Case study: molecular dynamics of solvent diffusion in polymers

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Eugene Hudson
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1 Course MP3 Lecture 11 29/11/2006 Case study: molecular dynamics of solvent diffusion in polymers A real-life research example to illustrate the use of molecular dynamics Dr James Elliott 11.1 Research case studies In the lecture 10, you looked at a research case study using energy minimisation and MD to study fast ion conduction in an inorganic crystal. In this lecture, we will be discussing how MD can be used to study solvent uptake and diffusion in glassy polymers. We will start by learning about the glass transition and touch on the issue of non-equilibrium simulations (which will be the main topic of lecture 12). The objective will be to obtain predictions of the glass transition temperature of polymers from MD simulation, and comment qualitatively on the solvent diffusion in these systems. 1

2 The glass transition The phenomenon of the glass transition can be observed experimentally by plotting the volume of a glass-forming substance as a function of temperature. V Dark line is ideal liquid (upper portion) and crystal (lower portion) cooling curve Red line is typical glass cooling curve T K T g T m T [K] The glass transition The glass transition temperature T g can fall anywhere in between the melting point T m (or even above it!) and the point at which the extrapolation of the liquid cooling curve meets the solid cooling curve. This lower limit is called the Kauzmann temperature T K, and the phenomenon of glasses possessing non-zero entropy as T 0 (implied by graph on previous slide) is called the Kauzmann paradox, as it contradicts the third law of thermodynamics (see lecture 1). The paradox is resolved by the hypothesis that a glassy system is not in thermal equilibrium, and that the laws of thermodynamics only apply to systems in equilibrium. 2

3 11.3 Glass transition using MD Can be studied by using either NVT or NpT simulations, which display discontinuous changes in the gradients of the pressure or volume, respectively. Specific Volume (cm 3 g -1 ) Tg=295K Temperature (K) Problems with using MD approach Unfortunately, it is found that the T g calculated from MD does not match the experimental T g, with an error of as much as 100 K. The reason for this is that T g depends on the thermal history of the sample, i.e. how fast it was cooled from the rubber state (T > T g ) into the glassy state (T < T g ). The faster the cooling rate, the higher the value of T g. This can be understood by realising that a greater amount of structural disorder is frozen into the system during a fast quench. The lower limit on T g is the Kauzmann temperature T K, and T g = T K in the limit of an infinitesimally slow quench. 3

4 Problems with using MD approach It should now be clear why, in general, MD tends to greatly overestimate T g. The reason is that the time scales of the simulations are very small indeed compared to the temperature changes which are being applied. This results in extraordinarily fast cooling rates compared with those which are possible in real experiments. For example, consider cooling a material by 1 K over a typical simulation of length 1 ps. This is a cooling rate of K/s, compared with a typical experimental rate of order 10 2 K/s! 11.5 Vögel-Fulcher law Another consequence of the kinetic arrest which occurs at the glass transition point is that the viscosity diverges there, and decreases exponentially away from it. B η = Aexp TV Tg + 50 K T TV This type of relationship is known as the Vögel-Fulcher law. The effects of the Vögel-Fulcher law can be observed in MD as a dramatic slowing down of the polymer chain relaxation. At first sight, obtaining a correspondence with an experimental T g from MD seems hopeless. 4

5 11.6 Scaling laws to the rescue However, it is possible to extrapolate to experimental measurement times and obtain a correspondence by using the principle of time-temperature superposition. Crudely speaking, this says that fast viscoelastic changes at high temperatures can be mapped onto to slow changes at low temperatures. Using this principle, Williams, Landel and Ferry derived their semi-empirical WLF equation: log a T C1( T Tg) = C + ( T T ) 2 g Young and Lovell (AN6a.40) where a T is the ratio of relaxation times at temperatures T and T g, which is known as the shift factor WLF extrapolation So, by fitting the WLF equation to our MD simulation data, we can extrapolate to experimental measurement times and predict a realistic T g. Tg (K) 400 Simulated Tg 350 WLF Extrapolation 300 Experimental Tg E+00 1.E+04 1.E+08 1.E+12 1.E+16 Cooling Time (ps) 5

6 11.8 Motion of solvents in a glassy polymer Although the motion of the glassy matrix is frozen out below the glass transition point, any solvent molecules in the material are still free to diffuse. Of course, they will find it easier to diffuse through the material when it is above its glass transition point (i.e. a rubber) rather than when it is below it (i.e. a glass). This is because the mechanism of diffusion is an activated hopping process where the solvent molecules jump between areas of occupiable free volume which are connected by transient pathways. Below T g, the opening of pathways is constrained by the the surrounding polymer, and there is a change in the slope of the variation of diffusion coefficient as a function of temperature Solvent motion above T g Using MD, plot the c.o.m. motion of a solvent molecule when T >> T g. y (Å) x (Å) 6

7 Solvent motion above T g Looking at the motion of a single molecule over a longer time scale, there is evidence for jump diffusion. M ean-squared-displacement (Å 2 ) Time (ps) Solvent motion below T g Contrast this with the behaviour when T < T g y (Å) x (Å) 7

11.11 Atomistic visualisation of solvent motion This

degrees of freedom of the polymer also has an influence on

8 11.11 Atomistic visualisation of solvent motion This example is polydimethylsulphoxane (PDMS) with nitrogen as solvent Dual mode sorption The freezing out of the translational degrees of freedom of the polymer also has an influence on the overall solvent uptake of the material, which increases below T g. This effect is called dual mode sorption, and is related both to the increase in free volume in the glassy state and the fact that the system is thermally disjoint. 8

9 Effective temperature Dual mode sorption can be rationalised by the concept of effective temperature. The effective temperature is a local measure of temperature which is divorced from the true thermodynamic temperature by a loss of ergodicity. In this case, the adsorption sites in the glass have a lower effective temperature than the overall thermodynamic temperature, which leads to an increase in the amount of adsorbate which can be accommodated. Fitting the adsorption isotherm with an effective temperature that is not the true temperature gives the correct amount of adsorbate Effective temperature The effective temperature, as calculated from the quantity of adsorbate, scales with the real temperature. Effective Temperature (K) Real Temperature (K) 9

10 11.14 Simulating out of equilibrium The problem of a local effective temperature arises because the simulations are out of equilibrium. Intrinsically, there is no difficulty in principle using MD techniques out of equilibrium. However, we expect to run into problems when defining quantities like temperature. Also, we cannot use Einstein s formula to calculate diffusion coefficients of the glassy material. It is OK to use this for adsorbed solvent, because it is equilibrated with respect to the glassy system. In the next lecture, we will discuss in more detail techniques for simulating out of equilibrium Summary In this lecture, we started off by introducing the glass transition, which occurs at a temperature that is dependent on the rate at which the material is quenched from the melt. However, there is a fundamental lower limit called the Kauzmann temperature. We saw how it is possible to predict glass transition temperatures of materials using molecular dynamics, combined with WLF extrapolation to experimental time scales. We also considered solvent uptake. Again, the crucial steps in the modelling process were (i) validation of model from experiment, (ii) mapping of simulation results onto experimental time scales and (iii) application to industrially relevant problems. 10

From L. H. Sperling, "Introduction to Physical Polymer Science, 2'nd Ed."

From L. H. Sperling, Introduction to Physical Polymer Science, 2'nd Ed. PDF File: (Click to Down Load): Chapter1.pdf > => Back to TOC => To Syllabus => Back to Chapter 1 Free Volume and T g In a plot of volume versus temperature through the glass transition, the thermal expansion