Associations and dissociations with time-dependent reaction coefficients in finite polymer mixtures: The model and analytical-numerical method for solution by successive approximations

• The work deals with the association and dissociation reactions with time-dependent coef- ficients in finite mixtures of polymers dispersed in fluid media with solid components. The polymers are regarded to be formed by identical units, polymer-forming units (PFUs) and, thus, present homopolymers. The model takes into account the porosity of the dispersion- medium/polymer-mixture system. The work derives the model for the reactions in the fi- nite mixtures. The model presents a non-autonomous quadratic finite ODE system in a time-independent hyperplane and is based on the conservation law for the total num- ber of PFUs. A variety of engineering applications of the derived finite-mixture model are discussed. The simplest case of the finite mixtures, i.e., the monomer-dimer mix- tures with time-independent reaction coefficients is completely analyzed. An analytical- numerical (AN) method of the successive-approximations (SA) type is proposed for solv- ing the derived model. The AN/SA method includes explicit analytical expressions for each of the approximations in terms of the preceding approximation. The method is exact in the dissociation-only case. The approximations are expected to converge if the association- reaction coefficients are not too large and the zeroth approximations are not very far from the solution. The AN/SA method comprises two sequences of the approximations. If the first one converges uniformly in the entire time axis, then the limit function is a steady- state (or “dynamic equilibrium”) solution of the non-autonomous quadratic ODE system. The second sequence presumes that the first sequence is convergent in the above men- tioned sense. The second sequence is intended for calculation of the solutions of initial- value problems for the above ODE system in a semi-infinite time interval. The main differ- ences from common computational methods are formulated. The AN/SA method is quan- titatively illustrated with a few examples of the settings in the aforementioned case of monomer-dimer mixtures, also in comparison with the explicit Euler method. The form of the AN/SA method allows especially efficient implementation on multi-processor/multicore personal computers with graphic processing units even if the dimension of the state space is large. The developed model and method form a constructive framework for analysis or design of polymer mixtures dispersed in fluid-solid media. An application to prospective manufacturing of spatially heterogeneous polymer products is noted. A few directions for future research are proposed as well.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

NATURVETENSKAP  -- Fysik -- Annan fysik (hsv//swe)

NATURAL SCIENCES  -- Physical Sciences -- Other Physics Topics (hsv//eng)

Nyckelord

Homopolymer mixture

Chemical reaction with time-dependent coefficients

Quadratic ordinary differential equation

Discrete Smoluchowski model

Analytical-numerical method

Steady-state solution

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