3.014 Materials Laboratory Sept. 18th – Sept. 22nd, 2006 Laboratory Week 1 – Module α1 Polymer Structures Instructor: Meri Treska OBJECTIVES - Understand structure of amorphous materials - Learn principles of x-ray scattering from amorphous materials - Determine the glass transition temperature of various methacrylate polymers,using differential scanning calorimetry (DSC) SUMMARY OF TASKS 1) Perform x-ray scattering measurements on a series of amorphous methacrylate polymers and semicrystalline polyethylene 2) Interpret observed peaks in the scattering patterns for methacrylate polymers 3) Determine the average interchain distances for polyethylene and methacrylate polymers from x-ray scattering patterns 4) Perform DSC measurements on methacrylate polymers, and determine Tg 5) Correlate the structure data with the glass transition temperature of the methacrylate polymers 1BACKGROUND Diffraction from Crystalline Materials of Finite Crystal Dimension For crystalline materials, we learned that the periodic arrangement of atoms gives rise to constructive interference of scattered radiation having a wavelength λ comparable to the periodicity d when Bragg’s law is satisfied1: 2sinndλθ= [1] where n is an integer and θ is the angle of incidence. Bragg’s law implies that constructive interference occurs only at the exact Bragg angle and the Intensity vs. 2θ curve exhibits sharp lines of intensity. In reality, diffraction peaks exhibit finite breadth, due both to instrumental and material effects. An important source of line broadening is finite crystal size. In crystals of finite dimensions, there is only partial destructive interference of waves scattered from angles slightly deviating from the Bragg angle.1 This is illustrated in Fig. 1 below, which shows XRD patterns for two polycrystalline V2O5 films with different average crystal sizes. 2Figure 1. XRD patterns from V2O5 thin films with average crystallite sizes of a) 250 nm and b) 80 nm. Peak broadening is observed with decreasing crystallite size. (data courtesy S. C. Mui)If we define the angular width of a particular peak as: (121222B)θθ=− [2] where θ1 and θ2 define the angular bounds of the peak in radians, then the average crystal size can be estimated from the Scherrer formula as1: 0.9cosBtBλθ= [3] X-ray Scattering from Amorphous Materials Above it was established that as the crystal size gets smaller, diffraction peaks get broader. What happens in the extreme when there is no crystallinity? Whereas crystalline materials exhibit long range order, amorphous materials such as glassy polymers, metallic glasses and oxide glasses exhibit only short range order.2 Their scattering pattern displays broad, low intensity peaks characteristic of the average local atomic environment, as shown in Fig. 2 for an amorphous zirconium phosphate3. Applying the Scherrer formula to the peaks in this pattern, what is the calculated “crystal size” for this material? What do the peak positions represent? Figure 2. X-ray scattering pattern of amorphous zirconium phosphate synthesized using a non-ionic surfactant template.3 3Figure removed due to copyright restrictions.To better understand amorphous x-ray patterns, we can calculate the predicted intensity using the structure factor for amorphous materials. The structure factor is defined as2: 1() exp2MnnnFsfis rπ=⎡=⎣∑rur⎤⋅⎦ [4] where is the scattering vector, fsrn is the atomic scattering factor (proportional to atomic number), and is the atomic position vector for the nth atom. For a crystalline system, the summation in [4] is taken over the unit cell, and the total intensity is determined from the contribution of all unit cells.nrur1 For an amorphous material, there is no unit cell, since the atomic positions are not strictly periodic. Hence we take the summation in [4] over all atoms in the material. Recalling the relationship between Icoh and F: *11() () exp2 exp 2NNcoh m n n mmnIFsFs ff isr isrππ==⎡⎤⎡∝= ⋅−⎤⋅⎣⎦⎣∑∑⎦rur r uur [5] and letting , nmnmrrr=−rurruur⎤⋅⎦ 11exp 2NNcoh m n nmmnIffisπ==⎡=⎣∑∑ruur [6] Using the assumption that the material is isotropic, i.e., that its structure has radial symmetry, the exponential can be replaced by its angular average2: ()200sin( )exp 2 cos sin( )nmnmnmqrir s d dqrπππαααφ=∫∫ [7] 11sin( )NNnmcoh m nmnnmqrIffqr===∑∑ [8] 4where 4sin 2qsπθπλ== [9] For simplicity, we consider the case where all atoms are of the same type: 211sin( )NNnmcohmnnmqrIfqr===∑∑ [10] If we consider the interaction of each atom with itself,2 2sin( )1nmcohnmnmqrIfNqr≠⎡⎤=+⎢⎣⎦∑⎥ [11] where the first term is individual atom scattering, obtained by letting rnm→0. Icoh can be expressed in continuum form by substituting in the radial distribution function, 4πr2ρ(r): 220sin( )14 ()cohqrIfN r r drqrπρ∞⎡⎤=+⎢⎣⎦∫⎥ [12] where ρ(r) is the atomic pair density function,3,6 which gives the average density of atoms (number/volume) at a distance r from the center of a reference atom.2,4 The quantity 4πr2ρ(r)dr in eq. 12 gives the number of atoms within a shell of thickness dr around the reference atom, as shown in Fig. 3. drdr Figure 3. Schematic representation of atomic distribution within an amorphous material. The quantity 4πr2ρ(r)dr is the number of atoms in a shell of thickness dr around a reference atom. 5The atomic pair density function is related to the pair distribution function g(r), by:2,4 ()()orgrρρ= [13] where ρo is the bulk atomic density of the material (atoms/volume). For amorphous materials and liquids, g(r) has a form such as that shown in Fig. 4.5 How would this function look for a crystalline system? g(r)r1g(r)r1Figure 4. Schematic illustration of the pair distribution function for amorphous materials. The maxima correspond to distances where there is a higher probability of finding a neighboring atom. The oscillatory behavior of g(r) gives rise similar oscillations in the scattering patterns from amorphous materials. From eq. [12] we can expect that the scattering intensity from an amorphous material will behave as a damped oscillatory function whose features depend on the average local spacing between atoms in the structure (Fig. 5). I