Macromolecular Dynamics Models PDF

26. Macromolecular dynamics models 27.1. Rouse model Description of macromolecular dynamics Rousse model: It assumes that the molecule is long enough to be represented by a number of molecular subunits each of which obeys the Gaussian distribution function (Section 10.3.1). It is envisaged that th...

26. Macromolecular dynamics models 27.1. Rouse model Description of macromolecular dynamics Rousse model: It assumes that the molecule is long enough to be represented by a number of molecular subunits each of which obeys the Gaussian distribution function (Section 10.3.1). It is envisaged that these subunits can be represented by a series of beads connected by springs as shown in Figure 16.7. The springs are assumed to obey Hooke’s law and also act as universal joints along the chain. It is thought that the dynamics of the polymer chains depends upon the interplay of three different forces acting on the repeat units. A frictional force, which is proportional to the relative velocity of the repeat unit and the surrounding medium. The force between adjacent repeat units along the chain maintaining connectivity. A random force on the chain from collisions with the surrounding medium leading to Brownian motion. One of the important predictions of this model is that the viscosity of a polymer melt η should be proportional to the number of repeat units along the chain, i.e., 𝜂∝𝑀 Such behavior is found at low values of molar mass. for all molten polymers that are above some critical value of molar mass, Mcr, there is a much stronger dependence of the viscosity upon molar mass. Typically, 𝜂 ∝ 𝑀3.5 This change in behaviour has been accounted for in terms of the.concept of chain entanglements. 1 26.2. Reptation of a single chain Coils trapped in a network Here we look at a system which is simple than a polymer melt but which still shows some nontrivial entanglement effects. This corresponds to a single, ideal, polymeric chain P (with N monomers) trapped in a three-dimensional network. Assume that we have one linear P chain moving in a given network. The network is described by fixed obstacles O1, O2, etc. The chain P is not allowed to cross any of them, but it can move in between in a wormlike fashion. We call this “reptation”. The basic reptation process is shown in Fig. VIII.4. 2 To understand the effect of the obstacle at one moment it is convenient to think of the chain as being trapped in a certain tube. The chain progresses by reptation. It leaves some parts of the tube, and it “creates” some new parts of the tube. The terminal time 𝜏𝑡 is essentially the time required for complete renewal of the tube. Let us assume for a moment that our chain is trapped inside one infinitely long tube. For motions along the tube we can introduce a “tube mobility” for the chain µ𝑡𝑢𝑏𝑒. This is defined by applying a steady force f to the chain (along the tube direction) and measuring the resulting chain velocity along the tube 𝑣 = µ𝑡𝑢𝑏𝑒 𝑓. The friction fore 𝑣/µ𝑡𝑢𝑏𝑒 is proportional to the length of the chain or N. Then, µ1 µ𝑡𝑢𝑏𝑒 = where µ1 is independent of N. “tube diffusion coefficient” 𝐷𝑡𝑢𝑏𝑒 is related to by an 𝑁 Einstein relationship µ1 𝑘𝑇 𝐷1 𝐷𝑡𝑢𝑏𝑒 = = 𝑁 𝑁 𝐿2 𝑁𝐿2 𝜏𝑡 ≅ ≅ 𝐷𝑡𝑢𝑏𝑒 𝐷1 Since L is linear in N, then, 𝜏𝑡 = 𝜏1 𝑁 3 Assuming that our melt is far above any glass transition temperature Tg, 𝜏1 may be of order 10-11 sec. If we have a long chain (N = 104), this leads to 𝜏𝑡 ~ 10 𝑠𝑒𝑐. Thus the reptation concept does give us a plausible feeling for the viscoelastic behavior of polymers. 3