Cloud point polydispersity

The critical point (Ij of the two-phase region encountered at reduced temperatures is called an upper critical solution temperature (UCST), and that of the two-phase region found at elevated temperatures is called, perversely, a lower critical solution temperature (LCST). Figure 2 is drawn assuming that the polymer in solution is monodisperse. However, if the polymer in solution is polydisperse, generally similar, but more vaguely defined, regions of phase separation occur. These are known as "cloud-point" curves. The term "cloud point" results from the visual observation of phase separation - a cloudiness in the mixture. 

The algorithm we used for solvent/polydisperse polymer equilibria calls for only one solvent/polymer interaction parameter. The interaction parameter (pto) i ed in the algorithm can be determined from essentially any type of ethylene/polyethylene phase equilibrium data. Cloud-point data have been used (18). while Cheng (16) and Harmony ( ) have done so from gas sorption data. 

The cloud point is close to, but not necessarily equal to the lower consolute solution temperature for polydisperse nonionic surfactants (97). These are equal if the surfactant is monodisperse. Since the lower consolute solution temperature is like a critical point for liquid—liquid mixtures, the dilute and coacervate phases have the same composition, and the volume fraction of solution which the coacervate comprises is a maximum at this temperature (98). If a coacervate phase containing a high concentration of surfactant is desired, the solution should be at a temperature well above the cloud point. 

Figure 5. Examples of moment free energy (70) for Flory-Huggins theory of length-polydisperse polymers, with one moment density, p, retained. The parent is of the Schulz form (65), with pf = 0.03, Lu = 100 (hence p = p, /Lv = 3 x 10-4), and a = 2 (hence Lw = 150) the point pt = pj° is marked by the filled circles. In plot (a), the value of x = 0.55 is sufficiently small for the parent to be stable The moment free energy is convex. Plot (b) shows the cloud point, % 0.585, where the parent lies on one endpoint of a double tangent the other endpoint gives the polymer volume fraction p, in the shadow phase. Increasing x further, the parent eventually becomes spinodally unstable [x 0.62, plot (c)]. Note that for better visualization, linear terms have been added to all free energies to make the tangent at the parent coincide with the horizontal axis.

As a result of polydispersity effects, the composition of the incipient 13-phase segregated at the cloud point is located on a shadow curve, outside the cloud-point curve (point (3 in Fig. 8.4). The effects of polydispersity on phase diagrams and phase compositions may be found in specialized reviews (Tompa, 1956 Kamide, 1990 Williams et al., 1997). Because < )Mo < ( M,crit(xcp), the incipient (3-phase, which is richer in the modifier, will be dispersed in the a-phase, which is richer in the growing thermosetting polymer. The opposite occurs when < )M0 > M,crit(xcp)- It has been shown both theoretically (Riccardi et al., 1994 and 1996 Williams et al., 1997), and experimentally (Bonnet et al., 1999) that... 

Supercritical fractionation of high molecular weight alkane mixtures with propane or LPG may be used to produce products with lower polydispersity that that of molecular distillation. Operating temperatures just above the cloud point of the mixtures can be used compared to the high temperatures needed in molecular distillation. It was also shown that an optimum reflux ratio exists for every set of operating conditions. For this system it was also found that the operating costs of a supercritical fraction unit is marginally less than that of a molecular distillation unit. 

Krause, S. Stroud, D. E., "Cloud-Point Curves for Polystyrenes of Low Polydispersity in Cyclohexane Solution," J. Polym. Sci., Polym. Phys. Ed., 11, 2253 (1973). 

Ratzsch, M. Kruger, B. Kehlen, H., "Cloud-Point Curves and Coexistence Curves of Several Polydisperse Polystyrenes in Cyclohexane," J. Macromol. Sci., Chem., A27, 683 (1990). 

My, oo). This new phase boundary for the cloud point of the rubber is shown as the upper curve in Figure 1.34 and will correspond to an extent of conversion pcvphase dispersed in the thermoset (a-)phase and, as shown in Figure 1.34, the theoretical composition of the P-phase will be given by the position on the cloud-point boundary as shown. (It has been noted that this is not strictly correct due to polydispersity effects and the actual composition lies outside this line (Pascault et al, 2002)). 

FIGURE 16.9 The cloud-point and shadow curves for poly(ethylene-octene)-hexane at450 K from experiment (points) and the SAFT equation of state (curves). The dotted curve shows the composition of the incipient phase at the cloud point. Results for monodisperse and polydisperse polymer are included. (From Jog, P.K. et al., Ind. Eng. Chem. Res., 41(5), 887, 2002. With permission.)... 

Figure 3.27 Cloud point behavior of the poly(ethylene-co-methyl acrylate) (69 mol% ethylene and 31 mol% methyl acrylate) in ethane, propane, ethylene, and propylene. The copolymer concentration is fixed at 5wt% and the weight-average molecular weight of the copolymer is 99,000 with a molecular weight polydispersity of 3.0. (Hasch et al., 1993.)...

Figure 15.5 Effect of molecular weight dispersity ( )) (formerly known as polydispersity) using schematic Gibbs triangle diagrams for polymer-solvent system, generation of cloud point curve and shadow curve in temperature-composition diagram.

HAN Han, S.J., Gregg, C.J., and Radosz, M., How the solute polydispersity affects the cloud-point and coexistence pressures in propylene and ethylene solutions of alternating poly(ethylene-co-propylene),/ /. Eng. Chem. Res., 36, 5520, 1997. 

Figure 1. Demixing diagrams for PS in 0-solvents and poor solvents (schematic). The variable X might be pressure, Mw D/H ratio in solvent or solute, etc. See text for a further discussion, (a, top left) PS in a 0-solvent (monodisperse approximation). For X=Mw - the X=0 intercepts of the upper and lower heavy lines drawn through the minima or maxima in the demixing curves define 0Land0u, respectively, (b, top right) PS in a poor solvent (monodisperse approximation). The heavy dot at thecenterlocates the hypercritical (homogeneous double critical) point. (c, bottom right) The effect of polydispersity. BIN=binoda] curve, CP=cloud point curve, SP=spinodal, SHDW=shadow curve. See text Modified from ref. 6 and used with permission.

The binodal curve is the boundary between thermodynamically stable and metastable solutions. The term binodal is used in truly binary systems while in actual polydisperse systems the correct denomination is cloud-point curve (CPC). Thus the experimental determination of this boundary always leads to a CPC. 

The simple thermodynamic model derived in Sect. 2.1 has been useful to get a qualitative insight into the phase separation process. When one intends to apply it to an actual system, the significant influence of polydispersity is clearly evidenced. For example, Fig. 13 shows the experimental cloud-point curve for a DGEBA-CTBN binary mixture, together with binodal and spinodal curves calculated by assuming monodisperse components [66] (curves are arbitrarily fitted to the critical point). The shape of the CPC and precipitation threshold temperature (maximum of the CPC) appearing at low modifier concentrations are a clear manifestation of the rubber polydispersity [77]. 

However, a major limitation of this model is the impossibility of fitting cloud-point curves for polydisperse systems. Moreover, it cannot deal with the fractionation effect accompanying phase separation, i.e. the dispersed phase will be enriched in the highest molar-mass fractions of modifier but in the lowest molar-mass fractions of the growing thermosetting polymer. This may produce variations in stoichiometry and conversion between both phases. These phenomena can be conveniently treated taking polydispersity of constituents into account. 

The occurrence of a secondary phase separation inside dispersed phase particles, associated with the low conversion level of the p-phase when compared to the overall conversion, explains the experimental observation that phase separation is still going on in the system even after gelation or vitrification of the a-phase [26-31]. A similar thermodynamic analysis was performed by Clarke et al. [105], who analyzed the phase behaviour of a linear monodisperse polymer with a branched polydisperse polymer, within the framework of the Flory-Huggins lattice model. The polydispersity of the branched polymer was treated with a power law statistics, cut off at some upper degree of polymerization dependent on conversion and functionality of the starting monomer. Cloud-point and coexistence curves were calculated numerically for various conversions. Spinodal curves were calculated analytically up to the gel point. It was shown that secondary phase separation was not only possible but highly probable, as previously discussed. 

Figure Dependence of the cloud point temperature T of a polydisperse polystyrene...

Given f P), these q I equations can, in principle, be solved for Tp, ", and f Py as functions of . The resulting relation between Tp and gives the cloud-point curve, while that between Tp and " gives a line called the shadow curve. The latter cannot be determined experimentally, since the second phase is too small in volume to be analyzed for the composition. It can be shown that the cloud-point and shadow curves coincide with the conjugate coexistence curves when and only when the solution is strictly binary. This fact is important, because some authors make no distinction between cloud-point curve and coexistence curve in describing phase equilibria of polydisperse solutions. 

Polydisperse (PPO-Z -PEO) allyl ether siloxane surfactants were synthesized by the hydrosilylation reaction of 1,1,1,3,5,7,7,7 octamethyltetrasiloxane (MD2 M) with oligo(PO-b-EO) allyl ethers. In this series, the surface tension increased with increasing the EO chain length and it decreased with increasing the PO ratio, while the sedimentation time of the aqueous solution showed opposite trend. The cloud point temperatures tended to increase with the increase in the EO chain length and decrease of the PO ratio [48]. 

Figure 4.4.19. Principles of liquid-liquid demixing in polymer solutions, a) - strictly binary polymer solution of a monodisperse polymer, b) - quasi-binary polymer solution of a polydisperse polymer which is characterized by a distribution function C - critical point, dashed lines - tie lines, T(l) - temperature/concentration in the homogeneous region, T(2) - temperature/concentrations of the cloud point (phase ) and the corresponding shadow point (phase "), T(3) - temperature in the heterogeneous LLE region, coexistence concentrations of phase and phase at T(3) are related to the starting concentration = cloud point concentration of (2).

For solutions of polydisperse polymers, such a procedure cannot be used because the critical concentration must be known in advance to measure its corresponding coexistence curve. Additionally, the critical point is not the maximum in this case but a point at the right-hand side shoulder of the cloud-point curve. T wo different methods were developed to solve this problem, the phase-volume-ratio method, e.g., Koningsveld, where one uses the fact that this ratio is exactly equal to one only at the critical point, and the coexistence concentration plot, e.g. Wolf, where an isoplethal diagram of values of tp 2 and tp 2 vs. (p 2 gives the critical point as the intersection point of cloud-point and shadow curves. 

KIM Kim, H.C. and Kim, J.-D., The polydispersity effect of distributed oxyethylene chains on the cloud points of nonionic surfactants, J. Colloid Interface Sci., 352,444, 2010. 

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