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Since the pioneering work of McFadden (1974), discretechoice random-utility models have become work horses in many areas in economics.In these models, the random variables enter additively or multiplicatively, and the noise distributions take a particular parametric form. We show that the same qualitative results, with closed-form choice probabilities, can be obtained for a wide class of distributions without such specifications. This class generalizes the statistically independent distributions where any two c.d.f.:s are powers of each others to a class that allows for statistical dependence, in a way analogous to how GEV distributions generalize Gumbel distributions. We show that this generalization is sufficient, and under statistical independence also necessary, for the following invariance property: all conditional random variables, when conditioning upon a certain alternative having been chosen, are identically distributed. In our general framework, proofs become simpler, more direct and transparent, well-known results are obtained as special cases, and one can characterize the Gumbel, Fréchet and Weibull distributions.

This work summarizes several studies that all concern cellulose ethers of the types methylcellulose (MC) hydroxypropylmethylcellulose (HPMC) and ethyl(hydroxyethyl)cellulose (EHEC). They share the feature of negative temperature response, as they are soluble in water at room temperature but phase separate and sometimes form gels at high temperatures. The different types of viscosity transitions occurring in these three cellulose ethers are well-known. However, earlier studies have not solved the problem of why both HPMC and EHEC, as the temperature increases, exhibit a viscosity decrease just before the viscosity increases, whereas MC only has one transition temperature where the viscosity increases. With our investigations we have aimed to compare the effect of temperature on bulk solutions and on adsorbed layers of the different polymers using a range of techniques.

Light scattering and cryo transmission electron microscopy (cryo-TEM) was employed to study aggregation of MC, HPMC and EHEC in solution. The solvent quality of water is reduced for all three polymers in solution as the temperature increases, and this infers an onset of aggregation at a certain temperature. The aggregation rate follows the order EHEC > HPMC > MC. Cryo-TEM pictures of solutions frozen from high temperatures showed closely packed fibrils forming dense networks in MC solution. Some fibrils were also found in HPMC solution above the transition temperature, but they did not interconnect readily. This is explained by the bulky and hydrophilic hydroxypropyl groups attached to HPMC. EHEC has similar substituents, while MC only has short and hydrophobic methyl groups attached to the main chain.

An amphiphilic liquid, diethyleneglycolmonobutylether (BDG) was used as an additive to change the properties of MC solutions in water. With 10 wt% BDG added, the effect was similar in viscosity and light scattering measurements as well as cryo-TEM pictures, inducing a temperature response resembling that of HPMC in pure water. 5 wt% of BDG was enough to change the aggregation type and induce a transition temperature with viscosity decrease. The effect of the additive is rationalized by BDG acting as a hydrophobic and bulky substituent in MC, similar to the large substituents in HPMC and EHEC.

Two instruments, a quartz crystal microbalance with dissipation (QCM-D) and an ellipsometer, were used in parallel to determine the changes with temperature on an adsorbed layer of MC and HPMC on silica kept in water and in polymer solution. The silica needed to be hydrophobized for significant adsorption to take place. Adsorption was similar for both polymers at low temperatures, whereas a sharp transition in several layer properties occurred for HPMC, but not for MC, close to the solution viscosity transition temperature. Atomic force microscopy (AFM) was used to measure attractive and repulsive forces and also friction forces between MC layers in polymer solution. The small changes in normal forces with temperature infer that the hydrophobic groups in MC are mostly depleted from the surface. The surface–polymer interactions increase with increasing temperature and the layer becomes more cohesive, which induces a higher load bearing capacity and lower friction when measured at high loads. AFM imaging was employed to obtain the height distribution in MC adsorbed layers. These images indicate that fibril-like structures were formed at a lower temperature in the surface layer than in bulk solution.

The different preferences for adsorption and for aggregation in MC and HPMC above the solution transition temperatures are explained by the fibril formation in MC shielding hydrophobic parts of the polymer from the solution, and thus counteracting adsorption, but also fast aggregation. The viscosity decrease in HPMC and EHEC is conferred to intra-chain contraction and aggregation into less extended structures.