Reactions at Polymer Interfaces: Transitions from Chemical to Diffusion-Control and Mixed Order Kinetics

We study reactions between end-functionalized chains at a polymer-polymer interface. For small chemical reactivities (the typical case) the number of diblocks formed, $R_t$, obeys 2nd order chemically controlled kinetics, $R_t \sim t$, until interfacial saturation. For high reactivities (e.g. radicals) a transition occurs at short times to 2nd order diffusion-controlled kinetics, with $R_t \sim t/\ln t$ for unentangled chains while $t/\ln t$ and $t^{1/2}$ regimes occur for entangled chains. Long time kinetics are 1st order and controlled by diffusion of the more dilute species to the interface: $R_t \sim t^{1/4}$ for unentangled cases, while $R_t \sim t^{1/4}$ and $t^{1/8}$ regimes arise for entangled systems. The final 1st order regime is governed by center of gravity diffusion, $R_t \sim t^{1/2}$.Comment: 11 pages, 3 figures, uses poliface.sty, minor changes, to appear in Europhysics Letter

Coating thermal noise for arbitrary shaped beams

Advanced LIGO's sensitivity will be limited by coating noise. Though this noise depends on beam shape, and though nongaussian beams are being seriously considered for advanced LIGO, no published analysis exists to compare the quantitative thermal noise improvement alternate beams offer. In this paper, we derive and discuss a simple integral which completely characterizes the dependence of coating thermal noise on shape. The derivation used applies equally well, with minor modifications, to all other forms of thermal noise in the low-frequency limit.Comment: 3 pages. Originally performed in August 2004. Submitted to CQG. (v2) : Corrections from referee and other

Kinetic Regimes and Cross-Over Times in Many-Particle Reacting Systems

We study kinetics of single species reactions ("A+A -> 0") for general local reactivity Q and dynamical exponent z (rms displacement x_t ~ t^{1/z}.) For small molecules z=2, whilst z=4,8 for certain polymer systems. For dimensions d above the critical value d_c=z, kinetics are always mean field (MF). Below d_c, the density n_t initially follows MF decay, n_0 - n_t ~ n_0^2 Q t. A 2-body diffusion-controlled regime follows for strongly reactive systems (Q>Qstar ~ n_0^{(z-d)/d}) with n_0 - n_t ~ n_0^2 x_t^d. For Q<Qstar, MF kinetics persist, with n_t ~ 1/Qt. In all cases n_t ~ 1/x_t^d at the longest times. Our analysis avoids decoupling approximations by instead postulating weak physically motivated bounds on correlation functions.Comment: 10 pages, 1 figure, uses bulk2.sty, minor changes, submitted to Europhysics Letter

ON THE MODE OF DETECTING NITRIC ACID, And the total inefficacy of the New Test of decolorizing the

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The Slowly Formed Guiselin Brush

We study polymer layers formed by irreversible adsorption from a polymer melt. Our theory describes an experiment which is a ``slow'' version of that proposed by Guiselin [Europhys. Lett., v. 17 (1992) p. 225] who considered instantaneously irreversibly adsorbing chains and predicted a universal density profile of the layer after swelling with solvent to produce the ``Guiselin brush.'' Here we ask what happens when adsorption is not instantaneous. The classic example is chemisorption. In this case the brush is formed slowly and the final structure depends on the experiment's duration, $t_{final}$. We find the swollen layer consists of an inner region of thickness $z^* \sim t_{final}^{-5/3}$ with approximately constant density and an outer region extending up to height $h \sim N^{5/6}$ which has the same density decay $\sim z^{-2/5}$ as for the Guiselin case.Comment: 7 pages, submitted to Europhysics Letter

Irreversible Adsorption from Dilute Polymer Solutions

We study irreversible polymer adsorption from dilute solutions theoretically. Universal features of the resultant non-equilibrium layers are predicted. Two cases are considered, distinguished by the value of the local monomer-surface sticking rate Q: chemisorption (very small Q) and physisorption (large Q). Early stages of layer formation entail single chain adsorption. While single chain physisorption times tau_ads are typically microsecs, for chemisorbing chains of N units we find experimentally accessible times tau_ads = Q^{-1} N^{3/5}, ranging from secs to hrs. We establish 3 chemisorption universality classes, determined by a critical contact exponent: zipping, accelerated zipping and homogeneous collapse. For dilute solutions, the mechanism is accelerated zipping: zipping propagates outwards from the first attachment, accelerated by occasional formation of large loops which nucleate further zipping. This leads to a transient distribution omega(s) \sim s^{-7/5} of loop lengths s up to a size s_max \approx (Q t)^{5/3} after time t. By tau_ads the entire chain is adsorbed. The outcome of the single chain adsorption episode is a monolayer of fully collapsed chains. Having only a few vacant sites to adsorb onto, late arriving chains form a diffuse outer layer. In a simple picture we find for both chemisorption and physisorption a final loop distribution Omega(s) \sim s^{-11/5} and density profile c(z) \sim z^{-4/3} whose forms are the same as for equilibrium layers. In contrast to equilibrium layers, however, the statistical properties of a given chain depend on its adsorption time; the outer layer contains many classes of chain, each characterized by different fraction of adsorbed monomers f. Consistent with strong physisorption experiments, we find the f values follow a distribution P(f) \sim f^{-4/5}.Comment: 18 pages, submitted to Eur. Phys. J. E, expanded discussion sectio