Motion of the Monomers

9 Motion of the Monomers Here, we consider the mean square displacement of monomers on the entangled chains. Over a long time, t the mean square displacement of the monomer, ([r (0 r (0)] ), becomes identical to the 

We focus our attention primarily on the time range of i where is the time for the mean square displacement of a monomer on the test chain to reach bt. Att the test chain wiggles within the mbe without feeling the presence of the geometrical constraint imposed by neighboring chains. At i t, the motion of the test chain is the same as that of the primitive chain. 

In Sections 4.3.2.3 and 4.3.2.4, we assumed that the primitive chain follows a simple one-dimensional diffusion along its contour. This view is correct only for 

The test chain would follow the dynamics of the unrestricted Rouse chain if the entanglements were absent, as would the primitive chain at f t. In Section 3.4.9, we considered the mean square displacement of monomers on the Rouse chain. We found that the dynamics is diffusional at r and Tj t, where % is the relaxation time of the Mth normal mode but not in between. When the motion of the Rouse chain is resnicted to the tube, the mean square displacement of monomers along the tube, ([ (t) - x(0)] ), will follow the same time dependence as the mean square displacement of the unrestricted Rouse chain in three dimensions. Thus, from Eqs. 3.240 and 3.243, 

In the time scale of t Ij, the effect of finding the new direction by the chain ends becomes dominant, and the mean square displacement of monomers will become equal to that of the center of mass. In the time scale of f the motion of the monomers is complicated. At sufficiently short times (t r ), the monomers will make a diffusional motion without feeling the presence of other monomers, as we have seen for both the Rouse chain and the Zimm model. We can at least say that the dependence of ([r (f) - r (0)] ) on t is sharper at t  

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In contrast, the second substituent (such as the CH3 group at the a-carbon) does not significantly affect the IMM of the polymer if the first substituent causes high steric hindrance to the transfer motion of the monomer unit about the —C-C- bond of the main chain as in PS. In Table 8 are compliled data on PS and poly (a-methylstyrene), P(a-CH3S), in toluene (r = 5.1 and 8.6 ns, respectively). 

To determine the value of the diffusivity that connects the two approaches, we follow Einstein s thermodynamic arguments given in Section 5.2 for evaluating the translational Brownian diffusion coefficient. The basis for this is the random Brownian motion of the monomer units in the gel, which translates into the gel osmotic pressure. If, as above, the flow through the gel is assumed to follow Darcy s law (Eq. 4.7.7), then we may write the applied hydrodynamic force per mole of solution flowing through the gel as... 

The short dielectric relaxation process (presented here by 12) is associated with the anisotropic motion of the monomer alcohol species in a chain cluster (149). In microemulsions, the short process is the superposition of several dielectric relaxation processes, which have similar relaxation times such as movement or rotation of the alcohol monomers, hydrate water, and surfactant polar head groups. The short relaxation time is barely affected by the alcohol concentration in the mixture since it is less sensitive to the aggregation process. 

In type I template polymerization, in its most extreme case when monomer units are connected with the template by covalent bonds, the Brownian motion of the monomer units is replaced by conformational changes of the template molecules. Moreover, the distance between two monomer units is virtually independent of the overall concentration, and cannot be changed by dilution of the template solution. 

Three assumptions have been made to estimate the molecular length required to achieve main chain scission (a) the simultaneous and concerted motion of the monomer units (b) the equal interaction energy between the corresponding monomer units in adjacent polymer chains (c) the flow activation energy for the corresponding monomer is an approximation for E. In the case of polyethylene, E is 1.01 kcal/mole and the bond energy is about 82 kcal/mole. 

Striking differences of their reactivities were reported before by Ch. Krohnke [54]. The limited amount of samples available was sufficient for the permittivity analysis. Figure 9.35 shows the behaviour of different topospecifically deuterated derivatives of TS at T = 60 °C [73]. It should be stressed that the induction periods of single crystals with nominally the same history did not differ by more than 10% at the same polymerization temperature. The toposelective modification of these substituted diacetylenes by the deu-teration of the methylene groups close to the triple bonds of the diacetylene monomer (that are engaged in the crankshaft-type motion of the monomer molecule around its center of mass during solid state polymerization) evidently has a drastic influence on the sohd state polymerization. 

The second conceivable mechanism has already been mentioned. Sometimes it is observed that a homogeneous mixing of two polymers results in a volume shrinkage. The related decrease in the free volume available for local motions of the monomers may lead to a reduced mobility and hence a lowering of the entropy. The effect usually increases with temperature and finally overcompensate the initially dominating attractive interactions. 

Although many other kinetic treatments have been proposed [192], that of Aniansson and co-workers is possibly the most comprehensive. Current developments of this theory have dealt with the dynamics and extent of partial motions of the monomers out into the aqueous environment and back again into the micelle [225]. A comprehensive list of kinetic studies on surfactants has been given by Muller [226]. 

The Poley-type absorption in the region of 100 cm" is due to the librational motion of the monomer units of the backbone chain and characterizes the -process in this nonpolar polymer (see below). The origin of dielectric loss in PE at microwaves is dipolar impurities, end groups, chain folds and branch points. 

In simulations, a direct way to study the motion of the monomers is to measure the mean-square displacement of a monomer gi(t) as a function of time t, which is given by... 

The Zimm model rests upon the Langevin equation for over-damped motion of the monomers, i.e., it applies for times larger than the Brownian time scale Tb 2> OTm/where is Stokes friction coefficient [12]. On such time scales, velocity correlation functions have decayed to zero and the monomer momenta are in equilibrium with the solvent Moreover, hydrodynamic interactions between the various parts of the polymer are assumed to propagate instantaneously. This is not the case in our simulations. First of all, the monomer inertia term is taken into account, which implies non-zero velocity autocorrelation functions. Secondly, the hydrodynamic interactions build up gradually. The center-of-mass velocity autocorrelation function displayed in Fig. 9 reflects these aspects. The correlation function exhibits a long-time tail, which decays as (vcm(t)vcm(O)) on larger time scales. The... 

The polycondensation of linear monomers is characterized by elimination of one small molecule (e.g., H2O, HCl, or CH3OH) in each growing step. The liberation of the byproduct compensates for the loss of entropy resulting from the loss of three-dimensional translational motion of the monomers. Hence, the reaction entropy is close to zero (the exact value depends on the extent of rotational motion in monomer and repeating unit). A typical RO-PC is based on two different growing steps. The first one is an addition reaction and elimination of a byproduct only occius in the second step which is a condensation reaction as illustrated for the polycondensation of an a,co-alkanediol with maleic anhydride (see Formula 9.1). Therefore, RO-PCs have a negative reaction entropy which needs to be overcompensated by a negative reaction enthalpy (i.e., by an exothermic... 

The kinetics of the random motion of individual monomers inside polymer coils of double and single stranded DNA (dsDNA and ssDNA) could be unraveled using FCS. dsDNA was used as a model of a semiflerible polymer, while ssDNA was used as a model for a flexible polymer. To follow the motion the end monomers of the individual DNA strands were labeled with carboxyrhodamine 6G. For ssDNA, the motion of the monomers could be described by the Zimm model, while for dsDNA Rouse behavior was observed. This could be explained because of weak hydrodynamic interactions in the case of dsDNA attributed to the stiffness. 

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