)
Based on Eq. (5), the suspended bead has a tendency to rise or sink in the glycerol–water mixture if b is not equal to m . The beads were buoyed up in aqueous solutions when their densities were less than those of the mixture (Fig. 1a). The beads were stably suspended in solutions as their densities were accurately equal to those of the
Fig. 3. Estimation of the density of (a) polymer beads and (b) commercial magnetic beads by Eq. (8), the apparent density of beads, b is equal to the m of the point where the fitted linear curve intercepts the line x = 0. Table 2 The flow cytometric data of polymer beads shown in Fig. 4. Mass fraction of glycerol [wt%]
4a 4b 4c 4d 4e
0 10 20 24 30
FS in R1
SS in R1
The proportion of beads in
Mean
RSD [%]
Mean
RSD [%]
R1 [%]
R2 [%]
436.22 458.66 483.35 541.42 746.31
6.28 5.53 7.38 9.71 20.9
427.33 423.74 430.75 425.56 445.70
13.57 13.33 13.22 13.71 25.42
81.48 89.42 89.52 – –
15.78 7.36 2.97 – –
The ratio of doublet beads (in R2) to singlet beads (in R1) [%]
19.4 8.2 3.3 – –
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Q. Zhang et al. / Talanta 80 (2010) 1681–1685
mixture (Fig. 1b), and the beads were sunk to the bottom of the vial while their densities were larger than those of the mixture (Fig. 1c). 3.3. Estimation of bead density The floating velocity is defined as positive, and sedimentation velocity as negative. As shown in Eq. (5), the velocity (floating or sedimentation) of beads reaches the maximum at equilibrium states (Fb = 0), and the equation of equilibrated floating or sedimentation velocity is given as follows (transformation of Eq. (5)),
v=
2R2 g(m − b ) 9
(6)
Since t is equal to d/v, the time (t) of floating or sedimentation is derived by, t=
9d =k· m − b − b )
2R2 g(m
k=
9d 2R2 g
(7)
where the distance (d) of floating or sedimentation is a constant, and consequently, k is a constant. According to Eq. (7), the floating or sedimentation time depends on the viscosities of aqueous glycerol solutions and the densities difference between the beads and the solutions [8]. Fig. 2 showed that the detected floating or sedimentation time of polymer beads is in accordance with Eq. (7). Moreover, the density of polymer beads can be obtained by using the floating or sedimentation time. Changing the expression of Eqs. (7) and (8), the result is represented as follows, m − b = k ×
⇒ m = b + k × t t
(8)
where the m and are shown in Table 1, and the motion time (t) is illustrated in Fig. 2. The density, m and the /t are illustrated in Fig. 3a. The b fitted by linear least square method is 1.0570 g/cm3 ((1.0563 + 1.0576)/2). The calculated RSD was 0.07%. As shown in Table 1, the densities of 25 and 26 wt% glycerol solutions were 1.0606 and 1.0634 g/cm3 , respectively. The bead density provided by the vendor (b,vendor = 1.062 g/cm3 ) stands between the densities of 25 wt% (b,vendor > m,25% ) and 26 wt% glycerol solutions (b,vendor < m,26% ). However, in experiments, the bead were buoyed up in both 25 wt% and 26 wt% glycerol solutions (see Fig. 2), which indicated that the density (apparent) of beads was lower than those (b,apparent < m,25% < b,vendor < m,26% , see Eq. (5)). In other words, by adopting the vendor-provided density, the movement of beads in aqueous solution cannot be precisely predicted. In contrast, the motion of hybrid beads in solution can be more accurately estimated by using the detected apparent density (see Eqs. (6) and (7) and Fig. 2). Similarly, as shown in Fig. 3b, the detected density of magnetic beads, 1.0941 g/cm3 , is a little bit smaller than that provided by vendor. The RSD obtained was 0.11%. The smaller measured densities are probably due to surface hydrophilic liquid layer [8] and inner nano-scale porous [9]. 3.4. Influence of the motion of beads on flow cytometric analysis The motion of beads plays a crucial role in the flow cytometric analysis. As illustrated in Fig. 4 and Table 2, the mean of forward light scatter (FS) signals are increased with increasing of glycerol mass fraction. In the mean time, those of side light scatter (SS) signals are close to a constant. The light signals are generated as beads pass through the laser beam in a fluid stream. And the FS signal is proportional to the projected spot area or size which is related to the size of beads and the distance between the beads and the sensor [5b]. When the motion tendency of beads was changed from
Fig. 4. Flow cytometry analysis of polymer beads. The dispersion solutions of a–e are pure water, 10, 20, 24 and 30 wt% glycerol solutions, respectively. (a1 –e1 ) The bead subpopulations based on side light scatter (SS) vs. forward light scatter (FS); (a2 –e2 ), the histogram of size distribution. The singlet beads were counted in the region 1 (R1, a1 –e1 ), the doublet beads were recorded in the red region 2 (R2, a1 –c1 ) and marked by red arrows in the diagrams a2 –c2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
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beads in aqueous solution than the general method. This work is beneficial to micro-total analysis systems, separation, suspension stability [15], particle analysis systems and so forth. Acknowledgment This work was supported by the Center for Nanostructured Application and Nanobiotechnology Initiative at the University of Minnesota and by the NSF BME 0730825. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.talanta.2009.10.004. References
Fig. 5. The microphotograph of polymer beads dispersed in a solution containing 5 wt% glycerol and 0.05 wt% SDS, and the doublet beads were marked by arrows.
sedimentation to floating (Fig. 4a–e), the distance was increased, resulting in a larger FS signal. In addition, the increase of FS signals is not mainly due to the diffusion of glycerol from the core flow to the sheath flow [12], otherwise, the mean and RSD of SS signals should be also changed (see Table 2). Moreover, the proportion of doublet beads is decreased with the prolongation of suspension time (Fig. 4a–c). The doublet beads suspended in a liquid are shown in Fig. 5. Their signals might affect the accuracy of flow cytometry detection [13]. The numbers of doublet beads depend on previous treatment of the suspension (e.g., sonication and vortex), age, and solvent [14]. Since all the samples were performed with the same operations, the proportion of doublet beads can be reduced by extension of their suspension time. 4. Conclusions Theoretical description for the motion, including floating or sedimentation of polymer bead in aqueous solution was deduced by analysis of the net force of a liquid-suspended bead. Furthermore, the apparent density of beads can be easily calculated by using the measured floating or sedimentation time. For polymer beads and commercial magnetic beads, the detected apparent densities in the present study were 1.0570 and 1.0941 g/cm3 , respectively. The analysis showed that this milligram-scale method provided a more facile and precise approach to predict the movement of hybrid
[1] G.M. Whitesides, Nature 442 (2006) 367. [2] (a) C.-Y. Chang, Y. Takahashi, T. Murata, H. Shiku, H.-C. Chang, T. Matsue, Lab. Chip 9 (2009) 1185; (b) J.G. Kralj, A. Player, H. Sedrick, M.S. Munson, D. Petersen, S.P. Forry, et al., Lab. Chip 9 (2009) 917; (c) D.H. Yoon, J.B. Ha, Y.K. Bahk, T. Arakawa, S. Shoji, J.S. Go, Lab. Chip 9 (2009) 87. [3] (a) X. Llopis, M. Pumera, S. Alegret, A. Merkoc¸i, Lab. Chip 9 (2009) 213; (b) N. Pamme, Lab. Chip 6 (2006) 24; (c) K.S. Kim, J.-K. Park, Lab. Chip 5 (2005) 657. [4] (a) A. Funfak, M. Fischlechner, E. Donath, J.M. Köhler, The 2007 Spring National Meeting Houston, 2007, p. T1011; (b) K.D. Vorlop, D. Estapé, F. Gòdia, Biotechnol. Tech. 7 (1993) 287. [5] (a) B. Srinivasan, Y. Li, Y. Jing, Y. Xu, X. Yao, C. Xing, et al., Angew. Chem. Int. Ed. 48 (2009) 2764; (b) Introduction to Flow Cytometry: A Learning Guide, BD Biosciences, Manual Part Number: 11-11032-01, 2000. http://www.cancer.umn.edu/exfiles/ research/fcintro.pdf (accessed Oct 27, 2009). [6] T.W. Work, E. Work, Laboratory Techniques in Biochemistry and Molecular Biology, North Holland Publ. Comp., Amsterdam, 1976. [7] Test Method for Density and Relative Density of Liquids by Digital Density Meter, ASTM D 4052, 2002. [8] X.M. Sun, S.M. Tabakman, W.-S. Seo, L. Zhang, G.Y. Zhang, S. Sherlock, et al., Angew. Chem. Int. Ed. 48 (2009) 939. [9] X. Gao, S. Nie, Anal. Chem. 76 (2004) 2406. [10] P.N. Shankar, M. Kumar, Proc. R. Soc. Lond. A 444 (1994) 573. [11] V.G. Zarnitsyn, A.G. Fedorov, Langmuir 23 (2007) 6245. [12] C. Wang, Y. Gao, N.-T. Nguyen, T.N. Wong, C. Yang, K.T. Ooi, J. Micromech. Microeng. 15 (2005) 2289. [13] (a) S. Hansen, R. Dahl, H.J. Hoffmann, J. Immunol. Meth. 338 (2008) 58; (b) A.G. Schrum, D. Gil, E.P. Dopfer, D.L. Wiest, L.A. Turka, et al., Sci. STKE 389 (2007) pl2. [doi:10.1126/stke.3892007pl2]; (c) S.E. Weigum, P.N. Floriano, N. Christodoulides, J.T. McDevitt, Lab. Chip 7 (2007) 995; (d) N.K.Christensen, J. Laursen, L. Winther, USP 2008/0194508 A1 (2006). [14] A.K. Gaigalas, L. Wang, Springer Ser. Fluoresc. 6 (2008) 371. [15] J. Chang, Q. Zhang, W. Wang, Y. Han, P. Wang, X. Dong, CN Patent CN101109689 (2007) (in Chinese).
Contents lists available at ScienceDirect
Talanta journal homepage: www.elsevier.com/locate/talanta
A new and facile method for measurement of apparent density of monodisperse polymer beads Qi Zhang a,b , Balasubramanian Srinivasan c , Yuanpeng Li a , Ying Jing a , Chengguo Xing c , Jin Chang b,∗ , Jian-Ping Wang a,∗ a b c
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA School of Materials Science & Engineering, Tianjin University, Tianjin 300072, PR China Department of Medicinal Chemistry, College of Pharmacy, University of Minnesota, Minneapolis, MN 55455, USA
a r t i c l e
i n f o
Article history: Received 28 July 2009 Received in revised form 2 October 2009 Accepted 5 October 2009 Available online 12 October 2009 Keywords: Apparent density Viscosity Hybrid beads Polymer beads Micro-total analysis systems Suspension stability
a b s t r a c t The apparent density, an intrinsic physical property of polymer beads, plays an important role in the application of beads in micro-total analysis systems and separation. Here we have developed a new, facile and milligram-scale method to describe the motion of beads in aqueous solution and further detect the apparent density of beads. The motion of beads in solutions is determined by the viscosity of solutions and the density difference between beads and solutions. In this study, using various glycerol aqueous solutions with certain viscosities and densities, the motion time (i.e. floating or sedimentation time) of hybrid polymer beads was experimentally measured and theoretically deduced, and consequently, the apparent density of monodisperse beads can be quickly and easily calculated. The results indicated that the present method provided a more precise way to predict the movement of hybrid beads in aqueous solution compared with the approach for commercial use. This new method can be potentially employed in flow cytometry, suspension stability, and particle analysis systems. © 2009 Elsevier B.V. All rights reserved.
1. Introduction In micro-total analysis systems (TAS), monodisperse hybrid polymer beads have been widely used in many fields of biological analysis, high-throughput screening, and immunoassay and among others [1–3]. However, due to the unmatched densities of hybrid polymer beads and aqueous solutions [4], a well suspension of polymer beads in aqueous solutions cannot be achieved, resulting in a higher measurement error and lower signal-to-noise ratio [5]. Obviously, a feasible solution is adjusting either the densities of polymer beads [4] or those of aqueous solutions (e.g. an addition of glycerol or sucrose [6]). Although, the densities of polymer beads can be measured by a density meter (ASTM D4052) [7], the measurement of apparent densities of hybrid beads is still a grand challenge due to hybridization of nanoparticles, surface hydrophilic liquid layer [8] and inner nano-scale pores [9]. Here a simple and robust method that can precisely describe the movement of hybrid polymer beads in glycerol aqueous solution, was demonstrated, and further applied to the measurement of apparent density. And it also has potential use in flow cytome-
∗ Corresponding authors. E-mail addresses: [email protected] (J. Chang), [email protected] (J.-P. Wang). 0039-9140/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.talanta.2009.10.004
try, microfluidics chips, bead suspensions, and particle analysis systems. 2. Materials and methods 2.1. Materials Monodisperse polymer beads (PC06N/6182, with carboxyl group, size 9.77 m) were purchased from Bangs Laboratories, Inc. Uniform magnetic beads were obtained from Micromod (micromod Partikeltechnologie GmbH, Micromer® -M, Prod.-No. 08-01-104, with amine group, size 10 m). The densities of polymer beads and magnetic beads provided by vendor are 1.062 and 1.1 g/cm3 , respectively. Sodium dodecyl sulfate (SDS) and glycerol (≥99% (GC)) was procured from Sigma. The water used in all experiments was de-ionized water. 2.2. Motion of beads in glycerol–water mixtures 0.5 mg of polymer beads or commercial magnetic beads was dried under air at room temperature and re-dispersed in a 1.5 mL vial containing 0.5 mL of glycerol–water mixtures. Owing to the lower or higher densities of beads compared with those of glycerol solutions, the beads were buoyed up or sunk down in the
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Fig. 1. Schematic description of bead motion in various glycerol–water mixtures (the darker color means higher mass fraction of glycerol in mixture); (a) 50 wt% glycerol solution, the digital pictures of a1 and a2 were taken at 1.2 and 2.7 h, respectively. (b) 23 wt% glycerol solution, b1 and b2 at 3.5 and 7.5 h, respectively. (c) 10 wt% glycerol solution, c1 and c2 at 0.7 and 1.4 h, respectively. The beads were obtained from Bangs Laboratories, Inc.
solution. The total time taken for entire-bead floating to the surface or sedimentation to the bottom of glycerol solutions was defined as floating or sedimentation time of beads. The accuracy of the measurements was evaluated from 5 repetitive determinations (repeatability) of the motion time of polymer beads in the same glycerol aqueous solution, and the relative standard deviations (RSD) were also calculated. The schematic process and digital photos are shown in Fig. 1. We have included more digital photos taken at different time in supporting information (see Fig. S1). 2.3. Characterization The images of beads were taken using a microscope (Olympus, IX 51). Flow cytometry (Coulter, Epics XL) was utilized to measure the bead’s relative size (forward light scatter, FS signal), relative granularity or internal complexity (side light scatter, SS signal). Flow cytometry was set up according to the standard operating procedure. During detection, a total of 20000 events were recorded for each time point, and the beads dispersed in water or glycerol solutions were detected at the low flow rate with a concentration of 1 mg/mL. The populations of single beads and doublet beads were
selected based on forward and side scatter. The mean and RSD of these populations were also measured. All flow cytometric data were performed using FlowJo 5.7 software (Tree Star, San Carlos, CA, USA). 3. Results and discussions 3.1. Densities and viscosities of glycerol–water mixtures The densities of glycerol–water mixtures were measured by comparing their mass to their volume. The viscosities were calculated using the following empirical equation [10]. ln m − ln w 2 = Cm [1 + (1 − Cm ){a + bCm + cCm + · · ·}] ln g − ln w
(1)
where is kinematic viscosity ( = /), is dynamic viscosity, is the density, Cm is the mass fraction of glycerol in the mixture, and subscripts m, g and w denote the glycerol–water mixtures, glycerol and water, respectively. The constants a, b and c at 20 ◦ C are assigned to −0.76728, 0.12153 and −1.41519, respectively. As shown in Table 1, the detected densities are close to those provided by the vendor, and the interpolated viscosities are in accordance
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with those published [10] (the comparison between the densities and viscosities in the present study and those published were shown in Tables S1 and S2 in supporting information, respectively). 3.2. Theory of bead movement in aqueous solution The net force, Fb , of a bead in aqueous solution is composed of following two contributions, Fb = Fb,1 − Fb,2
(2)
where Fb,1 and Fb,2 are the sum of the buoyant force and the bead’s weight, and the force of movement resistance (Stokes’ drag force), respectively. According to Archimedes’ principle, the Fb,1 is given by, Fb,1 = Vb × (m − b ) × g Fig. 2. The floating and sedimentation time of polymer beads in various glycerol–water mixtures, the positive and negative values indicate that the beads are floating and sedimentating, respectively.
Table 1 The density and viscosity of glycerol–water mixtures. Mass fraction [g/g]
Density [g/mL]
viscosity [mPa s]
0.0000 0.0996 0.1510 0.1758 0.2002 0.2084 0.2202 0.2301 0.2402 0.2502 0.2601 0.3006 0.3802 0.5000
0.9984 1.0218 1.0323 1.0392 1.0449 1.0512 1.0537 1.0558 1.0594 1.0606 1.0634 1.0727 1.0925 1.1251
1.0085 1.2804 1.4920 1.6178 1.7546 1.8123 1.8868 1.9527 2.0260 2.0974 2.1744 2.5252 3.4571 5.9486
(3)
where Vb is the volume of a bead, m and b are the density of the aqueous solution and hybrid polymer beads (or magnetic beads), respectively, and g is the gravitational acceleration. The force of the movement resistance (Stokes’ drag force [11]) Fb,2 is obtained by the following equation. Fb,2 = 6Rv
(4)
where is the viscosity of aqueous solution, R is the radius of a bead, and v is the velocity. Hence, the Fb is deduced by, Fb = Fb,1 − Fb,2 = Vb × (m − b ) × g − 6Rv =
4 3 R × (m − b ) × g − 6Rv 3
(5)
Based on Eq. (5), the suspended bead has a tendency to rise or sink in the glycerol–water mixture if b is not equal to m . The beads were buoyed up in aqueous solutions when their densities were less than those of the mixture (Fig. 1a). The beads were stably suspended in solutions as their densities were accurately equal to those of the
Fig. 3. Estimation of the density of (a) polymer beads and (b) commercial magnetic beads by Eq. (8), the apparent density of beads, b is equal to the m of the point where the fitted linear curve intercepts the line x = 0. Table 2 The flow cytometric data of polymer beads shown in Fig. 4. Mass fraction of glycerol [wt%]
4a 4b 4c 4d 4e
0 10 20 24 30
FS in R1
SS in R1
The proportion of beads in
Mean
RSD [%]
Mean
RSD [%]
R1 [%]
R2 [%]
436.22 458.66 483.35 541.42 746.31
6.28 5.53 7.38 9.71 20.9
427.33 423.74 430.75 425.56 445.70
13.57 13.33 13.22 13.71 25.42
81.48 89.42 89.52 – –
15.78 7.36 2.97 – –
The ratio of doublet beads (in R2) to singlet beads (in R1) [%]
19.4 8.2 3.3 – –
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Q. Zhang et al. / Talanta 80 (2010) 1681–1685
mixture (Fig. 1b), and the beads were sunk to the bottom of the vial while their densities were larger than those of the mixture (Fig. 1c). 3.3. Estimation of bead density The floating velocity is defined as positive, and sedimentation velocity as negative. As shown in Eq. (5), the velocity (floating or sedimentation) of beads reaches the maximum at equilibrium states (Fb = 0), and the equation of equilibrated floating or sedimentation velocity is given as follows (transformation of Eq. (5)),
v=
2R2 g(m − b ) 9
(6)
Since t is equal to d/v, the time (t) of floating or sedimentation is derived by, t=
9d =k· m − b − b )
2R2 g(m
k=
9d 2R2 g
(7)
where the distance (d) of floating or sedimentation is a constant, and consequently, k is a constant. According to Eq. (7), the floating or sedimentation time depends on the viscosities of aqueous glycerol solutions and the densities difference between the beads and the solutions [8]. Fig. 2 showed that the detected floating or sedimentation time of polymer beads is in accordance with Eq. (7). Moreover, the density of polymer beads can be obtained by using the floating or sedimentation time. Changing the expression of Eqs. (7) and (8), the result is represented as follows, m − b = k ×
⇒ m = b + k × t t
(8)
where the m and are shown in Table 1, and the motion time (t) is illustrated in Fig. 2. The density, m and the /t are illustrated in Fig. 3a. The b fitted by linear least square method is 1.0570 g/cm3 ((1.0563 + 1.0576)/2). The calculated RSD was 0.07%. As shown in Table 1, the densities of 25 and 26 wt% glycerol solutions were 1.0606 and 1.0634 g/cm3 , respectively. The bead density provided by the vendor (b,vendor = 1.062 g/cm3 ) stands between the densities of 25 wt% (b,vendor > m,25% ) and 26 wt% glycerol solutions (b,vendor < m,26% ). However, in experiments, the bead were buoyed up in both 25 wt% and 26 wt% glycerol solutions (see Fig. 2), which indicated that the density (apparent) of beads was lower than those (b,apparent < m,25% < b,vendor < m,26% , see Eq. (5)). In other words, by adopting the vendor-provided density, the movement of beads in aqueous solution cannot be precisely predicted. In contrast, the motion of hybrid beads in solution can be more accurately estimated by using the detected apparent density (see Eqs. (6) and (7) and Fig. 2). Similarly, as shown in Fig. 3b, the detected density of magnetic beads, 1.0941 g/cm3 , is a little bit smaller than that provided by vendor. The RSD obtained was 0.11%. The smaller measured densities are probably due to surface hydrophilic liquid layer [8] and inner nano-scale porous [9]. 3.4. Influence of the motion of beads on flow cytometric analysis The motion of beads plays a crucial role in the flow cytometric analysis. As illustrated in Fig. 4 and Table 2, the mean of forward light scatter (FS) signals are increased with increasing of glycerol mass fraction. In the mean time, those of side light scatter (SS) signals are close to a constant. The light signals are generated as beads pass through the laser beam in a fluid stream. And the FS signal is proportional to the projected spot area or size which is related to the size of beads and the distance between the beads and the sensor [5b]. When the motion tendency of beads was changed from
Fig. 4. Flow cytometry analysis of polymer beads. The dispersion solutions of a–e are pure water, 10, 20, 24 and 30 wt% glycerol solutions, respectively. (a1 –e1 ) The bead subpopulations based on side light scatter (SS) vs. forward light scatter (FS); (a2 –e2 ), the histogram of size distribution. The singlet beads were counted in the region 1 (R1, a1 –e1 ), the doublet beads were recorded in the red region 2 (R2, a1 –c1 ) and marked by red arrows in the diagrams a2 –c2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
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beads in aqueous solution than the general method. This work is beneficial to micro-total analysis systems, separation, suspension stability [15], particle analysis systems and so forth. Acknowledgment This work was supported by the Center for Nanostructured Application and Nanobiotechnology Initiative at the University of Minnesota and by the NSF BME 0730825. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.talanta.2009.10.004. References
Fig. 5. The microphotograph of polymer beads dispersed in a solution containing 5 wt% glycerol and 0.05 wt% SDS, and the doublet beads were marked by arrows.
sedimentation to floating (Fig. 4a–e), the distance was increased, resulting in a larger FS signal. In addition, the increase of FS signals is not mainly due to the diffusion of glycerol from the core flow to the sheath flow [12], otherwise, the mean and RSD of SS signals should be also changed (see Table 2). Moreover, the proportion of doublet beads is decreased with the prolongation of suspension time (Fig. 4a–c). The doublet beads suspended in a liquid are shown in Fig. 5. Their signals might affect the accuracy of flow cytometry detection [13]. The numbers of doublet beads depend on previous treatment of the suspension (e.g., sonication and vortex), age, and solvent [14]. Since all the samples were performed with the same operations, the proportion of doublet beads can be reduced by extension of their suspension time. 4. Conclusions Theoretical description for the motion, including floating or sedimentation of polymer bead in aqueous solution was deduced by analysis of the net force of a liquid-suspended bead. Furthermore, the apparent density of beads can be easily calculated by using the measured floating or sedimentation time. For polymer beads and commercial magnetic beads, the detected apparent densities in the present study were 1.0570 and 1.0941 g/cm3 , respectively. The analysis showed that this milligram-scale method provided a more facile and precise approach to predict the movement of hybrid
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