17. Crystallization Kinetics PDF

17. Crystallization kinetics 17.1. General considerations. Crystallization has two steps: nucleation and growth. The ordered regions from nucleation are called nuclei. These nuclei are stable only below the melting temperature of the polymer since they are disrupted by thermal motion above this tem...

17. Crystallization kinetics 17.1. General considerations. Crystallization has two steps: nucleation and growth. The ordered regions from nucleation are called nuclei. These nuclei are stable only below the melting temperature of the polymer since they are disrupted by thermal motion above this temperature. Classification of nucleation: homogeneous and heterogeneous. Growth: crystal nuclei grow by the addition of further chains. During homogeneous nucleation in a polymer melt or solution, it is envisaged that small nuclei form randomly throughout the melt. Although this process has been analysed in detail from a theoretical viewpoint, it is thought that, in the majority of cases of crystallization from polymer melts and solutions, nucleation takes place heterogeneously on foreign bodies such as dust particles or the walls of the containing vessel. When all other factors are kept constant, the number of nuclei formed depends upon the temperature of crystallization. At low undercooling, nucleation tends to be sporadic and during melt crystallization, a relatively small number of large spherulites form. On the other hand, when the undercooling is increased many more nuclei form and a large number of small spherulites are obtained. An important experimental observation that simplifies the theoretical analysis is that the change in linear dimensions of the growing entities at a given temperature of crystallization usually is linear with time. This means that the spherulite radius r will be related to time t through an equation of the form r = vt where v is the growth rate. This equation usually is valid until the spherulites become so large that they touch each other. The growth rate v is strongly dependent upon the crystallization temperature. as the supercooling is increased there is a rapid increase in v. However, it is found that eventually there is a peak in v 1 and further lowering of the crystallization temperature produces a reduction in v. The reason for the peak in v is thought to be due to two competing effects. The thermodynamic driving force for crystallization will increase as the crystallization temperature is lowered. But as the temperature is reduced, there will be an increase in viscosity, and transport of material to the growth point will be more difficult and so v peaks and eventually decreases as the temperature is reduced even though the driving force continues to increase. 17.2. Overall crystallization kinetics If a polymer melt of mass W0 is cooled below the crystallization temperature, then spherulites will nucleate and grow over a period of time. Assuming that nucleation is homogeneous and that at a given temperature the number of nuclei formed per unit time per unit volume (i.e. the rate of nucleation) N is a constant, then the total number of nuclei formed in time interval dt will be NW0dt/ ρL where ρL is the density of the liquid polymer. After a length of time t, these nuclei will have grown into spherulites of radius r. The volume of each spherulite will be 4πr3/3 or, 4πv3t3/3. If the density of spherulitic material is ρS, then the mass of each spherulite will be 4πv3t3ρS/3. The total mass of spherulitic material, dWS, present at time t, and grown from the nuclei formed in the time interval, dt, will be given by 4 𝑑𝑡 𝑑𝑊𝑠 = 𝜋𝑣 3 𝑡 3 𝜌𝑠 𝑁𝑊0 3 𝜌𝐿 𝑡 4𝜋𝑣 3 𝜌𝑠 𝑁𝑊0 𝑡 3 𝑊𝑠 = ∫ 𝑑𝑡 3𝜌𝐿 0 𝑊𝑠 𝜋𝑁𝑣 3 𝜌𝑠 𝑡 4 = 𝑊0 3𝜌𝐿 2  𝑊𝐿 𝜋𝑁𝑣 3 𝜌𝑠 𝑡 4 = 1− 𝑊0 3𝜌𝐿 This analysis is highly simplified and the equations are valid only for the early stages of crystallization. It also follows that if the nuclei are formed instantaneously, then a t3 dependence would be expected as only the change in spherulite volume is time dependent. The analysis is slightly incorrect because, during crystallization, there is a reduction in the overall volume of the system and the center of the spherulites move closer to each other. When this impingement is taken into account, it is found that WL/W0 is related to t through an equation of the form WL/W0 = exp(-zt4) This type of equation is generally known as an Avrami equation. If types of nucleation and growth other than those considered here are found, the Avrami equation can be expressed as WL/W0 = exp(-ztn) From an experimental viewpoint, it is much easier to follow the crystallization process by measuring the change in specimen volume rather than the mass of spherulitic material. If the initial and final specimen volumes are defined as Vo and V∞, respectively, and the specimen volume at time t is given by Vt, then it follows that 𝑊𝐿 𝑊𝑆 𝑊0 1 1 𝑉𝑡 = + = + 𝑊𝐿 ( − ) 𝜌𝐿 𝜌𝑆 𝜌𝑠 𝜌𝐿 𝜌𝑆 𝑊0 𝑊0 Since 𝑉0 = and 𝑉∞ = 𝜌𝐿 𝜌𝑠 Then 𝑉0 𝑉∞ 𝑉𝑡 = 𝑉∞ + 𝑊𝐿 ( − ) 𝑊0 𝑊0 𝑊𝐿 𝑉𝑡 − 𝑉∞ = = exp⁡(−𝑧𝑡 𝑛 ) 𝑊0 𝑉0 − 𝑉∞ This equation then allows the crystallization process to be monitored by measuring how the specimen volume changes with time. In practice, this normally is done by dilatometry. The 3 crystallizing polymer sample is enclosed in a dilatometer and the change in volume is monitored from the change in height of a liquid in the dilatometer, which follows the specimen volume 𝑉𝑡 − 𝑉∞ ℎ𝑡 − ℎ∞ = = exp⁡(−𝑧𝑡 𝑛 ) 𝑉0 − 𝑉∞ ℎ0 − ℎ∞ The Avrami exponent n can be determined from the slope of a plot of log{ln[(ht−h∞)/(h0−h ∞)]} against log t. It often is difficult to estimate n from such plots because its value can vary with time. Also, non-integral values can be obtained and care must be exercised in using the Avrami analysis, as interpretation of the value of n in terms of specific nucleation and growth mechanisms can sometimes be ambiguous. 4