Amorphous Polymers: Polymer Conformation Laboratory 1: Module 1

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1 D E P A R T M E N T O F M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G M A S S A C H U S E T T S I N S T I T U T E O F T E C H N O L O G Y Materials Laboratory Fall 2008 Amorphous Polymers: Polymer Conformation Laboratory 1: Module 1 Objectives: Agathe Robisson - robisson@mit.edu Tracey Brommer - tbrommer@mit.edu Review random walk model and self-avoiding walk model for polymer chains Introduce dynamic light scattering as a technique to obtain dimensions of polymer chains in solution Investigate solvent influence on polymer coil dimension Tasks: Prepare dilute polymer solutions Perform dynamic light scattering on all solutions to obtain hydrodynamic radius Determine scaling relationship between radius and molecular weight for polymers Investigate solvent effect on chain dimensions INTRODUCTION Polymer Conformations: Polymers are covalently bonded long chain molecules composed of repeating units, called monomers, comprised of carbon and hydrogen, and sometimes oxygen, nitrogen, sulfur, silicon and/or fluorine atoms. The polymer structure is generated during polymerization. The number of monomers covalently bonded in the chain is called the degree of polymerization. Typically, this value varies from 500 to 10,000. The polymer microstructure depends on the way atoms are organized along the chain: monomers can assemble in different isomers (organization that cannot be changed without breaking covalent bonds). Fig. 1 displays different stereoisomers of polystyrene (PS). The polymer microstructure and its physical properties strongly depends on the stereoregularity of the chain pendant groups. Atactic polystyrene, which is widely used commercially, is entirely amorphous due 1

2 to the random arrangement of the pendant phenyl groups, while syndiotactic and isotactic polystyrene, having more regular structures, exhibit crystallinity. Isotactic PS (highly crystalline) H C C H H Styrene monomer Syndiotactic PS (semi-crystalline) Atactic PS (amorphous) Fig. 1 Tacticities of PS After polymerization, a single molecular chain can adopt many different conformations. The number of conformations depends on three characteristics: 1) the chain flexibility (some chains are stiff like piano wires, some others are flexible like silk thread), 2) the interactions between monomers in the chains (attractive or repulsive) 3) interactions with the surroundings. Figure 2 illustrates a polymeric chain with bonds (one of the largest DNA molecules), magnified 100 million times (10 8 ): Bond length is around 1 cm (versus 1 Å). 5 If attractions between monomers are strong, the polymer is a dense objects (collapsed globule), Fig 2a. If there is no interaction between monomers, the chain conformation is a random walk, Fig. 2b. If attractions between monomers are repulsive, the conformation is a self-avoiding walk, Fig. 2c. Finally, if long range repulsions (such as electrostatic) between monomers exist, the conformation is extended, Fig. 2d. 2

3 Fig. 2 Illustration of conformations for a polymer with bonds of length 1cm. 5 Polymer Melts We will focus here on the case where there is no interaction between monomers. This ideal chain can be described using a random walk statistics (also called ideal chain statistics). A random walk denotes a path of successive steps in which the direction of each step is uncorrelated with or independent of the previous steps: Steps forward and backward, left and right, up and down are all equally probable. Figure 3 describes two possible states of a chain with 10 segments. The flexibility of the chain is mainly due to torsion angles: the chain adopts gauche and trans bond conformations along the backbone (a single chain can adopt many different conformations). Fig. 3 Two possible conformations of a 10-segment chain Random walk statistics are also used to describe Brownian motion. 3

4 Fig. 4 Realization of self-random walk on square-lattice: ideal random walk step simulation. 1 Let s describe an ideal chain of N+1 backbone atoms. Because of the equal probability to step in opposite directions, the average end-to-end vector of a random walk of N steps, taken over many possible conformations, is identically zero. <R> = 0 The root-mean-square (rms) end-to-end distance, however, is finite, and characterizes the average spatial dimension traversed. If the length of each step is l, the rms end-to-end distance can be shown to be 3 l (1) <R 2 > 1/2 = N 1/2 l R This model is valid for freely jointed chains. In most typical polymers, restrictions on bond angles occur and mathematical developments lead to introduce a characteristic ratio C. The rms end-to-end distance R 0 for the polymer chain of N bonds is given by: (2) R 0 = <R 2 > 1/2 = N 1/2 C 1/2 l R 0 4

5 l is ~1.54 Å for a C-C bond. The characteristic ratio C depends on the particular polymer. For polystyrene, C is ~10.8 at room temperature. 1 A polystyrene chain of N = 10,000 monomers has 2N C-C bonds. Its fully extended length would be 2.N.l = 30,800 Å = 3,800 nm and its rms end-to-end distance would be expected to be R o 716 Å 72 nm in the melt or glassy state. An important feature of this result is that the polymer chain has a characteristic size, which scales with the number of monomers in the chain to the one-half power: (3) R o ~ N 1/2 The average volume occupied by the coil is therefore much greater than the volume of the chain itself, so that in the melt or glassy state, many other chains will be found intermingled with a single chain. This is very different than in the case of small molecule systems and leads to the unique rheological and mechanical properties exhibited by polymers. (4) V coil R o 3 ~ N 3/2 (5) V chain = N.v monomer ~ N (6) V coil/v chain ~ N 1/2 = Approximate available volume for other chains to interpenetrate the coil, creating entanglements between chains. Note that entanglement increases with N: the higher the molecular weight, the higher to polymer melt viscosity. Polymers in solution When polymers are dissolved in good solvents, the chains similarly take on coil-like conformations, but are often somewhat swelled, as a result of favorable interactions between monomer and solvent. The conformations of polymer chains in dilute solutions are well described as self-avoiding walks (SAWs). Whereas a random walk is able to cross its own trajectory, a SAW disallows such configurations the chain is said to have excluded volume. 1,5 This gives rise to an effective swelling. The rms end-to-end distance would then follow: 5

6 (7) R SAW ~ N 3/5 The rms end-to-end distance is larger for a self-avoiding walk than for a self-random walk. Compare Figure 5 to Figure 4. Fig. 5 Realization of self-avoiding random walk on square-lattice 100 step simulation. 1 Note that the degree of swelling (and therefore the exponent) is also a function of the solution temperature raising the temperature typically causes the chain dimensions to expand, while at low temperatures and in poor solvents, chains can collapse into a dense, globular state, before precipitating out of solution. 6 Figure 6 compares swollen and globular states. Fig. 6 - Monte Carlo simulation of swollen vs. globular states for N =

7 The above theories developed by Flory were generalized to a universal power law: (8) R ~ N n Where n is the scaling exponent. n = 0.5 for a theta-solvent (excluded volume = 0: the chain adopts the same conformation as in the melt), n 3/5 for a good-solvent, n < 0.5 for a poor solvent. Measurement of Coil Dimensions by Dynamic Light Scattering There are a number of experimental techniques by which the coil dimensions for a polymer chain can be measured. The hydrodynamic radius R h of a polymer chain in dilute solution can be determined from dynamic light scattering (DLS) measurements. 5,7 The hydrodynamic radius is directly proportional to the rms end-to-end distance for linear chains: (9) R h = const. <R 2 > 1/2 Dynamic light scattering measures the fluctuation in intensity of scattered light that occurs due to Brownian motion of the particles. The velocity of those fluctuations is related to the diffusion coefficient of the particles, which is used to calculate particle size using Stokes Einstein equation. When incident light scatters from many polymer coils, destructive (case A, Fig. 7) and constructive (case B, Fig. 7) interferences can be observed. 7

8 Fig. 7 - Schematics of two scattered beams interfering case A: destructive interference, case B: constructive interference. 2 The intensity of the scattered light fluctuates because the particles are moving in the suspension. The intensity fluctuates slower for large particles due to their slower diffusion rate. The decay of these fluctuations is related to the rate at which the polymer chains diffuse through solution as described by the Stokes-Einstein relation: (10) R h kt 6 D D is the diffusion coefficient, h s is the solvent viscosity, k is Boltzmann s constant, and T is temperature in Kelvin. s The decay in intensity is measured by comparing the intensity of fluctuations at time t, I(q,t) with the intensity at time t + t, I(q,t+t). This can be done by using the correlation function <I(q,t)I(q, t+t)>. The correlation function can be written for dilute monodisperse solution of coils: (11) < I(q,t) I(q,t+t) > = A + B exp (-2Gt) G is the characteristic decay rate, defined as G = D q 2, with D diffusion coefficient, and q is the scattering vector defined by: (12) 4 q n s sin where q is the scattering angle (90), l is the radiation wavelength (657 nm), and n s is the refractive index of the solvent. Those values need to be entered in the software. From this expression, we observe the auto-correlation function takes an initial value of 8

9 A+B but decays to A for times much larger than the characteristic correlation time t. Physically, the memory of the initial location of chains in the measured volume at t=0 is lost for times t >> t, when most chains have diffused out of the scattering volume. A + B A Because the chains behave as Brownian particles, the correlation time t is related to the distance traveled (1/q) by: time (13) 1 q 2D where D is the diffusion coefficient for the coil in solution. Fitting experimental curves allows the calculation of the diffusion coefficient, D which can be related to the hydrodynamic radius R h if the viscosity of the solvent is known. 9

10 EXPERIMENTAL (see also Lab Procedure) Materials to be investigated: Polystyrenes of several molecular weights Solvent: Tetrahydrofuran THF T T in centapoise, with T in C THF Refractive index = Measure hydrodynamic radius (R h) using DLS and find the scaling law between R h and N (number of monomers) by plotting a log-log curve. REFERENCES 1. Allen, Samuel M., and Edwin L. Thomas. The Structure of Materials. New York: John Wiley and Sons, 1999 pp DLS Technical Note, Malvern Instruments 3. Flory. Principles of Polymer Chemistry. Cornell UP, Grosberg, A. Y., and A. R. Khokhlov. Statsitical Physics of Macromolecules. New York: AIP, 1994 pp Rubinstein, M., and R. H. Colby. Polymer Physics. New York: Oxford UP, pp. 8, 49-54, Serhatli, E., M. Serhatli, B. M. Baysal, and F. E. Karasz. "Coil-globule transition studies of sodium poly(styrene sulfonate) by dynamic light scattering." Polymer 42 (2002): Young, R. J., and P. A. Lovell. Introduction to Polymers. 2nd ed. Cheltenham: Stanley Thornes, UK pp