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Polymer Conformations
Published in Timothy P. Lodge, Paul C. Hiemenz, Polymer Chemistry, 2020
Timothy P. Lodge, Paul C. Hiemenz
So far, we have considered chain dimensions solely in terms of the average end-to-end distance. However, there are two severe limitations to this approach. First, the end-to-end distance is generally very difficult to measure experimentally. Second, for many interesting polymer structures (e.g., stars, rings, combs, dendrimers, etc.) it cannot even be defined unambiguously. The end-to-end distance assigns particular significance to the first and last monomers, but all monomers are of importance. A useful way to incorporate this fact is to calculate the average distance of all monomers from the center of mass. We denote the instantaneous vector from the center of mass to monomer i as s→i, as shown in Figure 6.8. The center of mass at any instant in time for any polymer structure is the point in space such that ∑i=1Nmis→i=0
Mechanical Properties of Polymer Materials
Published in Yichun Zhou, Li Yang, Yongli Huang, Micro- and MacroMechanical Properties of Materials, 2013
Yichun Zhou, Li Yang, Yongli Huang
Curl significantly shortens the straight-line distance between the two end points of the polymer chain (the end-to-end distance). The more severe the curl, the shorter is the end-to-end distance h. Therefore, we can use the end-to-end distance of polymer chains to describe or characterize the shape of the polymer chain. Of course, because the rotation in the molecule changes the conformation, it is appropriate to use the statistical averaging method-the mean square of the end-to-end distance h‾2 (the average of the square of the end-to-end distance h of polymer chains)-to describe the average size of polymer chains.
Influence of polymers on flow and heat transfer due to peristaltic waves: a molecular approach
Published in Waves in Random and Complex Media, 2022
Maria Athar, Khalid Saeed, Adeel Ahmad, Junaid Anjum
Now, we consider , the dimensionless conformation tensor due to polymers [6]. Let denotes the polymer end-to-end distance vector. Also, let and be the polymer radius in the coiled formation and the orientation angle at equilibrium, respectively. Then, and are the dimensionless components of polymer end-to-end distance. , where is uniformly distributed in . That is, the ensemble average over polymers can be interchanged by the average over . Neglecting thermal noise, and satisfy the following Langevin equations:
Effect of functional-group distribution on the structure of a polymer in non-aqueous solvent
Published in Molecular Physics, 2018
Rui F. G. Apóstolo, Philip J. Camp, Beatrice N. Cattoz, Peter J. Dowding, Andrew D. Schwarz
This work is focused on the size of a functionalised polyethylene/polypropylene (PE/PP) random copolymer in a non-aqueous solvent (n-heptane); the PE/PP backbone is in good-solvent conditions. (Future experimental work will involve small-angle neutron scattering (SANS) studies, and deuterated heptane is a convenient means of improving contrast between solute and solvent.) The functional groups contain aromatic and polar chemical species, and hence in an aliphatic solvent, there are attractive interactions between them; the functional groups are proprietary (belonging to Infineum UK Ltd), and so more details cannot be given here. The main task is to determine how the effective interactions and the distribution of functional groups on the backbone affect the physical properties of an FP. The simplest property of a single polymer is its size, measured either by the radius of gyration () or its end-to-end distance (). Of course, chemical details are important, and the precise way in which backbone and functional groups interact cannot easily be predicted. But once something is known about the effective interactions between these groups, then a much simpler coarse-grained model can be sought that captures the essential features of the effective interactions. A lot of computational effort is expended in determining effective, coarse-grained interactions in polymer molecules [21,22], and in favourable cases, the interaction potentials are transferable [23]. But even the most primitive models, such as chains of soft-spheres, can yield some valuable insights, and it is then possible to survey a wider range of molecular variables at a fraction of the computational cost.
Atomistic simulations of long-chain polyethylene melts flowing past gold surfaces: structure and wall-slip
Published in Molecular Physics, 2020
A. P. Sgouros, D. N. Theodorou
In this work, three systems are examined with molten C260H522 capped by two FCC (100) surfaces of Au. C260H522 is a moderately entangled polymer melt; it exhibits entanglements per chain (using a mass between entanglements Me = 1150 g/mol at T = 443 K) [39] and lies in the crossover region from the Rouse to the entangled regime [40]. The surfaces are subjected to planar Couette flow over a broad range of shear rates, from = 0.18 ns–1 up to 1.92 ns–1. To better quantify the shear rates imposed on the samples, hereafter the results will be displayed with respect to the Weissenberg number (Wi) instead of , where ; τD was set to 48.09 ns, the terminal relaxation time derived from the autocorrelation function of the end-to-end vector of C260H522 in the melt at the simulation temperature of T = 450 K whilst maintaining the normal pressure at 1 atm, as shown in Figure 4 [23]. A value of Wi = 1 means that the shear rate is low enough that the equilibrium structure of the melt is more or less maintained; increasing Wi corresponds to increased departure from the linear regime[41]. Alternatively, one can define a Weissenberg number based on the Rouse time (WiR) in order to discern whether the chain orientation and/or stretching are the dominant factors affecting slip; here, we define the Rouse-Weissenberg time as . The Rouse time can be estimated from the relation: . For Nmon = 260, a monomeric friction coefficient [42] , and a root mean square end-to-end distance [41], the Rouse time of the C260 chains becomes, ; therefore, .