WO2021249897A1

WO2021249897A1 - Method for determining a coefficient in a thermoplastic polymer viscosity calculation

Info

Publication number: WO2021249897A1
Application number: PCT/EP2021/065089
Authority: WO
Inventors: Christoph BONTENACKELS
Original assignee: Covestro Deutschland Ag
Priority date: 2020-06-12
Filing date: 2021-06-07
Publication date: 2021-12-16

Abstract

By using measurement data of a high pressure capillary rheometer for the reverse engineering approach a wide range of temperatures, shear rates, and pressures can be covered by the optimization with high resolutions of the simple geometry of a high pressure capillary rheometer within reasonable simulation times. This is a big difference to more common approaches of the reverse engineering of production processes like injection molding which cover only small ranges of shear rates and temperatures and often use complex geometries which need long simulation times with only limited resolution of polymer specific effects. It is concluded that the reverse engineering of the measurement device (high pressure capillary rheometer) promises higher accuracy for a wider range of processes within smaller simulation times (especially when using the advanced pre-fitting methods and realistic limits for pressure dependency of viscosity at low shear rates as described above).

Description

Method for determining a coefficient in a thermoplastic polymer viscosity calculation

The present invention relates to a computer-implemented method for determining at least one empirical coefficient used in the calculation of the viscosity of a thermoplastic polymer melt.

In times of highly competitive markets, product and process development for technical applications face steadily increasing demands for faster development times and higher quality requirements. In the context of applications for technical plastic products, process simulation (such as injection molding simulation) is a key factor to fulfill the rising requirements in the early stages of application development processes with digital engineering methods. Due to continuously rising requirements towards the accuracy of these simulations, the quality of the material description as an input for the simulation is gaining in importance as a key factor for development processes of high quality technical applications.

In this context, shear viscosity is an important property for describing and designing polymer processing processes like injection molding with help of numerical simulations. These simulations require the viscosity as continuous function of shear rate, temperature and pressure as input factor for the simulation. In general, the measurement of polymer viscosity is commonly done with help of a high pressure capillary rheometer and afterwards fitted into a continuous viscosity model for usage in numerical simulation. However, the results obtained in simulations of polymer processing (such as in injection molding simulation) by the use of common viscosity fitting methods suffer from poor prediction quality of the simulated pressure. This leads to high security factors being applied during process and tool design.

A high pressure capillary rheometer comprises a stamp and capillary (defined by its diameter and its length) where a polymer melt is forced through the capillary with a defined stamp speed at a defined polymer temperature. In order to include the pressure dependency of the viscosity into the measurement, special high pressure capillary rheometers allow applying a counter pressure at the end of the capillary such that the general pressure level is increased during the measurement. As a high pressure capillary rheometer only measures a pressure loss along a defined capillary geometry, the polymer viscosity and shear rates need to be derived from the measurement data (pressure losses) and the input parameters (stamp speed, length & diameter of the capillary). In order to do so, the Hagen-Poiseuille equations based on Newtonian flow behavior are commonly used to calculate an apparent shear rate and an apparent viscosity.

The accuracy of the apparent shear rate and the apparent viscosity calculation suffers from simplifications since thermoplastic polymer melts in general show a different flow behavior than Newtonian fluids. Simplifications include neglecting visco-elastic properties, shear thinning, temperature dependency of viscosity and pressure dependency of viscosity. Furthermore, geometric simplifications of the analysis are applied such as neglecting inflow/outflow effects and wall-slip.

To overcome some of these limitations, analytical corrections of the apparent shear rate and the apparent viscosity are commonly used:

• Mooney analysis: Correction of wall-slip during the measurement

• Bagley correction: Correction of the apparent viscosity due to inflow and outflow effects

• Weissenberg-Rabinowitsch: Correction of the apparent shear rate at the capillary wall due to the shear thinning behavior of polymer melts

For a continuous description of viscosity as function of shear rate, temperature and pressure, the coefficients of a viscosity model are then fitted to the measurement data, such that the error between the viscosity calculated by the viscosity model and the viscosity derived from the measurement data (corrected or not) is minimized by an appropriate optimization algorithm.

To include the pressure dependency of the viscosity into the fitting, the counter pressure at the end of the capillary needs to be integrated into the analysis, such that a representative pressure in the capillary is being used during the fitting. One method to do so is to use the average pressure in the capillary by calculating the average between the counter pressure at the end of the capillary and the measured pressure loss as input for the viscosity fitting. Other methods are to use the pressure dependency of viscosity calculated by a Barus equation or the pressure dependent shift of the glass transition temperature evaluated by other polymer properties like pvT (pressure, volume and temperature) data.

While the mathematical fitting for conventional viscosity approximations in general reaches very high accuracy, as polymer specific effects and properties are left out, the real accuracy of the data for usage in numerical simulation is limited. Additionally, by the use of capillary rheometry, high counter pressures to capture pressure dependency of viscosity during the measurement requires high flow rates in the capillary, as the counter pressure conventionally is applied by narrowing the exit of the capillary. This leads to further restrictions in that the pressure dependency of viscosity at high counter pressures can only be measured at high shear rates and thus the pressure dependency at low shear rates (such as in a hot runner system of injection molding tools) can only be extrapolated from the existing measurement data. The poor prediction quality of these common approaches and corrections for viscosity fitting lead to the need for superior methods and tools to describe polymer viscosity as function of shear rate, temperature and pressure more appropriate for the use in numerical simulation of polymer processing (in particular an injection molding simulation).

WO 2013/164587 A1 discloses a rheometer and a rheometric method. An apparatus responsive to constitutive parameters of the rheological properties of complex fluids comprises: a flow path for the fluid that comprises a shear feature that results in multiple shear patterns between adjacent elements of the flow stream of the fluid as it transitions the shear feature; means to sense flow parameters of the flow stream along the flow path, wherein said flow parameters are selected to be sensitive to changes in rheological parameters of complex fluids; wherein said flow stream is pulsed, whereby time-dependent transient effects as a result of the complex rheological properties of the fluid affect the flow parameters sensed over and after the period of the pulse and are detected by the apparatus. The apparatus also determines constitutive parameters of the rheological properties of complex fluids by including computational means to calculate on the basis of said sensed flow parameters by inverse interpolation according to appropriate models of said rheological properties.

US 2018/373821 A1 describes the characterization of a fluid flow at field conditions. An apparatus to perform tests on fluid flow and configured to operate at field conditions includes one or more vessels and one or more sets of fluid injecting devices corresponding to respective ones of the one or more vessels. Each set of fluid injecting devices includes one or more fluid injecting devices each configured to inject a respective fluid through its respective vessel. The apparatus further includes one or more measurement devices operatively coupled to respective ones of the one or more vessels and configured to measure data associated with fluid flow of the one or more fluids injected into its respective vessel. The measured data comprises one or more of pressure gradient data and flow rate data. The apparatus is in communication with at least one processor configured to calculate a model based on the measured data. In calculating the model, the at least one processor is configured to infer one or more parameters for the model from the measured data.

It would be desirable to have an improved method for determining an empirical coefficient used in the viscosity modelling of a thermoplastic polymer melt.

Proposed is a method according to claim 1. Advantageous embodiments are the subject of the dependent claims. They may be combined freely unless the context clearly indicates otherwise. Accordingly, a computer-implemented method for determining at least one empirical coefficient used in the modelling of the viscosity of a thermoplastic polymer melt comprises:

A) Receiving at least one empirical coefficient, the at least one empirical coefficient being determined by a method comprising:

A1) At least once, experimentally determining, via a capillary rheometer having an inlet and an outlet, at least one rheological property of the polymer melt;

A2) Calculating, using an analytically solvable generalized Newtonian fluid (GNF) model, the at least one rheological property determined in step A1) of the polymer melt; wherein the calculation includes: at least one geometrical boundary representing the capillary rheometer of step A1) and at least a volume rate, a pressure at the outlet of the capillary and a temperature at the inlet of the capillary of the polymer melt and at least one empirical coefficient;

A3) Calculating the difference between the at least one rheological parameter obtained in step A1) and the at least one rheological parameter obtained in step A2);

A4) If the absolute value of the difference in step A3) exceeds a pre-determined threshold, modifying the at least one empirical coefficient used as an input for the GNF model calculation in step A2) and repeating steps A2) to A4);

B) Calculating, using a numerically solvable Navier-Stokes model in a computational fluid dynamics (CFD) calculation, the same rheological property or properties of the polymer melt as experimentally determined in step A1); wherein the input of the CFD calculation includes: at least one geometrical boundary representing the capillary rheometer of step

A1), at least the volume rate, the pressure at the outlet of the capillary and the temperature at the inlet of the capillary of the polymer melt and at least one empirical coefficient received in step A) or obtained as the result of a previous modification in step D);

C) Calculating the difference between the at least one rheological property obtained as an output from the CFD calculation in step B) and the same rheological property or properties as experimentally determined in step A1);

D) If the absolute value of the difference in step C) exceeds a pre-determined threshold, modifying at least one empirical coefficient in step B) and repeating steps B) to D).

The method according to the invention is able to improve the prediction quality and/or the prediction speed of viscosity data used for numerical simulation of polymer processing processes. In this method, the measurement data of a high pressure capillary rheometer can be translated into viscosity data as function of shear rate, extension rate, temperature and pressure based on a numerical simulation of the high pressure capillary rheometer and a reverse engineering approach to determine optimized model coefficients for commonly used viscosity models such as the Cross-WLF equation.

While the method is not restricted to a certain type of polymer, it is preferred that the polymer is a thermoplastic polymer, in particular a polycarbonate, a polycarbonate-ABS blend, a polycarbonate -polyester blend or a thermoplastic polyurethane. The polymer may be filled or unfilled and contain customary additives used in the art.

In step A) at least one empirical coefficient is received. The mode of the reception is not particularly limited. For example, reception may occur by informing an operator who is about to carry out the subsequent steps of the method using a computer. Likewise, reception may occur by accessing a file on a computer or a data storage medium, a local area computer network, a wide area computer network, the internet or a cloud computing environment.

The empirical coefficient may be determined in advance and stored in a database. It is possible that a separate entity, such as testing laboratory, determines the coefficient and then transmits the coefficient to the entity performing the method according to the invention. Alternatively, the coefficient may be determined ad hoc for a specific purpose.

In determining the empirical coefficient, sub-steps A1) to A4) of step A) are executed. According to step A1), at least one rheological property of the polymer melt is determined via a capillary rheometer. It goes without saying that the polymer material used in this measurement is the same material for which the calculation of its viscosity in the molten state shall be calculated in the method according to the invention. The experimental determination may be in accordance with DIN 54811, ASTM D3835, ISO 17744, ASTM D5930, ASTM D5099 or ISO 11443. Counter-pressure viscosimeters may also be used. Examples for internal diameters of a cylindrical capillary are 1 mm or 2 mm. Suitable capillary lengths may be from 10 mm to 30 mm.

In a preferred embodiment, the method for determining the at least one empirical coefficient as recited in step A) which comprises steps A1) to A4) is executed and the at least one empirical coefficient that has or have been determined is or are used as input for the CFD calculation in step B). Thereby also the “receiving at least one empirical coefficient” as recited in step A) is automatically fulfilled. In verbose terms this embodiment reads:

A computer-implemented method for determining at least one empirical coefficient used in the modelling of the viscosity of a thermoplastic polymer melt comprising:

Determining at least one empirical coefficient by a method comprising:

A2) Calculating, using an analytically solvable generalized Newtonian fluid (GNF) model, the at least one rheological property determined in step A1) of the polymer melt; wherein the calculation includes: at least one geometrical boundary representing the capillary rheometer of step A1) and at least a volume rate, a pressure at the outlet of the capillary and a temperature at the inlet of the capillary of the polymer melt and at least one empirical coefficient; A3) Calculating the difference between the at least one rheological parameter obtained in step A1) and the at least one rheological parameter obtained in step A2);

A1), at least the volume rate, the pressure at the outlet of the capillary and the temperature at the inlet of the capillary of the polymer melt and at least one empirical coefficient determined by a method comprising steps A1) to A4) after the calculated difference in step A3) is below the pre-determined threshold as recited in step A4) or obtained as the result of a previous modification in step D);

D) If the absolute value of the difference in step C) exceeds a pre-determined threshold, modifying at least one empirical coefficient in step B) and repeating steps B) to D). In another preferred embodiment, in step A1) the at least one rheological property includes the pressure drop over the entire capillary rheometer at a specified volume rate, counter-pressure and temperature of the polymer melt. If there are no counter-pressure means implemented in a measurement, the counter-pressure is taken to be the ambient air pressure. The absolute pressure in the measurement can be derived from the pressure drop and the counter-pressure. Without wishing to be bound to this embodiment, the invention may be described in the following using this pressure drop as an example for the rheological property.

Step A2) is a calculation of the experimentally determined rheological property of the polymer melt, for example of the pressure drop. The calculation is performed using an analytically solvable combination of a GNF (Generalized Newtonian Fluid) model as constitutive equation for the polymer melt and a physical equation for the flow field, where “analytically solvable” means that a solution to the equation can be found by using algebraic operations and concepts such as addition, multiplication, associativity and commutativity. Examples for such GNF models include the power-law model, the Spriggs truncated power-law model, the Carreau model, the Bingham model or the Casson model. Examples for the physical equation of the flow field are the Navier Stokes equations and any simplification based on those.

The input of the physical model furthermore includes at least one geometrical boundary representing the capillary rheometer of step A1). Examples for such boundaries are the length of the capillary and the diameter of the capillary. The volume rate as input is the volume rate of the capillary rheometer measurement.

In a preferred embodiment, in step A2) the GNF model incorporates, as a physical equation, the Flagen-Poiseuille equation:

with

where dp is the pressure difference, dL is the length difference,

is the volume rate,

is the shear rate, η is the melt viscosity, D is the diameter of the capillary and where η can be (preferably: is) a function of parameters including the shear rate, the temperature, the pressure, the extension rate and at least one empirical coefficient. The shear rate may be the apparent shear rate or the true shear rate. Without wishing to be bound to this embodiment, the invention may be described in the following using the Fiagen-Poiseuille equation as an example for the physical equation for the flow field.

The input of the physical equation for the flow field includes a volume rate, a pressure at the outlet of the capillary and a temperature at the inlet of the capillary of the polymer melt and at least one empirical coefficient. This is to provide for the fact that a pressure (including a pressure drop) and a temperature are able to be determined and recorded in an experimental setup.

The at least one empirical coefficient may be part of the GNF model. In a preferred embodiment, in the GNF model in step A2) any dependency of a viscosity of the polymer melt on shear rate, extension rate and/or temperature is described using the Cross-WLF equation:

where η is the melt viscosity, η₀ is the zero shear viscosity,

is the shear rate,

is the critical stress level at the transition to shear thinning and n is the power law index in the high shear rate regime and where η₀ is defined as:

with T being the temperature, T_* being the glass transition temperature, A₂ = A₃ + D₃ p, p being the pressure, D₁, A₁, A₃ and D₃ being empirical coefficients and T_* = D₂ + D₃ p with D₂ being an empirical coefficient and the at least one empirical coefficient received in step A) is selected from n, A₁, A₃, D₁, D₂, D₃,

or a combination of at least two of the aforementioned coefficients. Without wishing to be bound to this embodiment, the invention may be described in the following using the Cross-WLF equation as the origin of the at least one empirical coefficient. The Cross-WLF equation as contemplated in the present disclosure may be the standard equation outlined above or may be extended to incorporate further models or phenomena. In another preferred embodiment, in step A2) a virtual capillary rheometer of length L, having an inlet and an outlet, is divided into a plurality of discrete sections of the same length dL, pressure loss and shear heating are calculated in each section separately and the pressure loss and shear heating calculations are repeated until the calculated pressure drop over the entire capillary rheometer converges. In the context of the present invention the term “converges” is to be understood as that the difference of a value of a present iteration to the value of the preceding iteration is below a pre-determined threshold. In particular, the convergence of the pressure drop indicates that the presently calculated pressure drop differs by less than a pre determined value from the pressure drop of the preceding calculation.

Moreover, inflow effects can also be integrated into the pre-fitting of model coefficients of viscosity models by modelling the respective inflow area by discrete, cylindrical segments and adding the pressure loss in these segments to the pressure loss of the whole capillary rheometer.

According to step A3) the difference between the experimentally determined rheological parameter and the calculated rheological parameter is determined. For example, one could determine the difference in pressure drop as obtained in the capillary rheometer and as calculated using the Hagen-Poiseuille equation with the Cross-WLF equation as an expression for the viscosity as function of the shear rate and the temperature.

In step A4) it is decided whether the difference is acceptable or not. In other words, it is decided whether the calculation of step A2) with its at least one empirical coefficient describes the experimentally determined result accurately enough. Then it may be inferred that the at least one empirical coefficient in the GNF model is within acceptable boundaries. The at least one empirical coefficient can then be passed on for reception according to step A) of the method.

If, on the other hand, the absolute value of the difference in step A3) exceeds a pre-determined threshold, the at least one empirical coefficient used as an input for the calculation in step A2) is modified and the sequence A2) to A4) is run again with the new coefficient(s). If there are several empirical coefficients, it is possible to modify one, a plurality or all of them.

Preferably, all coefficients are modified. For this the individual coefficients may be weighted with the derivative of the viscosity model with respect to this coefficient. This has the effect that all coefficients have the same influence on the result.

A pre-determined threshold may, for example, be a difference of 1 bar or less in the pressure drop or a deviation of 10% or less between experimental results and calculated results. Another pre-determined threshold may be the accuracy limit of the rheometer. This may be listed in a norm governing the capillary rheometer measurement or may be stated by the manufacturers of the rheometer.

Step B) of the method according to the invention involves a CFD calculation using the pre- optimized at least one empirical coefficient received in step A) as an initial input. The pre- optimization via convergence of experimental results and analytically solvable equations can save a considerable amount of computing time in the CFD calculation itself.

In step B) the same rheological property or properties of the polymer melt that have been determined experimentally in step A1) are computed. Preferably this is the pressure drop at a specified initial pressure and temperature of the polymer melt.

The input of the CFD calculation includes at least one geometrical boundary representing the capillary rheometer of step A1). Examples for such boundaries are the length of the capillary and the diameter of the capillary. The CFD input also includes a volume rate, the pressure at the outlet of the capillary and the temperature at the inlet of the capillary of the polymer melt. Without wishing to be bound by this statement, the CFD calculation can be seen as creating a virtual capillary rheometer which attempts to reproduce experimental results. Then, one may assume that the convergence of calculated and experimental data equates to other calculated parameters of the polymer melt which cannot be measured in principle or in practice are also correct within acceptable error margins.

A useful expression of a Navier-Stokes model for a GNF is given below:

where u is the velocity vector, T is the temperature, t is the time, p is the pressure, σ is the total stress tensor, I is the identity tensor, p is the density, η is the viscosity, k is the thermal conductivity, C_p is the specific heat and

is the shear rate. The CFD calculation may address the following aspects:

- Shear thinning and flow field: In all cases (analytical viscosity fitting, fitting based on CFD simulation of a high pressure capillary rheometer), the effect of shear thinning is included in the commonly used continuous viscosity models for polymers. In this context, all analytical approaches based on Hagen- Poiseuille equations calculate the pressure in the capillary only on basis of the shear rate and shear stress at the wall, neglecting that polymer melts show a characteristic shear distribution perpendicular to the flow direction. By doing so, all analytic approaches average polymer viscosity over the diameter of the capillary and assign this value of viscosity to the shear rate at the capillary wall (or in case of Rabinowitsch correction to the corrected shear rate), while a CFD simulation of a high pressure capillary rheometer can take the characteristic shear distribution with correct assignment of viscosity values into account.

- Shear heating and temperature dependency: To calculate the temperature dependency of the viscosity, all analytical approaches based on Hagen-Poiseuille equations average the temperature in the capillary perpendicular to the flow direction and assign this temperature to a viscosity value and shear rate obtained at the wall of the capillary. By using the approach of a CFD simulation of the high pressure capillary rheometer, the whole flow field in the capillary (velocity distribution perpendicular to the flow direction) is calculated and the effect of shear heating can be simulated along the length of the capillary and especially perpendicular to the flow direction in the regions of high shear rates close to the wall of the capillary. By doing so, the temperature distribution in the capillary can be explicitly calculated and assigned to shear rates and viscosity values calculated in every point of the simulation model.

- Pressure dependency of viscosity: In general, the pressure dependency of viscosity is also a function of temperature and shear rate in common viscosity models (e.g., Cross-WLF equation). Due to neglecting the flow field and temperature distribution in the high pressure capillary rheometer by analytical approaches based on Hagen-Poiseuille equations, the pressure dependency of viscosity is averaged over the flow field and temperature distribution in the capillary.

- In- and Outflow effects: By modelling a high pressure capillary rheometer in CFD software, in- and outflow effects can be described by modelling the respective geometric areas within the simulation model.

- Influence of the measurement device and environmental conditions: By modelling the entire high pressure capillary rheometer in a CFD simulation model, the measurement device can be simulated, such that also the influence of the measurement equipment and its environment can be included into the simulation.

By modelling and solving all of these effects by coupled differential equations, all interactions of polymer specific effects can be captured in the CFD simulation and the real behavior of the polymer can be described within the CFD simulation. As CFD simulations require high computational effort, the analytical pre-fitting as part of step A) helps to find a good starting point for the optimization of the complex CFD simulation model such that the range of potential coefficient variations in the viscosity model can be limited and fewer iterations are needed for the CFD optimization. In a preferred embodiment, in the CFD calculation of step B) any dependency of the viscosity on shear rate, temperature, pressure, strain rate and/or extension rate is described using the Cross-WLF equation. Without wishing to be bound to this embodiment, the invention may be described in the following using the Cross-WLF equation as the viscosity model in the Navier- Stokes equations. The at least one empirical coefficient used as an input for the CFD calculation may be selected from n, A₁, A₃, D₁, D₂, D₃

, or a combination of at least two of the aforementioned coefficients.

In a preferred embodiment, in the CFD calculation of step B) any dependency of the viscosity on an extension rate is described using an extended Cross-WLF equation.

The Cross-WLF equation as used in the present disclosure may be the standard Cross-WLF equation or an extended Cross-WLF equation. For example, if the viscosity dependency on the extension rate is considered, the Cross-WLF equation may be extended by a juncture-loss model or an extension viscosity model. Including the extension rate in the viscosity model allows to calculate the pressure drop at the inlet and the outlet of the capillary rheometer.

The influence of the extension rate on viscosity can be described by the following extension of the Cross-WLF equation:

With being the shear viscosity expressed by the standard Cross-WLF equation,

being the extension rate and A, B being material specific fitting parameters.

In another embodiment, the additional junctures loss at the inlet of the capillary rheometer can be taken into account indirectly by correlating the juncture loss to the wall shear stress in the capillary:

With Δp being the total pressure loss in the capillary rheometer, Δp_c being the pressure loss in the capillary, Δp_e the juncture loss at the inlet of the capillary, being the shear stress in

the capillary, L being the capillary length, D being the capillary diameter and C₁, C₂ being material specific fitting parameters. According to step C) the difference between the experimentally determined rheological parameter and the CFD-calculated rheological parameter is determined. For example, one could determine the difference in pressure drop as obtained in the capillary rheometer and as calculated using the Navier-Stokes equations having the Cross-WLF equation as an expression for the viscosity as function of the shear rate and the temperature.

In step D) it is decided whether the difference is acceptable or not. Then it may be inferred that the at least one empirical coefficient in the GNF model is within acceptable boundaries. Then the method according to the invention may terminate.

If, on the other hand, the absolute value of the difference in step C) exceeds a pre-determined threshold, the at least one empirical coefficient used as an input for the calculation in step B) is modified and the sequence B) to D) is run again with the new coefficient(s). If there are several empirical coefficients, it is possible to modify one, a plurality or all of them.

In a preferred embodiment, after completion of step D) the at least one empirical coefficient is communicated to a user and/or a computer program. The computer program may be a CAE program such as Autodesk Moldflow. The communication may take place on the same computer, over a local area network (LAN), a wide area network (WAN) or a distributed computing environment (“cloud”). Communication protocols include FTP, HTTP, HTTPS, e- mail, direct messaging, paper mail, verbal communication, etc. The communication may be effected automatically or via a person-to-person interaction.

In a preferred embodiment, after completion of step D) an injection molding simulation is performed using the at least one empirical coefficient. Using the one or more optimized empirical coefficients obtained in the method according to the invention will increase the accuracy of the injection molding simulation performed in software. The capillary rheometer used in step A1) and subsequently modeled in step B) may have a rotational symmetry. It is also possible to use other cross-sectional geometries for the capillary such as a square or a rectangular geometry. In a preferred embodiment, the capillary rheometer of step A1) has a cross-section with a ratio of length to width of 10:1 or greater. Then the capillary rheometer may be classified as a slit. For example, the ratio of length to width may be 15:1 or greater. In cases of slit geometries in the capillary rheometer, the analytically solvable Hagen-Poiseuille relationship may be stated as:

with B (width) and H (height) being the dimensions of the slit (B > H) and K being a constant depending on the aspect ratio of the slit.

In a preferred embodiment, in the calculation of step A2) and/or step B) a correction factor is included to reduce a pressure dependency of the polymer melt viscosity if the shear rate is below a pre-determined threshold. This can overcome the restriction that high pressure capillary rheometers can capture the pressure dependency of viscosity only at high flow or shear rates. This characteristic of high pressure capillary rheometers is especially seen in the fitting of viscosity models using a WLF equation for pressure dependency (like the Cross- WLF equation and the Carreau-WLF equation) where the pressure dependency of viscosity itself is an exponential function of the pressure and pressure dependency of viscosity increases with lower shear rates.

Therefore, an additional restriction to the viscosity fitting may be added to limit pressure dependency of viscosity at a shear rate of zero for high pressures (e.g., 200 MPa as realistic maximum pressure for applications such as injection molding) for the analytical pre-fitting and/or the reverse engineering approach based on a CFD simulation. To define a limit for the pressure dependency of the viscosity at a shear rate of zero for high pressures, special measurement equipment and molecular dynamics simulation can be used to define an appropriate limit for the pressure dependency (e.g., the ratio of viscosities or derivative of the viscosity with respect to the pressure) with low additional effort for further testing or evaluation. It is possible to introduce a penalty term to for exceeding a preset limit during the optimization. For example, at a temperature of 300 °C, a shear rate of 0/s and a pressure of 200 MPa a factor may be defined for the viscosity increase relative to a pressure of 0 MPa, 300 °C and 0/s. This factor may be, for instance, 150. The limiting condition in this example would be: η (300 °C, 0/s, 200 MPa) < 150 η (300 °C, 0/s, 0 MPa). A penalty factor S can be postulated with which the error function in the optimization is multiplied by. Here, S could formulated as:

This allows to minimize the deviation of the measured values from the simulated values. The optimizer attempts to conform to the artificial boundary for the viscosity increase.

The invention also encompasses a data processing apparatus system comprising means for carrying out the method according to the invention, a computer program product comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method according to the invention and a computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to carry out the method according to the invention.

The present invention will be further described with reference to the following figures without wishing to be limited by them.

FIG. 1 shows a flow chart describing a method according to the invention.

FIG. 2 shows a flow chart describing the analytical pre-fitting of a viscosity coefficient.

FIG. 3a shows a sequence of operations for calculating an inlet pressure.

FIG. 3b shows a nested loop structure using the sequence of FIG. 3a.

FIG. 4 shows a system according to the invention.

FIG. 5 shows testing data obtained by a method according to the invention.

FIG. 1 shows a flow chart describing a method according to the invention. This corresponds to steps A) to D) of the method. In step 100 the analytically pre-optimized empirical coefficients of the Cross-WLF equation are received. How these coefficient have been pre optimized is the subject of the flow chart in FIG. 2. According to step 200 a computer model of a capillary rheometer is provided. The order of steps 100 and 200 may be reversed without negatively impacting the method according to the invention. Likewise, steps 100 and 200 may be executed concurrently. The computer model of the capillary rheometer mirrors the capillary rheometer used for the testing of the polymer melt according to step A1) of the method. In particular, the lengths and the internal diameters are identical. In step 300 a computational fluid dynamics (CFD) model is provided which includes the Navier-Stokes equations and the Cross-WLF equation as a model for the dependency of the polymer melt’s viscosity on the shear rate, temperature, pressure and optionally also the extension rate. The CFD model also incorporates the geometry of the capillary rheometer as provided in step 200. According to step 400 the CFD model is used to calculate the pressure drop which would occur in the computer model using the temperature and volume rate data of the real-world capillary rheometer test. Step 500 compares the measured (real-world) and calculated pressure drop. If the difference is below a pre-determined threshold, the method may end as the empirical coefficient(s) of the Cross-WLF equation are satisfactory. Otherwise, the coefficient(s) is/are modified in step 600 and the CFD calculation of step 400 is re-run.

FIG. 2 shows a flow chart describing the analytical pre -fitting of a viscosity coefficient. This corresponds to the steps A1) to A4) of the method according to the invention. Step 700 reflects the measurement of the pressure drop for a polymer melt in an capillary rheometer. The temperature of the polymer melt at the inlet of the capillary, the measured pressure drop over the capillary, the pressure at the outlet of the capillary, the stamp speed (volume rate), and the geometric dimensions of the capillary rheometer, such as the length and the inner diameter of the actual capillary, may be noted. Step 800 involves the calculation of the pressure drop for the same polymer that has been tested in the capillary rheometer while using the same geometry as in the rheometer, in particular the length and inner diameter of the capillary, by the Hagen-Poiseuille equation and the generalized Newtonian fluid (GNF) model. Furthermore, the temperature of the polymer melt at the inlet of the capillary in the rheometry testing is used in step 800. The equation used in this step to model the dependency of the viscosity on the shear rate, temperature, pressure and optionally also the extension rate is the Cross-WLF equation. Step 900 compares the measured (step 700) and calculated (step 800) pressure drops. If the difference is below a pre-determined threshold, the empirical coefficients of the Cross-WLF equation are ready to be transmitted to the CFD calculation in step B) of the method according to the invention. Otherwise, the coefficients are modified in step 1000 and the calculation of step 800 is re-run.

FIGs. 3a and 3b show operations used in a preferred pre-fitting of a coefficient. The operations refer to the preferred embodiment where in step A2) a virtual capillary rheometer of length L, having an inlet and an outlet, is divided into a plurality of discrete sections of the same length dL, pressure loss and shear heating are calculated in each section separately and the pressure loss and shear heating calculations are repeated until the calculated total pressure drop over the entire capillary rheometer converges.

This can be realized by a structure where an inner loop is nested inside an outer loop. The inner loop can be used for calculating the shear heating and pressure loss, whereas in the outer loop the coefficients of the Cross-WLF equation are modified until a satisfactory result has been achieved.

FIG. 3a shows procedures P_calc and T_calc which may be run in the inner loop. T_calc concerns the virtual capillary rheometer when the internal temperature is considered. The capillary has a total length L and is divided into sections all having the same length dL. At the outlet on the right-hand side of the drawing the temperature is designated as T₀. At the inlet on the left-hand side the temperature is designated T_L. This is deemed to be a known quantity. In each segment between T₀ and T_L there are internal temperatures T₁, T₂, etc. The designations T₁, T₂ are counted starting from T₀. Equivalently, one may use the designations T_L-1, T_L-2, etc. which counts backwards starting from T_L.

P_calc concerns the same virtual capillary rheometer when the internal pressure is considered. The capillary has a total length L and is divided into sections all having the same length dL. At the outlet on the right-hand side of the drawing the pressure is designated as p₀ which can be the ambient pressure (1013 mbar) or a counter-pressure applied via the rheometer. In any event, the pressure po is deemed to be a known quantity. At the inlet on the left-hand side the pressure is designated p_L. In each segment between p₀ and p_L there are internal pressures p₁, p₂, etc. The designations p₁, p₂ are counted starting from p₀. Equivalently, one may use the designations p_L-1, p_L-2, etc. which counts backwards starting from p_L.

The temperatures T_L-1, T_L-2, ... T₀ and the pressures p_L, p_L-1, p_L-2, ... p₁ are deemed to be quantities that need to be calculated.

FIG. 3b shows a nested loop structure with an inner loop comprising P_calc and T_calc and an outer loop. The sequence may begin with an initial guess of coeff to send to the inner loop. This initial guess can use literature values, for example.

In the inner loop, in an initial pressure calculation run P_calc the pressure drop in the discrete segments over the length of the capillary is calculated starting at the segment next to the outlet where the pressure p₀ is known. The pressure p₁ is expressed as the sum of the pressure p₀ and a Hagen-Poiseuille term having the parameters of the length dL, the internal diameter of the capillary D, the shear rate

and the viscosity η which in itself is a function of the temperature T₀, the shear rate

, the pressure po and at least one empirical coefficient designated as coeff. Preferably, the viscosity model is the Cross-WLF equation and, also preferably, coeff represents all empirical coefficients of the Cross-WLF equation.

As T₀ is not measured, an initial guess is undertaken in this initial pressure calculation run. An example for a reasonable initial guess is that the outlet temperature equals the inlet temperature, in other words T₀ = T_L.

The pressure calculation then propagates one segment further towards the inlet of the capillary. The pressure p₂ is expressed as the sum of the pressure of the previous segment p₁ and a Hagen - Poiseuille term having the parameters of the length dL, the internal diameter of the capillary D, the shear rate

and the viscosity η which is described using the same viscosity model as in the previous step (e.g., the Cross-WLF equation) and which in itself is a function of the temperature of the previous segment (here, T₁), the shear rate , the pressure p₁ and the at least

one empirical coefficient coeff. As T₁ is not measured, an initial T₁ = T_L guess is undertaken in this first calculation run. This is repeated until the inlet of the capillary is reached where p_L is calculated.

The calculated p_L of the initial pressure calculation run is noted and serves as a starting point for determining the convergence of the calculation.

In an initial temperature calculation run the temperature change in the discrete segments over the length of the capillary is calculated starting at the segment next to the inlet where the temperature T_L is known. The temperature T_L-1 is expressed as the sum of the inlet temperature T_L and the quotient of the pressure difference between the calculated inlet pressure p_L and the previously calculated pressure in the current segment p_L-1 divided by the product of the polymer melt density p and the polymer’s heat capacity at constant pressure c_p. This calculation reflects the adiabatic shear heating of the polymer in the capillary.

The temperature calculation then propagates one segment further towards the outlet of the capillary. The temperature T_L-2 is expressed as the sum of the temperature of the preceding segment T_L-1 and the quotient of the difference between the pressure of the preceding segment p_L-1 and the pressure of the current segment p_L-2 (both pressures were previously calculated in the initial pressure calculation) divided by the product of the polymer melt density p and the polymer’s heat capacity at constant pressure c_p.

This propagation using temperature and pressure values from the preceding segment is repeated until the outlet is reached. The sequence of running P_calc and T_calc with mutual updating of the calculated pressure and temperature values for each segment dL is repeated until the calculated pressure drop p_L - p₀ converges, i.e. the difference to the calculated pressure drop of a previous cycle of P_calc and T_calc is below a pre-determined threshold.

As the individual segments dL become smaller and smaller, it may become convenient to use a temperature value from a neighboring segment in the pressure calculation or vice versa. The calculation error will likewise become smaller and smaller. For example in the calculation of p₁ the value for T₀ or for T₁ may be used.

After convergence the final value for p_L is outputted to the outer loop. If the calculated pressure drop p_L- p₀ is not sufficiently close to the measured pressure drop from the corresponding capillary rheometer measurement, coeff is/are modified and sent to the inner loop for updating of the viscosity model. Then the inner loop is run again until pressure drop convergence and the comparison of the outer loop is repeated.

These cycles continue until the difference between calculated pressure drop and measured pressured drop is sufficiently small. This is taken as an indication that the analytical pre-fitting of coeff is satisfactory.

FIG. 4 illustrates a data processing apparatus system 1000 configured for determining at least one empirical coefficient used in the calculation of the viscosity of a thermoplastic polymer melt. In some implementations, system 1000 may include one or more servers 1010. Server(s) 1010 may be configured to communicate with one or more client computing platforms 1090 according to a client/server architecture and/or other architectures. Client computing platform(s) 1090 may be configured to communicate with other client computing platforms via server(s) 1010 and/or according to a peer-to-peer architecture and/or other architectures. Users may access system 1000 via client computing platform(s) 1090.

Server(s) 1010 may be configured by machine-readable instructions 1040. Machine-readable instructions 1040 may include one or more instruction modules. The instruction modules may include computer program modules. The instruction modules may include one or more of a coefficient receiving module 1050, a computational fluid dynamics calculation module 1060, a difference calculation module 1070 and a coefficient modification module 1080 and/or other instruction modules.

The coefficient receiving module 1050 may be configured to execute step A) of the method according to the invention. The CFD calculation module 1060 may be configured to execute step B) of the method according to the invention. The difference calculating module 1070 may be configured to execute step C) of the method according to the invention. The coefficient modification module 1080 may be configured to execute step D) of the method according to the invention.

In some implementations, server(s) 1010, client computing platform(s) 1090, and/or external resources 1100 may be operatively linked via one or more electronic communication links. For example, such electronic communication links may be established, at least in part, via a network such as the Internet and/or other networks. It will be appreciated that this is not intended to be limiting, and that the scope of this disclosure includes implementations in which server(s) 1010, client computing platform(s) 1090, and/or external resources 1100 may be operatively linked via some other communication media.

A given client computing platform 1090 may include one or more processors configured to execute computer program modules. The computer program modules may be configured to enable an expert or user associated with the given client computing platform 1090 to interface with system 1000 and/or external resources 1100, and/or provide other functionality attributed herein to client computing platform(s) 1090. By way of non-limiting example, the given client computing platform 1090 may include one or more of a desktop computer, a laptop computer, a handheld computer, a tablet computing platform, a NetBook, a Smartphone and/or other computing platforms.

External resources 1100 may include sources of information outside of system 1000, external entities participating with system 1000, and/or other resources. In some implementations, some or ah of the functionality attributed herein to external resources 1100 may be provided by resources included in system 1000.

Server(s) 1010 may include electronic storage 1020, one or more processors 1030, and/or other components. Server(s) 1010 may include communication lines, or ports to enable the exchange of information with a network and/or other computing platforms. Illustration of server(s) 1010 in FIG. 4 is not intended to be limiting. Server(s) 1010 may include a plurality of hardware, software, and/or firmware components operating together to provide the functionality attributed herein to server(s) 1010. For example, server(s) 1010 may be implemented by a cloud of computing platforms operating together as server(s) 1010.

Electronic storage 1020 may comprise non-transitory storage media that electronically stores information. The electronic storage media of electronic storage 1020 may include one or both of system storage that is provided integrally (i.e., substantially non-removable) with server(s) 1010 and/or removable storage that is removably connectable to server (s) 1010 via, for example, a port (e.g., a USB port, a firewire port, etc.) or a drive (e.g., a disk drive, etc.). Electronic storage 1020 may include one or more of optically readable storage media (e.g., optical disks, etc.), magnetically readable storage media (e.g., magnetic tape, magnetic hard drive, floppy drive, etc.), electrical charge-based storage media (e.g., EEPROM, RAM, etc.), solid-state storage media (e.g., flash drive, etc.), and/or other electronically readable storage media. Electronic storage 1020 may include one or more virtual storage resources (e.g., cloud storage, a virtual private network, and/or other virtual storage resources). Electronic storage 1020 may store software algorithms, information determined by processor(s) 1030, information received from server(s) 1010, information received from client computing platform(s) 1090, and/or other information that enables server(s) 1010 to function as described herein.

Processor(s) 1030 may be configured to provide information processing capabilities in server(s) 1010. As such, processor(s) 1030 may include one or more of a digital processor, an analog processor, a digital circuit designed to process information, an analog circuit designed to process information, a state machine, and/or other mechanisms for electronically processing information. A1though processor(s) 1030 is shown in FIG. 4 as a single entity, this is for illustrative purposes only. In some implementations, processor(s) 1030 may include a plurality of processing units. These processing units may be physically located within the same device, or processor(s) 1030 may represent processing functionality of a plurality of devices operating in coordination. Processor(s) 1030 may be configured to execute modules 1040, 1050, 1060, 1070 and/or 1080, and/or other modules by software; hardware; firmware; some combination of software, hardware, and/or firmware; and/or other mechanisms for configuring processing capabilities on processor(s) 1030. As used herein, the term “module” may refer to any component or set of components that perform the functionality attributed to the module. This may include one or more physical processors during execution of processor readable instructions, the processor readable instructions, circuitry, hardware, storage media, or any other components.

It should be appreciated that although modules 1040, 1050, 1060, 1070 and/or 1080 are illustrated in FIG.4 as being implemented within a single processing unit, in implementations in which processor(s) 1030 includes multiple processing units, one or more of modules 1040, 1050, 1060, 1070 and/or 1080 may be implemented remotely from the other modules. The description of the functionality provided by the different modules 1040, 1050, 1060, 1070 and/or 1080 described below is for illustrative purposes, and is not intended to be limiting, as any of modules 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, and/or 130 may provide more or less functionality than is described. For example, one or more of modules 1040, 1050, 1060, 1070 and/or 1080 may be eliminated, and some or all of its functionality may be provided by other ones of modules 1040, 1050, 1060, 1070 and/or 1080. As another example, processor(s) 1030 may be configured to execute one or more additional modules that may perform some or all of the functionality attributed below to one of modules 1040, 1050, 1060, 1070 and/or 1080.

FIG. 5 shows simulation and testing data for a polycarbonate of the type Makrolon LED2245. According to its publicly available data sheet this material has a melt volume-flow rate MVR (300 °C, 1.2 kg, ISO 1133) of 34 cm³/10 min and a glass transition temperature (10 °C/min, ISO 11357-1/-2) of 145 °C. The curves of FIG. 5 display the change in injection pressure p over time t.

The measured curve M was recorded using an injection molding tool with a pressure sensor. The inlet temperature of the polymer melt was 300 °C. The curve S, as a comparison example, reflects an injection molding simulation using the Cross-WLF equation as the viscosity model. The empirical coefficients in the Cross-WLF equation were:

The curve O reflects a mold flow simulation akin to the simulation for curve S but with empirical coefficients in the Cross-WLF equation optimized by a method according to the invention. The empirical coefficients in the Cross-WLF equation were:

It can be seen that the standard model S results in a far too high pressure after 2 seconds, whereas the method according to the invention results in a pressure much closer to the experimentally determined value.

In summary, by using measurement data of a high pressure capillary rheometer for the reverse engineering approach a wide range of temperatures, shear rates, and pressures can be covered by the optimization with high resolutions of the simple geometry of a high pressure capillary rheometer within reasonable simulation times. This is a big difference to more common approaches of the reverse engineering of production processes like injection molding which cover only small ranges of shear rates and temperatures and often use complex geometries which need long simulation times with only limited resolution of polymer specific effects. It is concluded that the reverse engineering of the measurement device (high pressure capillary rheometer) promises higher accuracy for a wider range of processes within smaller simulation times (especially when using the advanced pre-fitting methods and realistic limits for pressure dependency of viscosity at low shear rates as described above).

Claims

Patent Claims:

1. A computer-implemented method for determining at least one empirical coefficient used in the modelling of the viscosity of a thermoplastic polymer melt comprising:

2. The method according to claim 1, wherein the method for determining the at least one empirical coefficient as recited in step A) which comprises steps A1) to A4) is executed and the at least one empirical coefficient that has or have been determined is or are used as input for the CFD calculation in step B).

3. The method according to claim 1 or 2, wherein in step A1) the at least one rheological property includes the pressure drop over the entire capillary rheometer at a specified volume rate, counter-pressure and temperature of the polymer melt.

4. The method according to one of claims 1 to 3, wherein in step A2) the GNF model incorporates, as a physical equation, the Hagen-Poiseuille equation:

with

where d_p is the pressure difference, dL is the length difference,

V is the volume rate,

g is the shear rate, η is the melt viscosity, D is the diameter of the capillary and where η can be a function of parameters including the shear rate, the temperature, the pressure, the extension rate and at least one empirical coefficient.

5. The method according to one of claims 1 to 4, wherein in the GNF model in step A2) any dependency of a viscosity of the polymer melt on shear rate, extension rate and/or temperature is described using the Cross-WLF equation:

where η is the melt viscosity, η₀ is the zero shear viscosity,

is the shear rate,

with T being the temperature, T_* being the glass transition temperature, A₂ = A₃ + D₃ p, p being the pressure, D₁, A₁, A₃ and D₃ being empirical coefficients and T_* = D₂ + D₃ p with D₂ being an empirical coefficient and the at least one empirical coefficient received in step A) is selected from n, A₁, A₃, D₁, D₂, D₃, or a combination of at least two of the aforementioned coefficients.

6. The method according to one of claims 1 to 5, wherein in step A2) a virtual capillary rheometer of length L, having an inlet and an outlet, is divided into a plurality of discrete sections of the same length dL, pressure loss and shear heating are calculated in each section separately and the pressure loss and shear heating calculations are repeated until the calculated pressure drop over the entire capillary rheometer converges.

7. The method according to one of claims 1 to 6, wherein in the CFD calculation of step B) any dependency of the viscosity on shear rate, temperature, pressure, strain rate and/or extension rate is described using the Cross-WLF equation.

8. The method according to one of claims 1 to 7, wherein in the CFD calculation of step B) any dependency of the viscosity on an extension rate is described using an extended Cross- WLF equation.

9. The method according to one of claims 1 to 8, wherein after completion of step D) the at least one empirical coefficient is communicated to a user and/or a computer program.

10. The method according to one of claims 1 to 9, wherein after completion of step D) an injection molding simulation is performed using the at least one empirical coefficient.

11. The method according to one of claims 1 to 10, wherein the capillary rheometer of step

A1) has a cross-section with a ratio of length to width of 10:1 or greater.

12. The method according to one of claims 1 to 11, wherein in the calculation of step A2) and/or step B) a correction factor is included to reduce a pressure dependency of the polymer melt viscosity if the shear rate is below a pre-determined threshold.

13. A data processing apparatus system comprising means for carrying out the method of one of claims 1 to 12.

14. A computer program product comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method of one of claims 1 to 12.

15. A computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to carry out the method of one of claims 1 to 12.