Abstract:We present the experimental determination of the liquid–liquid coexistence curve of living poly-α-methylstyrene (initiated by n-butyllithium) in methylcyclohexane. We measured the coexistence curve by measuring the phase separation temperatures of a set of samples of different mole fractions of the initial monomer, x*m. All the samples had the same ratio, r(=0.008), of the mole fraction of the initiator to the mole fraction of the monomer. We also measured the polymerization line by measuring the temperatures … Show more
“…Previous measurements for sulfur and poly͑␣-methylstyrene͒ solutions identifiy the polymer transition temperature by a sharp increase in the viscosity as measured by a falling ball viscometer. 50,51 The initiation probability ͑the analog of initiator concentration͒ in sulfur solutions is extremely small 2,3 ͑yielding one of the few cases where the living polymerization transition has the appearance of a real second order phase transition͒. While the viscometric method provides a reasonable estimate for the polymerization temperature of sulfur solutions, we expect the polymerization to initiate at a…”
Section: Resultsmentioning
confidence: 97%
“…Thus, the crossover temperature T x can occur well above T p ͑see the example in Fig. 9͒, and therefore great caution must be exercised in deducing 51 T p from viscometric or other transport measurements.…”
Section: Fig 9 Transition Temperatures Of Living Polymer Solutions mentioning
A Flory-Huggins type lattice model of living polymerization is formulated, incorporating chain stiffness, variable initiator concentration r, and a polymer-solvent interaction. Basic equilibrium properties ͓average chain length L, average fraction of associated monomers ⌽, specific heat C P , entropy S, polymerization temperature T p , and the chain length distribution p(N)͔ are calculated within mean-field theory. Our illustrative calculations are restricted to systems that polymerize upon cooling ͓e.g., poly͑␣-methylstyrene͔͒, but the formalism also applies to polymerization upon heating ͑e.g., sulfur, actin͒. Emphasis is given to living polymer solutions having a finite r in order to compare theory with recent experiments by Greer and co-workers, whereas previous studies primarily focused on the r→0 ϩ limit where the polymerization transition has been described as a second order phase transition. We find qualitative changes in the properties of living polymer solutions for nonzero r: ͑1͒ L becomes independent of initial monomer composition m 0 and temperature T at low temperatures ͓L(TӶT p)ϳ2/r͔, instead of growing without bound; ͑2͒ the exponent describing the dependence of L on m 0 changes by a factor of 2 from the r→0 ϩ value at higher temperatures (TуT p); ͑3͒ the order parametertype variable ⌽ develops a long tail with an inflection point at T p ; ͑4͒ the specific heat maximum C P * at T p becomes significantly diminished and the temperature range of the polymer transition becomes broad even for small r ͓rϳO(10 Ϫ3)͔. Moreover, there are three characteristic temperatures for rϾ0 rather than one for r→0: a ''crossover temperature'' T x demarking the onset of polymerization, an r-dependent polymerization temperature T p defined by the maximum in C P ͑or equivalently, the inflection point of ⌽͒, and a ''saturation temperature'' T s at which the entropy S of the living polymer solution saturates to a low temperature value as in glass-forming liquids. A measure of the ''strength'' of the polymerization transition is introduced to quantify the ''rounding'' of the phase transition due to nonzero r. Many properties of living polymer solutions should be generally representative of associating polymer systems ͑thermally reversible gels, colloidal gels, micelles͒, and we compare our results to other systems that self-assemble at equilibrium.
“…Previous measurements for sulfur and poly͑␣-methylstyrene͒ solutions identifiy the polymer transition temperature by a sharp increase in the viscosity as measured by a falling ball viscometer. 50,51 The initiation probability ͑the analog of initiator concentration͒ in sulfur solutions is extremely small 2,3 ͑yielding one of the few cases where the living polymerization transition has the appearance of a real second order phase transition͒. While the viscometric method provides a reasonable estimate for the polymerization temperature of sulfur solutions, we expect the polymerization to initiate at a…”
Section: Resultsmentioning
confidence: 97%
“…Thus, the crossover temperature T x can occur well above T p ͑see the example in Fig. 9͒, and therefore great caution must be exercised in deducing 51 T p from viscometric or other transport measurements.…”
Section: Fig 9 Transition Temperatures Of Living Polymer Solutions mentioning
A Flory-Huggins type lattice model of living polymerization is formulated, incorporating chain stiffness, variable initiator concentration r, and a polymer-solvent interaction. Basic equilibrium properties ͓average chain length L, average fraction of associated monomers ⌽, specific heat C P , entropy S, polymerization temperature T p , and the chain length distribution p(N)͔ are calculated within mean-field theory. Our illustrative calculations are restricted to systems that polymerize upon cooling ͓e.g., poly͑␣-methylstyrene͔͒, but the formalism also applies to polymerization upon heating ͑e.g., sulfur, actin͒. Emphasis is given to living polymer solutions having a finite r in order to compare theory with recent experiments by Greer and co-workers, whereas previous studies primarily focused on the r→0 ϩ limit where the polymerization transition has been described as a second order phase transition. We find qualitative changes in the properties of living polymer solutions for nonzero r: ͑1͒ L becomes independent of initial monomer composition m 0 and temperature T at low temperatures ͓L(TӶT p)ϳ2/r͔, instead of growing without bound; ͑2͒ the exponent describing the dependence of L on m 0 changes by a factor of 2 from the r→0 ϩ value at higher temperatures (TуT p); ͑3͒ the order parametertype variable ⌽ develops a long tail with an inflection point at T p ; ͑4͒ the specific heat maximum C P * at T p becomes significantly diminished and the temperature range of the polymer transition becomes broad even for small r ͓rϳO(10 Ϫ3)͔. Moreover, there are three characteristic temperatures for rϾ0 rather than one for r→0: a ''crossover temperature'' T x demarking the onset of polymerization, an r-dependent polymerization temperature T p defined by the maximum in C P ͑or equivalently, the inflection point of ⌽͒, and a ''saturation temperature'' T s at which the entropy S of the living polymer solution saturates to a low temperature value as in glass-forming liquids. A measure of the ''strength'' of the polymerization transition is introduced to quantify the ''rounding'' of the phase transition due to nonzero r. Many properties of living polymer solutions should be generally representative of associating polymer systems ͑thermally reversible gels, colloidal gels, micelles͒, and we compare our results to other systems that self-assemble at equilibrium.
“…The nature of the coexisting phases will depend on whether the temperature is above or below the intersection of the polymerization line with the coexistence curve. 2 Equilibrium polymerization of α-methylstyrene: (a) the phase diagram in methylcyclohexane, (b) the extent of polymerization, (c) the mass density, and (d) the heat capacity . For (a), the solid triangles (▴) are points on the polymerization line; the open squares (□) are points on the coexistence curve; and the solid square (▪) is the experimentally determined liquid−liquid critical point, and the lines are drawn to guide the eye.…”
Reversible polymerizations occur in inorganic, organic, and biological molecules. The thermodynamic states
at which monomer-to-polymer reactions become favorable can be viewed as second-order phase transitions
and can be profitably treated by the statistical mechanics of phase transitions. The nature and development
of the microscopic structure of a solution of active polymers in equilibrium with the monomers pose interesting
issues for the physical chemist.
“…This interpretation was further validated by Wheeler and Pfeuty in showing that Scott's generalization of the Tobolsky and Eisenberg model is equivalent to an Ising spin magnet in the limit that the spin vector dimension goes to 0. Several groups − have studied equilibrium distributions and phase diagrams of living polymers by exploiting the isomorphism between continuum and lattice models.…”
A generalization of the generalized Langevin equation (GLE), the so-called irreversible GLE (iGLE) [Hernandez, R.; Somer, F. L. J. Phys. Chem. 1999, 103, 1064, is further extended to describe non-stationary environments in which the non-stationarity is induced by the macroscopic behavior of the ensemble itself, rather than an external force. Such a formalism lends itself to the dynamical study of the length distributions of growing polymers.
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