Probing the Interplay of Size, Shape, and Solution Environment in

In the Laboratory

Probing the Interplay of Size, Shape, and Solution Environment in Macromolecular Diffusion Using a Simple Refraction Experiment Bijith D. Mankidy, Cecil A. Coutinho, and Vinay K. Gupta* Department of Chemical and Biomedical Engineering, University of South Florida, Tampa, Florida 33620 *[email protected]

Diffusion of polymers and biopolymers is of considerable significance in fields such as biology, biomedicine, chemical separations, and polymer technology (1-4). Polymeric diffusion in liquids has been widely studied owing to its relevance in inkprinting, cosmetics, soil conditioning, and coatings (4-6) as well as its fundamental scientific relevance (7). Recently, investigators at the National Institute of Standards and Technology have shown that consideration of size in the diffusion of nanosized oligomeric additives in concrete can be key in viscosity modification and can lead to doubling the service life of concrete structures (8). Diffusion of proteins, peptides, and polymeric carriers is also an important factor in their accumulation within pathological tissue, delivery to infected or inflamed sites, in ocular treatments, and in transdermal drug delivery (9, 10). With the increasing focus on nanoscience and nanotechnology, diffusion of macromolecules such as DNA and polypeptides has become an important aspect of nanofluidic devices (11, 12). It is, therefore, important to incorporate laboratory measurements that emphasize the concepts of macromolecular diffusion into the undergraduate curricula of physical chemistry, chemical engineering, and environmental and biological sciences. Methods to measure polymer diffusion coefficients have evolved over time from dynamic light scattering measurements (DLS) and pulsed nuclear magnetic resonance spectroscopy (NMR) to more sophisticated methods such as positron annihilation laser spectroscopy (PALS) and fluorescence correlation spectroscopy (FCS) (13-18). None of these methods are particularly suitable for an undergraduate laboratory given the cost, complexity, and instructional requirements. We demonstrate that a visually instructive laser refraction experiment based on Wiener's method (19, 20) can be readily implemented in the undergraduate laboratory to measure polymer diffusivities in solution. Even though the simplified setup of the refraction method provides approximate values for diffusion coefficients of a polymer, the ease and low cost of the setup make the approach attractive for instructional purposes. Past reports (21-23) have utilized this optical technique for undergraduatelevel demonstrations of diffusion of single solutes such as KCl or CsCl in liquids such as water or glycerol in a physical chemistry laboratory. In a recent study (38), we have shown that the method can be also used to analyze diffusion in ternary mixtures of simple species such as KCl and sucrose in water. Here, the refraction experiment is used to underscore the interplay of size, shape, and solution environment on diffusion coefficients of common

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linear and globular polymers. We also demonstrate incorporation of features such as digital capture of data using a webcam, data digitization, and multivariable data regression using common software (Microsoft Excel). Use of these tools in tandem with diffusion theory, polymer science concepts, and a hands-on laboratory experiment provides an opportunity to practice skills valuable throughout a student's academic program and professional career in science and engineering. The experiments reported here analyze two synthetic polymers, poly(vinyl alcohol) (PVA) and poly(ethylene glycol) (PEG), that are representative of flexible, linear macromolecules. PVA is widely used in industry in pigments, adhesives, or composite materials, whereas PEG is widely used in cosmetics as well as preparation of biocompatible surfaces (5, 6, 24). In addition to PVA and PEG, a globular protein such as bovine serum albumin (BSA) and an oligosaccharide such as chitosan are analyzed. BSA is widely used in immunoassays, microbiology, and affinity chromatography (24, 25) and chitosan is extremely useful in biomedicine, food applications, waste and water treatment, and agricultural applications (26, 27). Theory and Data Analysis Since past reports (21-23) have described the theoretical background in detail, only a brief summary is provided here. One-dimensional diffusion of simple, small molecular-weight solutes and polymers occurs from a region of higher concentration to a region of lower concentration. As shown in Figure 1A, a beam of laser is fanned at an angle of 45° onto a rectangular cuvette using a glass rod that is inclined at 45° from the horizontal. The transmitted beam is projected on a screen to observe the refraction of the laser light. If there is no refractive index gradient in the solution along the vertical direction, a straight baseline is obtained on the screen, pqr (Figure 1A). On the other hand, when a refractive index gradient exists along the vertical direction the laser, light is deflected to form a curve on the screen, pq0 r. Over time the solute diffuses toward the upper layers of the solvent and the decrease in the refractive index gradient results in the flattening and broadening of the deflection curve pq0 r. The one-dimensional diffusion is governed by Fick's law (20) and the spatial gradient of the concentration distribution has been shown to follow " # Dcðy, tÞ c0 ðy - y0 Þ2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð1Þ Dy 4Dðt - t0 Þ 4πDðt - t0 Þ

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r 2010 American Chemical Society and Division of Chemical Education, Inc. pubs.acs.org/jchemeduc Vol. 87 No. 5 May 2010 10.1021/ed800159k Published on Web 03/10/2010

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Here, Δn0 represents the difference in the refractive index between the solution and solvent at the initial time (t0). The value of Δn0 depends on the initial weight fraction of the solute and is typically available in literature (32) or can be measured using a refractometer. The deflection curve is an optical manifestation of eq 4 at any given time (t). To fit the experimental data, the x and y coordinates of the optical trace pq0 r and the baseline pqr are obtained. This raw data can then be baseline corrected by using the following equations: xd ¼ xcurve ¼ xbaseline

ð5Þ

yd ¼ ycurve - ybaseline The data points (xd, yd) are related to eq 4 by using scaling constants ψ1 and ψ2 that also incorporate distances A and B and the cuvette width W, shown in Figure 1A: " # ψ1 Δn0 ψ2 ðxd - x0 Þ2 yd ðxd , tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð6Þ 4Dðt - t0 Þ 4πDðt - t0 Þ Figure 1. (A) Schematic of experimental setup. (B) Illustration of refraction of light as it transmits though the cuvette. (C) Picture of the optical deflection curve formed on the screen.

Regression of the experimentally recorded deflection curve at various times using the eq 6 yields an estimate of the diffusion coefficient, D.

where c represents concentration of a diffusing solute as a function of time, t, and vertical position, y, in the cuvette; D is the diffusion constant of the solute; y0 is the position of the boundary between the solute solution and the pure solvent and c0 is the initial concentration of the polymer at time t0. The refractive index gradient that results from the concentration gradient leads to bending of visible light. This bending at one point in the cuvette is illustrated in Figure 1B. Snell's law of refraction gives

Materials and Equipment

n sinðjÞ ¼ ðn þ ΔnÞ sinðj þ ΔjÞ

ð2Þ

where n is the refractive index at one location, Δn is the change in the refractive index at an adjacent layer of liquid, j is the angle of the light path with respect to the surface normal, and Δj is the change in the angle. Jost (20) has shown that eq 2 can be used to obtain the radius of curvature (R) of the deflected light and thereby, the vertical deflection (ycurve) of the curve that forms on screen 1 Δj dn=dy BWn dn ¼ lim ¼ w ycurve ¼ ¼ BW R Δy f 0 Δz n R dy ð3Þ where B is the distance between the cuvette and the screen and W is the cuvette width (see Figure 1A). Thus, the experimentally recorded deflection curves are directly related to the refractive index gradient, which depends on the concentration gradient. Experiments and theory (28-30) have proven that under dilute conditions the refractive index of a liquid solution is a linear function of the concentration. Jin and co-workers (31) have analytically shown this linear dependence via a simple approach based on the relationship between the refractive index and molecular polarizability. A brief summary of this analysis is given in the supporting information. Because of this linear dependence, we can approximate the refractive index gradient as " # Dnðy, tÞ Δn0 ðy - y0 Þ2 µ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð4Þ Dy 4Dðt - t0 Þ 4πDðt - t0 Þ

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Chitosan (catalog no. 523682), PEG (catalog no. 202444), BSA (A3059), and phosphate buffer saline (PBS) were purchased from Aldrich. PVA (catalog no. 81381) was obtained from Fluka. Aqueous solutions with 10 mass % solute were prepared using deionized water from an EasyPure UV system (Barnstead, IA). For experiments using BSA, the pH of the solution was either maintained at 7.4 by using PBS or it was adjusted to 5.0 by addition of 0.01 M HCl. A 5 mW laser diode module (model 31-0508 Coherent Inc., Santa Clara) with wavelength of 670 nm was fixed to an optical breadboard and the laser light was fanned by using a glass rod (3 mm diameter) to project a thin band of laser light at 45° to the horizontal. A similar arrangement has been used in past reports (21, 22). The distances A and B shown in Figure 1A can be adjusted so that a clear magnified image will be formed on the screen. An acrylic cuvette (Cynmar Corporation, IL) of dimensions 60 mm  30 mm  62 mm was half-filled with a 10 mass % polymer solution. The deionized water was carefully added dropwise. Care has to be taken at the beginning of the experiment to form a sharp interface. Toward this end, a small piece of corkboard was used as a float to blunt the impact of the drops and minimize mixing and forced convection. A deflected curve pq0 r as shown in Figures 1A and C is developed immediately as the result of the refractive index gradient along the vertical direction in the diffusion cuvette. The cuvette was adjusted vertically by means of a lab jack to position the deflection curve on the screen. Images formed on the screen were photographed using a digital webcam (Rosewill RCM-32301). Pictures were taken automatically at regular intervals of time using Automouseclicker (version 2.10), which is a freely licensed software available for download on the Web (33). Hazards The polymers, PVA, BSA, PEG, and chitosan, do not present much of a health hazard, but may form combustible

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In the Laboratory

Figure 2. Optical deflection data for diffusion of BSA in pH 7.4 solution.

Figure 3. Baseline corrected data for BSA at pH 7.4 with the theoretical fit from eq 6.

dust concentrations in air. Students should be warned not to stare into the laser beam. A 0.01 M HCl solution should be prepared carefully and students should be advised to handle it with caution.

Table 1. Diffusivities of Polymers Diffusivity/(10-11 m2 s-1) Solute

Results and Discussion The optical deflection curves of 10 solution of BSA at pH 7.4 at different intervals of time obtained after digitization of the captured images are shown in Figure 2. The data are then baseline corrected using eq 5 and regressed to eq 6 as a function of variables for position, xd, and time, t. The fitting parameters are Δn0, x0, t0, and D. Prior to characterizing the polymeric systems, the scaling constants ψ1 and ψ2 were determined experimentally using potassium chloride KCl in water (in an undergraduate laboratory, this step may be completed beforehand by a teaching assistant). Data were collected using 10%, 15%, and 23% by mass solutions of KCl. For each solution, the value of Δn0 was taken from literature (32) and eq 6 was used to fit the yd(xd,t) data with a common set of fitting parameters ψ1, ψ2, and DKCl. Spreadsheet software, such as Microsoft Excel with the Solver feature, can be easily used for the regression analysis of data using the least sum-squared error method to obtain a best fit. Alternatively, students can use a programming environment such as Matlab or Mathematica. For BSA, the baseline corrected data and the fitted curves are shown in Figure 3. It is well-known that solute interactions with its environment and the molecular size and shape of the solute will affect the diffusion coefficient (15, 34). Therefore, diffusion measurements were performed on chitosan, PEG, PVA, and BSA (pH 7.4 and pH 5) to demonstrate the interplay of molecular size, shape, and environment in the diffusion of polymers (Table 1). Most common polymerization procedures lead to macromolecules of different sizes and control of this polydispersity in size is important for the functional properties of the polymeric material (34). For polymeric diffusion coefficients, size plays an important role and oligomers or chains with small molecular size diffuse faster than bulky, high molecular weight polymers. The results in Table 1 support this. Specifically, the higher molecular weight polymers such as PVA and BSA have smaller diffusion coefficients than the lower molecular weight materials PEG and chitosan.

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Average Molecular Weight/(103 Da)

Experimental

Literature

∼5

39.96

-

PEG

∼3-3.7

13.16

10.50a

PVA

∼ 22 (Mw); ∼ 8.5 (Mn)

5.27

4.40a

BSA

∼66

Chitosan

a

6.23 @ pH 7.4 6.00 (15), 6.81 20.42 @ pH 5 (1, 35) @ pH 7.4

DLS measurement using a Malvern Nanosizer instrument.

The impact of shape on diffusion coefficient is also highlighted in Table 1. Even though PEG and chitosan used in the experiments have comparable molecular weights, the diffusivity of chitosan is found to be almost three times the value for PEG. It is well-known that intramolecular and intermolecular hydrogen bonding in chitosan can lead to a more globular structure for the oligosaccharide, which results in more rapid translational diffusion than for a linear PEG chain (26, 27, 36). This impact of shape is further emphasized when the values for PVA and BSA (at pH 7.4) are compared. BSA has a much higher molecular weight than PVA, but its diffusivity is comparable to PVA because the protein is folded into a more compact structure relative to PVA, which remains in an extended conformation in aqueous solutions. In a polymer solution, the polymer-solvent interactions are extremely significant in controlling the conformation of a macromolecule structure. It is well-known that solution properties such as pH can alter the three-dimensional shape of proteins such as BSA (37). The effective size of BSA is altered by changes in pH because it influences the intramolecular electrostatic interactions. BSA shrinks to a more compact size at a lower pH, thus decreasing its effective size. At a pH of 5, the decrease in height of the optical deflection curve is more rapid than that in Figure 3 (pH = 7.4), which is consistent with an increase in diffusivity of BSA owing to decreasing pH of the solution. Regression of the deflection curves shows that the estimated

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diffusion coefficient (Table 1) increases by nearly three times upon lowering the pH. Summary The tabulated results and comparison of the diffusion coefficient values with literature data or the values measured with techniques such as DLS indicate a fairly good agreement in view of the simplicity of the refraction technique. From the perspective of undergraduate instruction, the experimental method offers a rapid, visually instructive, and easy method for studying diffusion of polymers. The low cost of the optical technique, the ease of the experiment, and the range of educational concepts that can be introduced add to the value. The laboratory experiment reinforces concepts in chemical and biological sciences such as molecular basis of macroscopic properties and diffusional mass transfer alongside mathematical concepts of partial differential diffusion equation, the Gaussian form of the concentration gradient, and multivariable functions. Because the estimate for D depends on factors such as the accuracy with which the deflection image is digitized, the care taken to minimize convection in the initial setup of the experiment, the number of time intervals chosen for capture of the deflection curve, and the regression criterion, the experiment described here can be extended to emphasize aspects such as error analysis of the data. The hands-on learning opportunity also functions as a pedagogical tool for practicing skills such as data acquisition, graphing, and regression using common software such as Microsoft Excel. Finally, investigations on impact of polymer solute concentration, temperature, type of solvent, and the nature of polymer can easily form the basis of mini-projects for teams of students. Acknowledgment This research was carried out with the support of the Department of Chemical and Biomedical Engineering at the University of South Florida. Graduate support in the form of a teaching assistantship from NSF grant on Curriculum Reform (EEC-0530444) to C.A.C. and B.D.M. is also acknowledged. Literature Cited 1. Geankoplis, C. J. Transport Processes and Unit Operations, 4th ed.; Prentice Hall: Englewood Cliffs, NJ, 1978. 2. Vrentas, J. S.; Duda, J. L. Encycl. Polym. Sci. Eng. 1986, 5, 36–68. 3. Peppas, N. A. Transdermal Delivery Drugs 1987, 1, 17–28. 4. Hansen, C. M. Prog. Org. Coat. 2004, 51, 55–66. 5. Polyvinyl Alcohol, Properties and Applications, 2nd ed.; Finch, C. A., Ed.; John Wiley & Sons, Inc.: New York, 1973. 6. Harris, J. M.; Zalipsky, S. Poly( ethylene glycol): Chemistry and Biological Applications, 1st ed.; American Chemical Society: Washington, DC, 1997. 7. Granick, S.; Bae, S. C. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 3434–3435.

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8. Bentz, D. P.; Snyder, K. A.; Cass, L. C.; Peltz, M. A. Cem. Concr. Compos. 2008, 30, 674–678. 9. Grassi, M.; Grassi, G. Curr. Drug Delivery 2005, 2, 97–116. 10. Siddique, S.; Khan, M. Y.; Verma, C. J.; Pal, T. K.; Khanam, J. Pharma Rev. 2008, 6, 144–147. 11. Stein, D.; van der Heyden, F. H. J.; Koopmans, W. J. A.; Dekker, C. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 15853–15858. 12. Hsieh, C.-C.; Balducci, A.; Doyle, P. S. Nano Lett. 2008, 8, 1683–1688. 13. Longsworth, L. G. Ann. N.Y. Acad. Sci. 1945, 46, 211–239. 14. Crank, J.; Park, G. S. Diffus. Polym. 1968, 1–39. 15. van Holde, K. E. Physical Biochemistry; Prentice-Hall: Englewood Cliffs, NJ, 1971. 16. Schiller, J.; Naji, L.; Trampel, R.; Ngwa, W.; Knauss, R.; Arnold, K. Methods Mol. Med. 2004, 101, 287–302. 17. Lv, H.-l.; Wang, B.-g.; Yang, J.-c. Desalination 2008, 234, 33–41. 18. Grabowski, C. A.; Mukhopadhyay, A. Macromolecules 2008, 41, 6191–6194. 19. Weiner, O. Ann. Phys. Chem. N. F. 1893, 49, 105. 20. Jost, W. Diffusion in Solids, Liquid, Gases. In Physical Chemistry: A Series of Monographs; Academic Press, Inc.: New York, 1960. 21. King, M. E.; Pitha, R. W.; Sontum, S. F. J. Chem. Educ. 1989, 66, 787–90. 22. Sattar, S.; Rinehart, F. P. J. Chem. Educ. 1998, 75, 1136–1138. 23. Rashidnia, N.; Balasubramaniam, R.; Kuang, J.; Petitjeans, P.; Maxworthy, T. Int. J. Thermophys. 2001, 22, 547–555. 24. Encyclopedia of Polymer Science and Technology, 3rd ed.; Mark, H. F., Ed.; John Wiley & Sons, Inc.: New York, 2004. 25. Handbook of Affinity Chromatography, 2nd ed.; Hage, D. S., Ed.; CRC Press: Boca Raton, FL, 2005. 26. Shigemasa, Y.; Minami, S. Biotechnol. Genet. Eng. Rev. 1996, 13, 383–420. 27. Dutta, P. K.; Dutta, J.; Chattopadhyaya, M. C.; Tripathi, V. S. J. Polym. Mater. 2004, 21, 321–333. 28. Glover, F. A.; Goulden, J. D. S. Nature 1963, 200, 1165–1166. 29. Miyake, Y.; Furuichi, J. J. Phys. Soc. Jpn. 1961, 16, 1263. 30. Matsumoto, M.; Oyanagi, Y. J. Polym. Sci. 1960, 31, 225–227. 31. Jin, Y. L.; Chen, J. Y.; Xu, L.; Wang, P. N. Phys. Med. Biol. 2006, 51, N371–N379. 32. International Critical Tables of Numerical Data, Physics, Chemistry and Technology, 1st Electronic Edition; Washburn, E. W., Ed.; Knovel: New York, 2003. 33. AutoMouseClicker. http://www.murgee.com/auto-mouse-clicker/ 34. Wu, Y.-H.; Wang, D.-M.; Lai, J.-Y. J Phys. Chem. B 2008, 112, 4604–4612. 35. Charlwood, P. A. J. Phys. Chem. 1953, 57, 125–128. 36. Tsaih, M. L.; Chen, R. H. J. Appl. Polym. Sci. 1999, 73, 2041–2050. 37. Michnik, A.; Michalik, K.; Drzazga, Z. J. Therm. Anal. Calorim. 2005, 80, 399–406. 38. Coutinho, C. A.; Mankidy, B. D.; Gupta, V. K., Chem. Eng. Edu., in press.

Supporting Information Available Detailed student handout. This material is available via the Internet at http://pubs.acs.org.

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