A Note on the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>U</mml:mi></mml:math>Operator

Neuman, Maurice

doi:10.1103/physrev.83.671

Cited by 4 publications

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References 3 publications

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“…The entropic effect induced by collapse of polymer coils is much different from that induced by coil extension, with the latter usually called elastic entropy or entropic elasticity, while the former is referred to as the spring entropic effect . It has been noted that the theoretical treatment model of spring entropy is much different from that of elastic entropy . The spring entropy effect plays a critical role in many soft matter and other processes .…”

Section: Introductionmentioning

confidence: 99%

Dual entropic and enthalpic processes in the lower critical solution temperature phase separation of poly(vinyl methyl ether) aqueous solutions

Song

Lin

et al. 2019

J Polym Sci B Polym Phys

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In the lower critical solution temperature phase separation of poly(vinyl methyl ether) aqueous solutions, the process corresponding to the weakening of the hydrogen bond interaction with increasing temperature is dominant and occurs over the entire concentration region of solutions and over a broad temperature range from 30 to 41°C, giving rise to the energetic enthalpic effect during phase separation, while the conformational change, that is, collapse of the swollen polymer coils, occurs only in the swelling polymer solution when the water concentration is above 38.3 wt %, giving rise to the entropic effect during phase separation. In addition, the entropic process corresponding to the collapse of the polymer coils occurs in a much narrow theta temperature range from 35.5 to 37°C. If the solution is held at a constant temperature for a sufficiently long time, 90% collapse of the polymer coils occurs in only the 0.5 °C temperature region between 35.5 and 36°C. Accordingly, in the enthalpic process, the most dramatic blueshift of the νC‐O bond peak occurs in the temperature range between 35 and 41°C, while this blueshift is only approximately 2 cm−1 in the temperature range from 30 to 35°C, prior to the collapse of the polymer coils due to the entropic effect. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019, 57, 323–330

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Section: Introductionmentioning

confidence: 99%

Dual entropic and enthalpic processes in the lower critical solution temperature phase separation of poly(vinyl methyl ether) aqueous solutions

Song

Lin

et al. 2019

J Polym Sci B Polym Phys

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Sull’operatoreU(t) di Dyson-Feynman

Fubini

1952

Nuovo Cim

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Bound States and the Interaction Representation

1952

Phys. Rev.

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The temporal development of the state vectors in the interaction representation is investigated for a quantum-mechanical system that has bound states. It is shown that there is a non-unitary operator which determines the variation of the state vectors of the free states. This operator satisfies the same differential equation and initial condition as the unitary operator which transforms a state vector at the remote past to the state vector at a finite time or in the remote future. The present investigation is relevant to a remark made by H. S. Snyder.No iteration process is made use of in the general investigation. Born's method of successive approximations is discussed at the end.

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A Note on theUOperator

Cited by 4 publications

References 3 publications

Dual entropic and enthalpic processes in the lower critical solution temperature phase separation of poly(vinyl methyl ether) aqueous solutions

Dual entropic and enthalpic processes in the lower critical solution temperature phase separation of poly(vinyl methyl ether) aqueous solutions

Sull’operatoreU(t) di Dyson-Feynman

Bound States and the Interaction Representation

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