4,615 research outputs found

    Interfacial Reaction Kinetics

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    We study irreversible A-B reaction kinetics at a fixed interface separating two immiscible bulk phases, A and B. We consider general dynamical exponent zz, where xt∌t1/zx_t\sim t^{1/z} is the rms diffusion distance after time tt. At short times the number of reactions per unit area, RtR_t, is {\em 2nd order} in the far-field reactant densities nA∞,nB∞n_A^{\infty},n_B^{\infty}. For spatial dimensions dd above a critical value dc=z−1d_c=z-1, simple mean field (MF) kinetics pertain, Rt∌QbtnA∞nB∞R_t\sim Q_b t n_A^{\infty} n_B^{\infty} where QbQ_b is the local reactivity. For low dimensions d<dcd<d_c, this MF regime is followed by 2nd order diffusion controlled (DC) kinetics, Rt≈xtd+1nA∞nB∞R_t \approx x_t^{d+1} n_A^{\infty} n_B^{\infty}, provided Qb>Qb∗∌(nB∞)[z−(d+1)]/dQ_b > Q_b^* \sim (n_B^{\infty})^{[z-(d+1)]/d}. Logarithmic corrections arise in marginal cases. At long times, a cross-over to {\em 1st order} DC kinetics occurs: Rt≈xtnA∞R_t \approx x_t n_A^{\infty}. A density depletion hole grows on the more dilute A side. In the symmetric case (nA∞=nB∞n_A^{\infty}=n_B^{\infty}), when d<dcd<d_c the long time decay of the interfacial reactant density, nAsn_A^s, is determined by fluctuations in the initial reactant distribution, giving nAs∌t−d/(2z)n_A^s \sim t^{-d/(2z)}. Correspondingly, A-rich and B-rich regions develop at the interface analogously to the segregation effects established by other authors for the bulk reaction A+B→∅A+B\to\emptyset. For d>dcd>d_c fluctuations are unimportant: local mean field theory applies at the interface (joint density distribution approximating the product of A and B densities) and nAs∌t(1−z)/(2z)n_A^s \sim t^{(1-z)/(2z)}. We apply our results to simple molecules (Fickian diffusion, z=2z=2) and to several models of short-time polymer diffusion (z>2z>2).Comment: 39 pages, 7 figures, uses fund2.sty, submitted to Eur. Phys. J. B, 1 figure added, for short version see cond-mat/980409

    Reaction Kinetics in Polymer Melts

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    We study the reaction kinetics of end-functionalized polymer chains dispersed in an unreactive polymer melt. Starting from an infinite hierarchy of coupled equations for many-chain correlation functions, a closed equation is derived for the 2nd order rate constant kk after postulating simple physical bounds. Our results generalize previous 2-chain treatments (valid in dilute reactants limit) by Doi, de Gennes, and Friedman and O'Shaughnessy, to arbitrary initial reactive group density n0n_0 and local chemical reactivity QQ. Simple mean field (MF) kinetics apply at short times, k∌Qk \sim Q. For high QQ, a transition occurs to diffusion-controlled (DC) kinetics with k≈xt3/tk \approx x_t^3/t (where xtx_t is rms monomer displacement in time tt) leading to a density decay nt≈n0−n02xt3n_t \approx n_0 - n_0^2 x_t^3. If n0n_0 exceeds the chain overlap threshold, this behavior is followed by a regime where nt≈1/xt3n_t \approx 1/x_t^3 during which kk has the same power law dependence in time, k≈xt3/tk \approx x_t^3/t, but possibly different numerical coefficient. For unentangled melts this gives nt∌t−3/4n_t \sim t^{-3/4} while for entangled cases one or more of the successive regimes nt∌t−3/4n_t \sim t^{-3/4}, t−3/8t^{-3/8} and t−3/4t^{-3/4} may be realized depending on the magnitudes of QQ and n0n_0. Kinetics at times longer than the longest polymer relaxation time τ\tau are always MF. If a DC regime has developed before τ\tau then the long time rate constant is k≈R3/τk \approx R^3/\tau where RR is the coil radius. We propose measuring the above kinetics in a model experiment where radical end groups are generated by photolysis.Comment: 24 pages, 5 figures, uses bulk.sty, submitted to Eur.Phys.J.B discussion section expande

    Dynamics of Living Polymers

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    We study theoretically the dynamics of living polymers which can add and subtract monomer units at their live chain ends. The classic example is ionic living polymerization. In equilibrium, a delicate balance is maintained in which each initiated chain has a very small negative average growth rate (``velocity'') just sufficient to negate the effect of growth rate fluctuations. This leads to an exponential molecular weight distribution (MWD) with mean Nbar. After a small perturbation of relative amplitude epsilon, e.g. a small temperature jump, this balance is destroyed: the velocity acquires a boost greatly exceeding its tiny equilibrium value. For epsilon > epsilon_c = 1/Nbar^{1/2} the response has 3 stages: (1) Coherent chain growth or shrinkage, leaving a highly non-linear hole or peak in the MWD at small chain lengths. During this episode, lasting time taufast ~ Nbar, the MWD's first moment and monomer concentration m relax very close to equilibrium. (2) Hole-filling (or peak decay) after taufill ~ epsilon^2 Nbar^2. The absence or surfeit of small chains is erased. (3) Global MWD shape relaxation after tauslow ~ Nbar^2. By this time second and higher MWD moments have relaxed. During episodes (2) and (3) the fast variables (Nbar,m) are enslaved to the slowly varying number of free initiators (chains of zero length). Thus fast variables are quasi-statically fine-tuned to equilibrium. The outstanding feature of these dynamics is their ultrasensitivity: despite the perturbation's linearity, the response is non-linear until the late episode (3). For very small perturbations, epsilon < epsilon_c, response remains non-linear but with a less dramatic peak or hole during episode (1). Our predictions are in agreement with viscosity measurements on the most widely studied system, alpha-methylstyrene.Comment: 16 pages, submitted to Eur. Phys. J.

    Strongly Charged Polymer Brushes

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    Charged polymer brushes are layers of surface-tethered chains. Experimental systems are frequently strongly charged. Here we calculate phase diagrams for such brushes in terms of salt concentration n_s, grafting density and polymer backbone charge density. Electrostatic stiffening and counterion condensation effects arise which are absent from weakly charged brushes. In various phases chains are locally or globally fully stretched and brush height H has unique scaling forms; at higher salt concentrations we find H ~ n_s^(-1/3), in good agreement with experiment.Comment: 5 pages, 3 Postscript figure

    Aspects of UGT2B15 in the human

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