One essential part of the work presented in this chapter is the design of a custom-made experimental setup fitting the task to investigate the self-assembly behavior of a selected 1,3,5-benzenetrisamide from solution. The process focused on temperature changes and simultaneously mechanical stirring during self-assembly. A change of the overall BTA concentration by evaporation of the solvent had to be avoided. Figure 3.2 displays a schematic representation of the developed custom-made setup. The chosen vessel was a screw-mountable laboratory glass bottle (250 mL; Duran Group) with an outer-diameter of 70 mm and a height of 143 mm. A customized cap allowed for the insertion of a mechanical stirrer (RZR 2051; Heidolph) and a thermocouple (type K), while keeping the vessel sealed and avoiding any evaporation of the solvent. The selected mechanical stirrer ensured a reproducible mechanical stirring by enabling precise digital control of the stirring velocity measured in revolutions per minute (rpm). The thermocouple was linked to a Datalogger (PCE-T390; PCE-Instruments) to record time dependent temperature profiles of the self-assembly system inside the glass during the process (Tinside).
Figure 3.2: Schematic representation of the custom-made experimental setup for thermal induced self-assembly experiments consisting of a temperature control bath (A), a thermocouple (B), a mechanical stirrer (C), a screw-mountable vessel (D), a customized cap (E) and a guide for the mechanical stirrer (F).
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The vessel can be heated from the outside with a water bath and cooled with a cooling bath of a cryostat (Haake F3; cooling agent: ethylene glycol/water (1:1)). The cooling agent inside the cryostat was circulated constantly to improve the heat exchange. The experimental setup is designed in a way that it was possible that the vessel could be placed quickly from the heating bath into the cooling bath.
Heating and cooling temperatures (Toutside) were being kept constant during each experiment, so no temperature profiles of the cooling agent outside the vessel were recorded. Since changes in temperature are one key parameter of this experimental setup, the mechanism of heat exchange between the BTA solution on the inside of the glass and the cooling agent on the outside has to be introduced.
Transport processes such as heat transfer are of great importance in many industrial operations.[92]
The rate of heat flow Q̇ in this experimental setup is based on the indirect heat exchange between two liquid media through a wall. If the experiment is performed without any mechanical stirring, the movement of the fluid inside the vessel is governed by natural convection and due to insufficient mixing a temperature gradient arises from the glass wall to the inside of the vessel. Mechanical stirring during the cooling process enhances the heat transport from inside the vessel leading to a more homogenous temperature distribution throughout the liquid medium. Assuming an ideal mixing of the fluids, Figure 3.3 displays a simplified schematic representation of the heat exchange through a wall for very small time intervals in which all parameters are assumed to be constant.[124] The process can basically be divided into three individual steps.[125] The first step consists of the heat transfer from the self-assembly system to the wall on the inside of the glass Q̇1. This transfer is directly proportional to the thermal conductivity coefficient λ1 of the BTA solution and to the surface area of the wall Aw. Tinside
represents the temperature of the self-assembly system inside the vessel and Toutside corresponds to the constant cooling temperature outside the vessel. Tw1 refers to the temperature of the wall on the inside of the laboratory glass. According to equation 3.1, the greater the difference between Tinside and Tw1, the greater is the resulting heat transfer Q̇1.
Q̇1=ë1
ä1∙ Aw∙ (Tinside− Tw1). (3.1)
δ1 represents a boundary layer in which the heat transfer is mainly governed by thermal conduction.[126] On the one hand, a high thermal conductivity λ1 of the fluid inside the vessel leads to high values for the heat transfer, whereas on the other hand, a thick boundary layer δ1 close to the wall reduces the heat flow Q̇1.
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Figure 3.3: Schematic representation of the heat transfer from the self-assembly system on the cooling agent. The transfer consists of three individual steps: 1. Heat transfer from the self-assembly system on the wall. 2. Heat conduction through the wall. 3. Heat transfer from the wall to the cooling agent.
λ1: Thermal conductivity coefficient of the self-assembly system. λ2: Thermal conductivity coefficient of the cooling agent. λwall: Thermal conductivity coefficient of the material of the wall. Tinside: Temperature inside the vessel. Toutside: Temperature of the cooling agent. Tw1: Temperature of the wall on the inside. Tw2: Temperature of the wall on the outside. δ1: Boundary layer between self-assembly system and wall. δ2: Boundary layer between cooling agent and wall. d: Thickness of the wall.
The second step in the cooling process is the transfer of heat through the wall of the vessel Q̇2 which is given by equation 3.2.
Q̇2 =ëwall
d ∙ Aw∙ (Tw1− Tw2). (3.2)
Even though the wall of the vessel is curved, differences in the surface area of the wall Aw between the inside and outside are assumed to be negligible. λwall is the thermal conductivity coefficient of the material of the wall and d corresponds to its thickness. Tw2 describes the temperature of the wall on the outside of the vessel.
The third step of the heat exchange describes the transfer from the outside of the wall of the vessel to the cooling agent Q̇3 through a boundary layer δ2. High thermal conductivity λ2 of the cooling agent allows for large values of heat being transferred.
cooling
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Q̇3 =ë2ä2∙ Aw∙ (Tw2− Toutside). (3.3)
The total amount of heat exchanged is equal for all three individual steps, so that the overall heat exchange Q̇ from the solution inside the glass to the cooling agent on the outside can be represented by equation 3.4.[127]
Q̇ = Q̇1= Q̇2= Q̇3. (3.4)
Transformation of equations 3.1 and 3.3 and substitution into equation 3.2 yields the overall heat flow rate Q̇ from the self-assembly system to the cooling agent under stationary conditions:
Q̇ =ä1 Aw
ë1+ d
ëwall+ä2
ë2
∙ (Tinside− Toutside). (3.5)
The temperature profile of the BTA solution during the cooling process is directly related to the overall heat flow Q̇. While most variables are directly accessible such as thermal conductivities λ1 and λ2 as well as the differences in temperature, the thickness of the boundary layers δ1 and δ2 on the inside and on the outside of the system have to be derived from dimensionless quantities such as the Nusselt number (Nu), the Reynolds number (Re) and the Prandtl number (Pr), which are frequently used in chemical reaction engineering. The value of the Nusselt number is given by equation 3.6.[126]
Nu =áë1D
1 (3.6)
á1=ë1
ä1 (3.7)
á1 is called the heat transfer coefficient and is given by the ratio of ë1and ä1 as shown equation 3.7. D corresponds to a characteristic length depending on the geometry of the experimental setup. For a mechanically stirred batch reactor, this length corresponds to the inner diameter of the reactor. The Nusselt number depends on varying equations depending on the experimental boundary conditions.
In case of no applied mechanical stirring inside the vessel, the Nusselt number is governed by a phenomenon called natural convection. Local differences in density of the fluid medium cause movement of the individual particles. However, most experiments were performed under constant stirring. In this case, Nusselt number can be described by equation 3.8, which can be used for forced flow conditions.[126]
Nu = CRemPrn(L
d)p (3.8)
The values of C, m, n and p have to be determined empirically for a certain experimental setup.
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The Reynolds number (Re) is a dimensionless quantity that is frequently used in fluid mechanics, whereas the Prandtl number (Pr) corresponds to the ratio of the kinematic viscosity of the fluid and its thermal diffusivity. L and d represent characteristic lengths depending on the geometry of the selected setup. Equation 3.9 describes a possible relation for the Reynolds number of a mechanically stirred batch reactor.[126]
Re = N∙bí2 (3.9)
N represents the stirring velocity in revolutions per minute and b is the geometric width of the stirrer.
í is the kinematic viscosity of the system.[126] Increasing the mechanical stirring velocity inside the vessel results in an increase of the Reynolds number and in consequence an increase of the Nusselt number.
According to equation 3.6, an increase in the Nusselt number leads to higher values of á1 and thereby a decrease of the boundary layer ä1. A low thickness of this boundary layer improves the cooling process of the BTA solution. In addition, high kinematic viscosities of the fluid result in high values for the boundary layer limiting the cooling rate. Therefore, a cooling agent with a rather low viscosity was used as cooling agent (water and ethylene glycol (1:1)). The viscosity of pure ethylene glycol is around 22 mPas (20 °C), whereas pure water only exhibits a viscosity of 1 mPas (20 °C).
During the experiment the temperature of the cooling agent is kept constant by the use of a cryostat, while the temperature of the self-assembly system changes with time. Figure 3.4 displays schematically an estimated theoretical temperature profile of the self-assembly system during the cooling process. Thereby, the cooling rate of the system changes continuously as the temperature difference between the BTA solution inside the vessel and the cooling agent on the outside becomes smaller. However, during the cooling process, the variation in temperature results also in significant changes of the viscosity of the BTA solution, which also influences the thickness of the boundary layer ä1 and yet the heat exchange.
Two series of experiments will be conducted in the course of this work. The first series will focus on the variation of mechanical stirring velocities during the self-assembly process to investigate the influence of different mechanical conditions on the resulting morphology of supramolecular structures. This change in stirring will also lead to a change in the temperature profiles. During the second series of experiments, the cooling temperatures on the outside of the glass will be varied and therefore change temperature profiles by constant influence of shear force.
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Figure 3.4: Schematic representation of an expected temperature profile of a liquid inside the glass based on equation 3.5 if the temperature of the cooling agent is kept at a constant value.
Toutside
temperature inside
cooling time Toutside
Tstart