Patent Publication Number: US-11378505-B1

Title: Method of measuring extensional viscosity of polymer melts and capillary injection system

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of prior-filed provisional application with application No. 63/253,743, filed Oct. 8, 2021, which is incorporated by reference in its entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to a method of measuring rheological characteristics and a capillary injection system, and more particularly, to a method and capillary injection system of measuring an extensional viscosity. 
     DISCUSSION OF THE BACKGROUND 
     A viscosity is an important characteristic in rheology. The viscosity includes a shear viscosity and an extensional viscosity. Conventionally, it is difficult to obtain an accurate measurement of the extensional viscosity. However, an inaccurate measurement of the extensional viscosity would affect a quality of a rheological analysis. Therefore, it is critical to obtain the accurate extensional viscosity measurement. 
     This Discussion of the Background section is provided for background information only. The statements in this Discussion of the Background are not an admission that the subject matter disclosed in this section constitutes prior art to the present disclosure, and no part of this Discussion of the Background section may be used as an admission that any part of this application, including this Discussion of the Background section, constitutes prior art to the present disclosure. 
     SUMMARY 
     One aspect of the present disclosure provides a method of measuring an extensional viscosity of a polymer melt. The method includes operations of: based on a weighted generalized Newtonian fluid (GNF) viscosity model, obtaining a viscosity profile of the polymer melt according to a transport equation, a Navier Stokes equation, and a Trouton function; measuring a pressure drop of the polymer melt; obtaining a general viscosity of the polymer melt from the viscosity profile according to the pressure drop, wherein the general viscosity comprises a shear viscosity of the polymer melt and the extensional viscosity of the polymer melt; and extracting the extensional viscosity from the general viscosity. 
     In some embodiments, the polymer melt flows through a capillary, and a difference between a pressure at an outlet of the capillary and a pressure at an inlet of the capillary is the pressure drop. 
     In some embodiments, based on the weighted GNF viscosity model, obtaining the viscosity profile of the polymer melt according to the transport equation, the Navier Stokes equation, and the Trouton function includes operations of: determining a plurality of conditional parameters of the transport equation and the Navier Stokes equation; determining a plurality of computational parameters of the Trouton function; obtaining a Trouton ratio of the Trouton function according to the plurality of computational parameters; based on the weighted GNF viscosity model, obtaining an estimated pressure drop of the polymer melt according to the Trouton ratio, the transport equation, and the Navier Stokes equation; determining whether an error of the estimated pressure drop is within a threshold; and when the error of the estimated pressure drop is within the threshold, obtaining the viscosity profile according to the plurality of conditional parameters and the plurality of computational parameters. 
     In some embodiments, when the error of the estimated pressure drop is not within the threshold, updating the plurality of computational parameters of the Trouton ratio function. 
     In some embodiments, when the error is a positive value, an initial temperature of the plurality of conditional parameters is increased 1 degree so as to update the predetermined temperature. 
     In some embodiments, when the error is a negative value, an initial temperature of the plurality of conditional parameters is decreased 1 degree so as to update the predetermined temperature. 
     In some embodiments, the plurality of conditional parameters comprises a predetermined velocity vector of the polymer melt, a predetermined temperature of the polymer melt, and a predetermined pressure drop of the polymer melt. 
     In some embodiments, based on the GNF viscosity model, obtaining the estimated pressure drop of the polymer melt according to the Trouton ratio, the transport equation, and the Navier Stokes equation includes operations of: obtaining an estimated velocity vector of the polymer melt according to the transport equation; obtaining an estimated strain rate of the polymer melt according to the weighted GNF viscosity model; obtaining an estimated temperature of the polymer melt and an estimated pressure of the polymer melt according to the Navier Stokes equation, the estimated velocity vector, and the estimated strain rate; and obtaining the estimated pressure drop according to the estimated pressure. The estimated velocity vector, the estimated temperature, and the estimated pressure drop converge to the predetermined velocity vector, the predetermined temperature, and the predetermined pressure drop, respectively. 
     In some embodiments, the error of the estimated pressure drop is equal to a difference between the estimated pressure drop and the predetermined pressure drop divided by the predetermined pressure drop. 
     In some embodiments, the threshold is about ±10%. 
     In some embodiments, the Tronton function is represented using an expression: 
                 T   r     ⁡     (     γ   .     )       =       η   E       η   S                       T   r     ⁡     (     γ   .     )       =     3   +       T   0         [     1   +       (       λ   T     ⁢     γ   .       )       -   2         ]       n   T                 
wherein T r ({dot over (γ)}) represents a Trouton ration with respect to a shear rate {dot over (γ)}, η E  represents the extensional viscosity, η S  represents a shear viscosity, T 0  represents an initial temperature, λ T  represents a relaxation time, and n T  represents a power index.
 
     Another aspect of the present disclosure provides a capillary injection system for measuring an extensional viscosity of a polymer melt, including a barrel, a capillary, a piston, a first pressure transducer, a second pressure transducer, and a rheometer. The barrel has a cavity configured to contain a polymer melt. The capillary is coupled to the barrel. The piston is coupled to the barrel, and configured to provide a force to the polymer melt to make the polymer melt flow through the capillary. The first pressure transducer is configured to measure a first pressure of the polymer melt at an inlet of the capillary. The second pressure transducer is configured to measure a second pressure of the polymer melt at an outlet of the capillary. The rheometer is configured to perform operations of: based on a weighted GNF viscosity model, obtain a viscosity profile of the polymer melt according to a transport equation, a Navier Stokes equation, and a Trouton function; obtain a general viscosity of the polymer melt from the viscosity profile according to the first pressure and the second pressure, wherein the general viscosity comprises a shear viscosity of the polymer melt and the extensional viscosity of the polymer melt; and extract the extensional viscosity from the general viscosity. 
     The foregoing has outlined rather broadly the features and technical advantages of the present disclosure in order that the detailed description of the disclosure that follows may be better understood. Additional features and advantages of the disclosure will be described hereinafter, and form the subject of the claims of the disclosure. It should be appreciated by those skilled in the art that the conception and specific embodiment disclosed may be readily utilized as a basis for modifying or designing other structures or processes for carrying out the same purposes of the present disclosure. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the spirit and scope of the disclosure as set forth in the appended claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete understanding of the present disclosure may be derived by referring to the detailed description and claims when considered in connection with the Figures, where like reference numbers refer to similar elements throughout the Figures. 
         FIG. 1  is a schematic diagram of a capillary injection system according to some embodiments of the present disclosure. 
         FIG. 2  is a flowchart of a method of measuring an extensional viscosity of a polymer melt according to some embodiments of the present disclosure. 
         FIG. 3  is a detailed flowchart of the method shown in  FIG. 2  according to some embodiments of the present disclosure. 
         FIG. 4  is a detailed flowchart of the method shown in  FIG. 2  according to some embodiments of the present disclosure. 
         FIG. 5  is a schematic diagram of a Trouton ratio against an extensional rate according to some embodiments of the present disclosure. 
         FIG. 6  is a schematic diagram of a uniaxial extensional viscosity and an experimental data against an extensional rate according to some embodiments of the present disclosure. 
         FIG. 7  is a schematic diagram of a pressure drop and an experimental data against a shear rate according to some embodiments of the present disclosure. 
         FIG. 8  is a functional block diagram of the rheometer shown in  FIG. 1  according to some embodiments of the present disclosure 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments, or examples, of the disclosure illustrated in the drawings are now described using specific language. It shall be understood that no limitation of the scope of the disclosure is hereby intended. Any alteration or modification of the described embodiments, and any further applications of principles described in this document, are to be considered as normally occurring to one of ordinary skill in the art to which the disclosure relates. Reference numerals may be repeated throughout the embodiments, but this does not necessarily mean that feature(s) of one embodiment apply to another embodiment, even if they share the same reference numeral. 
     It shall be understood that, although the terms first, second, third, etc. may be used herein to describe various elements, components, regions, layers or sections, these elements, components, regions, layers or sections are not limited by these terms. Rather, these terms are merely used to distinguish one element, component, region, layer or section from another element, component, region, layer or section. Thus, a first element, component, region, layer or section discussed below could be termed a second element, component, region, layer or section without departing from the teachings of the present inventive concept. 
     The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limiting to the present inventive concept. As used herein, the singular forms “a,” “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It shall be further understood that the terms “comprises” and “comprising,” when used in this specification, point out the presence of stated features, integers, steps, operations, elements, or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, or groups thereof. 
       FIG. 1  is a schematic diagram of a capillary injection system  10  according to some embodiments of the present disclosure. The capillary injection system  10  includes a rheometer  100 , a barrel  110 , a piston  120 , a capillary  130 , a pressure transducer  140 , and a pressure transducer  150 . 
     The capillary injection system  10  is configured to inject a polymer melt PM into a target object through the capillary  130 , so as to measure an extensional viscosity of the polymer melt PM. The rheometer  100  is configured to measure behaviors (such as rheological properties) of the polymer melt PM during the injection. 
     The barrel  110  has a cavity configured to contain the polymer melt PM. The capillary  130  is coupled to the barrel  110 , and configured to receive the polymer melt PM from an inlet  131  of the capillary  130  and dispense the polymer melt PM from an outlet  132  of the capillary  130 . The piston  120  is configured to provide a force F to the polymer melt PM in the cavity. When the piston  120  is controlled to push the polymer melt PM in the barrel  110 , the polymer melt PM is pushed toward the inlet  131  of the capillary  130 . The polymer melt PM has a volumetric flow rate Q at the outlet  132  of the capillary  130  according to the force F. In some embodiments, the force F of the piston  120  is controlled by the rheometer  100 . 
     The pressure transducers  140  and  150  are configured to measure a pressure P 1  and a pressure P 2  of the polymer melt PM at the inlet  131  and the outlet  132  of the capillary  130 , respectively. According to the pressure P 1  and the pressure P 2 , a pressure drop ΔP between two ends of the capillary  130  can be obtained. In some embodiments, some of the rheological properties are obtained according to the pressure drop ΔP. 
     A viscosity of the polymer melt PM is critical for analyzing the behaviors of the polymer melt PM. Therefore, one function of the rheometer  100  is obtaining the viscosity of the polymer melt PM. In general, the viscosity includes a shear viscosity and an extensional viscosity. In some conventional approaches, the shear viscosity can be obtained by the rheometer  100 ; however, the extensional viscosity has seldom been studied. Therefore, the extensional viscosity cannot be obtained accurately, and an inaccurate extensional viscosity affects a performance of the rheometer  100 . Compared to the conventional approaches, the present disclosure provides a method  20  (shown in  FIG. 2 ) to obtain the accurate extensional viscosity to improve a quality of analysis. In some embodiments, the rheometer  100  is configured to perform the method  20  to obtain the extensional viscosity of the polymer melt PM. 
     Reference is made to  FIG. 2 .  FIG. 2  is a flowchart of the method  20  according to some embodiments of the present disclosure. The method  20  is configured to measure the extensional viscosity of the polymer melt PM in a capillary, such as the capillary  130  shown in  FIG. 1 . More particularly, the method  20  is configured to measure the extensional viscosity of the polymer melt PM by measuring the pressure drop ΔP of the polymer melt PM. To facilitate understanding, the method  20  is described with same numerals as those shown in  FIG. 1 . The method  20  includes operations S 22 , S 24 , S 26 , and S 28 . 
     In operation S 22 , based on a weighted generalized Newtonian fluid (GNF) viscosity model (which will be described below), a viscosity profile η({dot over (γ)}) of the polymer melt PM is obtained according to a transport equation, a Navier Stokes equation, and a Trouton function. {dot over (γ)} is a strain rate. In some embodiments, the viscosity profile η({dot over (γ)}) is obtained by the rheometer  100 . 
     In operation S 24 , the pressure drop ΔP of the polymer melt PM is measured. In some embodiments, the pressure drop ΔP is measured by the pressure transducers  140  and  150 . 
     In operation S 26 , a general viscosity η of the polymer melt PM is obtained from the viscosity profile η({dot over (γ)}) according to the pressure drop ΔP. In some embodiments, the general viscosity η is also known as a fluid viscosity. In general, the general viscosity η includes the shear viscosity η S  and the extensional viscosity η E  of the polymer melt PM. 
     In operation S 28 , the extensional viscosity η E  is extracted from the general viscosity η. 
     The operation S 22  is configured to obtain the viscosity profile η({dot over (γ)}), and further to obtain a relationship between the viscosity profile η({dot over (γ)}) and the pressure drop ΔP. In other words, the viscosity profile η({dot over (γ)}) obtained by the operation S 22  can be rewritten to be the viscosity profile η(ΔP). After operation S 22 , the pressure drop ΔP is measured (operation S 24 ), and the general viscosity η can be obtained (operation S 26 ) from the viscosity profile η(ΔP) using the measured pressure drop ΔP. When the general viscosity T is obtained, the extensional viscosity η E  can be extracted (operation S 28 ). 
     Reference is made to  FIG. 3 .  FIG. 3  is a detailed flowchart of operation S 22  of the method  20  according to some embodiments of the present disclosure. Obtaining the viscosity profile η({dot over (γ)}) (operation S 22 ) includes operations S 222 , S 224 , S 226 , S 228 , S 230  and S 232 . 
     In operation S 222 , a plurality of conditional parameters of the transport equation and the Navier Stokes equation are determined. The transport equation is expressed by an equation (1). The Navier Stokes equation is expressed by an equation (2), and can be rewritten to be an equation (3). In some embodiments, the equation (1) is also known as an equation of continuity; the equation (2) is also known as an equation of momentum; and the equation (3) is also known as an equation of energy. 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         ρ 
                       
                       
                         ∂ 
                         t 
                       
                     
                     + 
                     
                       
                         ∇ 
                         
                           · 
                           ρ 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       u 
                     
                   
                   = 
                   0. 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           ρ 
                           ⁢ 
                           u 
                         
                         ) 
                       
                     
                     + 
                     
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                         · 
                         
                           ( 
                           
                             ρ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             uu 
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         ∇ 
                         P 
                       
                     
                     - 
                     
                       ∇ 
                       
                         · 
                         τ 
                       
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ρ 
                         ⁢ 
                         C 
                       
                       p 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ∂ 
                             T 
                           
                           
                             ∂ 
                             t 
                           
                         
                         + 
                         
                           u 
                           · 
                           
                             ∇ 
                             T 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∇ 
                       
                         · 
                         
                           ( 
                           
                             k 
                             ⁢ 
                             
                               ∇ 
                               T 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       τ 
                       ⁢ 
                       
                         : 
                       
                       ⁢ 
                       
                         D 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     ρ is a density of the polymer melt PM; u is a velocity vector; t is time; τ is an extra stress tensor; ∇u is a velocity gradient tensor; D is a rate-of-deformation tensor; g is the acceleration vector of gravity; P is a pressure; C P  is a specific heat of the polymer melt PM; T is a temperature; and k is a thermal conductivity. 
     In some embodiments, the conditional parameters include the temperature T, the force F (shown in  FIG. 1 ), and a material of the polymer melt PM, and such conditional parameters are associated with variables shown in equations (1) to (3). For example, when the force F is determined (which can be controlled by the rheometer  100 ), the volumetric flow rate Q is determined, which affects the flowing behavior of the polymer melt PM and consequently affects solutions of the equations (1) to (3). As such, the variables in the equations (1) to (3) are associated with the force F. In other embodiments, the conditional parameters include other parameters which may affect the transport equation and the Navier Stokes equation. For example, the conditional parameters further include geometry of the capillary  130 , such as a diameter and a length of the barrel  110  and the capillary  130 . In some embodiments, the diameter and the length of the barrel  110  are 12 mm and 10 mm, respectively. In some embodiments, the diameter and the length of the capillary  130  are 1 mm and 0.2 mm, respectively. For another example, the conditional parameters further include isothermal melting temperature of the polymer melt PM. In some embodiments, the isothermal melting temperature is given over a range of shear rate from 10 s −1  to 100 s −1 . 
     In operation S 224 , a plurality of computational parameters of the Trouton function are determined. In some embodiments, the computational parameters are determined according to experimental extensional viscosity data. 
     In operation S 226 , a Trouton ratio T r  of the Trouton function is obtained according to the computational parameters. The Trouton function is expressed by equations (4) and (5). 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       r 
                     
                     ⁡ 
                     
                       ( 
                       
                         γ 
                         . 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         η 
                         UE 
                       
                       
                         η 
                         S 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       T 
                       r 
                     
                     ⁡ 
                     
                       ( 
                       
                         γ 
                         . 
                       
                       ) 
                     
                   
                   = 
                   
                     3 
                     + 
                     
                       
                         
                           T 
                           0 
                         
                         
                           
                             [ 
                             
                               1 
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       λ 
                                       T 
                                     
                                     ⁢ 
                                     
                                       γ 
                                       . 
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   2 
                                 
                               
                             
                             ] 
                           
                           
                             n 
                             T 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The extensional viscosity η E  includes a uniaxial extensional viscosity η E , a planar extensional viscosity η PE , and a biaxial extensional viscosity η BE . In some embodiments, the extensional viscosity η E  is substantially equal to the uniaxial extensional viscosity η UE . The uniaxial extensional viscosity η UE  and the shear viscosity is can be expressed by the equation (4). The computational parameters include T 0 , λ T , and n T , wherein the T 0  is an anisotropic factor, the λ T  is a relaxation time, and n T  is a power index. In some embodiments, the computational parameters are fitted by experimental extensional viscosity data. In some embodiments, the experimental extensional viscosity data are obtained previously and stored in a storage device  110  (shown in  FIG. 8 ) of the rheometer  100 . 
     In some embodiments, the uniaxial extensional viscosity η UE  is obtained based on a Binding approximation. The Binding approximation is described in detail in “An Approximate Analysis for Contraction and Converging Flows” published in the Journal of Non-Newtonian Fluid Mechanics Vol. 27, Issue 2, Pages 173-189, 1988, which is incorporated herein by reference in its entirety. For the sake of simplicity, the present disclosure does not elaborate the Binding approximation herein. 
     The Binding approximation provides limitations among the shear rate {dot over (γ)}, an extensional rate {dot over (ε)}, the extra stress tensor τ, and the pressure drop ΔP. The Binding approximation is expressed by equations (6) to (10). 
     
       
         
           
             
               
                 
                   
                     η 
                     UE 
                   
                   = 
                   
                     s 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
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                           - 
                           1 
                         
                       
                       . 
                     
                   
                 
               
               
                 
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                         . 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
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                   ( 
                   9 
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                     I 
                     nk 
                   
                   = 
                   
                     
                       ∫ 
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                       1 
                     
                     ⁢ 
                     
                       
                         
                           { 
                           
                             
                                
                               
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                           } 
                         
                         
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                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     s is an index of flow consistency of extension; h is an index of and flow behavior of extension; τ W  is a wall stress; n is a power index; and {dot over (γ)} w  is a wall shear rate. 
     In operation S 228 , based on the weighted GNF viscosity model, an estimated pressure drop ΔP est  of the polymer melt PM is obtained according to the Trouton ratio T r , the transport equation, and the Navier Stokes equation. The details of operation S 228  will be described below with respect to  FIG. 4 . 
     The weighted GNF viscosity model is modified from the general GNF viscosity model. More specifically, the weighted GNF viscosity model includes more limitations on the basis of the general GNF viscosity model. The weighted GNF viscosity model is described in detail in “A Revisitation of Generalized Newtonian Fluids” published in the Journal of Rheology Vol. 64, Issue 3, 2020, and in U.S. Pat. No. 10,710,285B, which are incorporated herein by reference in their entireties. The GNF viscosity model is expressed by an equation (11), and the weighted GNF viscosity model is further expressed by equations (12) to (15). 
     
       
         
           
             
               
                 
                   τ 
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           η 
                           S 
                         
                         ⁡ 
                         
                           ( 
                           
                             T 
                             , 
                             P 
                             , 
                             
                               γ 
                               . 
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       2 
                       ⁢ 
                       
                         η 
                         E 
                       
                       ⁢ 
                       
                         D 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     η 
                     W 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           W 
                         
                         ) 
                       
                       ⁢ 
                       
                         η 
                         S 
                       
                     
                     + 
                     
                       W 
                       ⁢ 
                       
                         
                           η 
                           E 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     1 
                     - 
                     W 
                   
                   = 
                   
                     
                       
                         
                           γ 
                           . 
                         
                         S 
                         2 
                       
                       
                         
                           γ 
                           . 
                         
                         2 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   W 
                   = 
                   
                     
                       
                         
                           γ 
                           . 
                         
                         E 
                         2 
                       
                       
                         
                           γ 
                           . 
                         
                         2 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       γ 
                       . 
                     
                     2 
                   
                   = 
                   
                     
                       
                         γ 
                         . 
                       
                       S 
                       2 
                     
                     + 
                     
                       
                         
                           γ 
                           . 
                         
                         E 
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     η W  is a weighted shear/extensional viscosity; W is a weighting factor, which is also known as an extension fraction; and {dot over (Γ)} S , {dot over (Γ)} E , and {dot over (γ)} are a principal shear rate, a principal extensional rate, and the strain rate, respectively. (1-W) and W represent shear-rate percentage and extension-rate percentage, respectively. For example, when W is equal to 0, the weighted GNF viscosity model returns to the general GNF viscosity model. 
     For a uniaxial extensional flow, the uniaxial extensional viscosity η UE  is expressed by an equation (16), and the T and D can be expressed by equations (16) and (17), respectively. 
     
       
         
           
             
               
                 
                   
                     η 
                     UE 
                   
                   = 
                   
                     
                       
                         
                           τ 
                           11 
                         
                         - 
                         
                           τ 
                           22 
                         
                       
                       
                         ɛ 
                         . 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   τ 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               τ 
                               11 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               τ 
                               22 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               τ 
                               33 
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   D 
                   = 
                   
                     
                       
                         ɛ 
                         . 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     According to equations (4) to (19), the extensional viscosity η E  for uniaxial extensional flow is expressed by an equation (19). 
     
       
         
           
             
               
                 
                   
                     η 
                     E 
                   
                   = 
                   
                     
                       
                         η 
                         UE 
                       
                       3 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     According to the equations (1) to (19), the pressure drop ΔP and the velocity vector u can be obtained by solving the equations (1), (2), (3) and (11) using the determined conditional parameters and the computational parameters. However, some of the conditional parameters and the computational parameters are fitting parameters, which means at least some of the equations (1), (2), (3) and (11) may deviate from the true values if those parameters are not fitted perfectly. As such, an operation S 230  is performed to verify whether the equations (1), (2), (3) and (11) are fitted well. To facilitate understanding, before the calculation performed in the operation S 228  converges, the pressure drop ΔP, the strain rate {dot over (γ)}, the temperature T, the pressure P, and the velocity vector u obtained during the operation S 228  are referred to as an estimated pressure drop ΔP est , an estimated strain rate {dot over (Γ)} est , an estimated temperature T est , an estimated pressure P est , and an estimated velocity vector u est , respectively. 
     In operation S 230 , it is determined whether an error E of the estimated pressure drop ΔP est  is within a threshold E th . The error E of the estimated pressure drop ΔP est  is equal to a difference between the estimated pressure drop ΔP est  and the predetermined pressure drop ΔP pre  divided by the predetermined pressure drop ΔP pre . In some embodiments, the threshold E th  is about ±10%. In some embodiments, the predetermined pressure drop ΔP pre  is measured by the pressure transducers  140  and  150 . Alternatively stated, the predetermined pressure drop ΔP pre  is a measured value or an experimental data. 
     When the error E of the estimated pressure drop ΔP est  is not within the threshold E th , the method  20  returns to the operation S 222 , and the operations S 222  to S 228  are performed again. When the error E is not within the threshold E th , the conditional parameters and computational parameters are updated, and the Trouton ratio T r  is obtained again according to the updated conditional parameters and computational parameters. 
     In some embodiments, when the error E is a positive value and greater than the threshold E th , T 0  (in the equation (5)) of the computational parameters is increased 1 degree to update the computational parameters (operated in operation S 226 ), wherein T 0  is an initial temperature. In other embodiments, when the error E is a positive value and greater than the threshold E th , λ T  of the computational parameters is increased to update the computational parameters (operated in operation S 226 ). In various embodiments, when the error E is a positive value and greater than the threshold E th , n T  of the computational parameters is increased to update the computational parameters (operated in operation S 226 ). 
     In some embodiments, when the error E is a negative value and less than the threshold E th , T 0  of the computational parameters is decreased 1 degree to update the computational parameters. In other embodiments, when the error E is a negative value and less than the threshold E th , λ T  of the computational parameters is decreased to update the computational parameters (operated in operation S 226 ). In various embodiments, when the error E is a negative value and less than the threshold E th , n T  of the computational parameters is decreased to update the computational parameters (operated in operation S 226 ). 
     When the error E of the estimated pressure drop ΔP est  is within the threshold E th , the method  20  proceeds to operation S 232 . In operation S 232 , the viscosity profile η({dot over (γ)}) is obtained according to the conditional parameters and the computational parameters. When the error E is within the threshold E th , the equations (1), (2), (3) and (11) are fitted well, and the viscosity profile η({dot over (γ)}) obtained from equations (1) to (19) is reliable. 
     Based on the above, initial values of the computational parameters of the Trouton function can be determined according to the Binding approximation. By operating the loop formed by the operations S 222  to S 230 , the computational parameters of the Trouton function can be updated to be convergence (fitted well). In some embodiments, To is a dominant parameter among the computational parameters of the Trouton function. 
     Reference is made to  FIG. 4 .  FIG. 4  is a detailed flowchart of operation S 228  according to some embodiments of the present disclosure. After the conditional parameters and the computational parameters are determined, the estimated pressure drop ΔP est  can be obtained from the equations (1), (2), (3) and (11). Obtaining the estimated pressure drop ΔP est  (operation S 228 ) includes operations S 2282 , S 2284 , S 2286  and S 2288 . 
     In operation S 2282 , the estimated velocity vector u est  of the polymer melt PM is obtained according to the transport equation. 
     In operation S 2284 , the estimated strain rate fest of the polymer melt PM is obtained according to the weighted GNF viscosity model. 
     In operation S 2286 , the estimated temperature T est  and the estimated pressure P est  of the polymer melt PM are obtained according to the Navier Stokes equation, the estimated tensor u est , and the estimated strain rate {dot over (γ)} est . 
     In operation S 2288 , the estimated pressure drop ΔP est  is obtained according to the estimated pressure P est . In some embodiments, the operation S 228  is iteratively performed. In other words, after a first round of the operations S 2282  to S 2288  is performed, a second round of the operations S 2282  to S 2288  is performed successively. In some embodiments, the operation S 228  is repeated until the estimated velocity vector u est , the estimated temperature {dot over (γ)} est , and the estimated pressure drop ΔP est  converge to a predetermined velocity vector u pre , a predetermined temperature T pre , and the predetermined pressure drop ΔP pre , respectively. 
     Reference is made to  FIG. 5 ,  FIG. 6  and  FIG. 7 .  FIG. 5  is a schematic diagram of the Trouton ratio T r  against the extensional rate {dot over (ε)} according to some embodiments of the present disclosure.  FIG. 6  is a schematic diagram of the uniaxial extensional viscosity η UE  and an experimental data against the extensional rate {dot over (ε)} according to some embodiments of the present disclosure.  FIG. 7  is a schematic diagram of the pressure drop ΔP and an experimental data against the shear rate {dot over (γ)} according to some embodiments of the present disclosure. The Trouton ratio T r , the uniaxial extensional viscosity η UE , and the pressure drop ΔP shown in  FIG. 5  to  FIG. 7  are obtained by the method  20 . 
     In this embodiment, the Trouton ratio T r  is divided into two sections, including a high strain rate section and a low strain rate section. The two-sectional Trouton ratio T r  is expressed by equations (20) and (21). 
     
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         r 
                       
                       ⁡ 
                       
                         ( 
                         
                           γ 
                           . 
                         
                         ) 
                       
                     
                     = 
                     
                       3 
                       + 
                       
                         
                           T 
                           L 
                         
                         
                           
                             [ 
                             
                               1 
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       λ 
                                       L 
                                     
                                     ⁢ 
                                     
                                       γ 
                                       . 
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   2 
                                 
                               
                             
                             ] 
                           
                           
                             n 
                             L 
                           
                         
                       
                     
                   
                   , 
                   
                     
                       γ 
                       . 
                     
                     &lt; 
                     
                       
                         γ 
                         . 
                       
                       * 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         T 
                         r 
                       
                       ⁡ 
                       
                         ( 
                         
                           γ 
                           . 
                         
                         ) 
                       
                     
                     = 
                     
                       3 
                       + 
                       
                         
                           T 
                           H 
                         
                         
                           
                             [ 
                             
                               1 
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       λ 
                                       H 
                                     
                                     ⁢ 
                                     
                                       γ 
                                       . 
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   2 
                                 
                               
                             
                             ] 
                           
                           
                             n 
                             H 
                           
                         
                       
                     
                   
                   , 
                   
                     
                       γ 
                       . 
                     
                     ≥ 
                     
                       
                         γ 
                         . 
                       
                       * 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     T L , T H , n L , n H , λ H , and λ L  are fitting parameters, and are the computational parameters. More specifically, T L  and T H  are a low strain rate portion and a high strain rate portion of T 0 , respectively; n L  and n H  are a low strain rate portion and a high strain rate portion of n T , respectively; and λ H , and λ L  are a high strain rate portion and a low strain rate portion of λ T , respectively. 
     Based on the data shown in  FIG. 5  and  FIG. 6 , the pressure drop ΔP against the shear rate {dot over (γ)} is obtained, and the viscosity profile η(ΔP) is thus obtained using the viscosity profile η({dot over (γ)}) and a correspondence between the pressure drop ΔP and the shear rate {dot over (γ)} shown in  FIG. 7 . 
     In addition, the uniaxial extensional viscosity η UE  and the pressure drop ΔP are substantially aligned with the experimental data, i.e., the uniaxial extensional viscosity η UE  and the pressure drop ΔP obtained by the method  20  are accurate and reliable. Therefore, the viscosity profile η(ΔP) is also reliable. 
     In some embodiments, the rheometer  100  shown in  FIG. 1  is configured to execute CAE simulation software (such as the calculations performed in method  20 ). Please refer to  FIG. 8 .  FIG. 8  is a functional block diagram of the rheometer  100  according to some embodiments of the present disclosure. 
     In some embodiments, the rheometer  100  has an internal-implemented computer, and includes a processing module  102  such as a processor adapted to perform a computer-implemented simulation method for use in capillary injection, an input/output (I/O) interface  104  electrically coupled to the processing module  102 , a read-only memory (ROM)  106 , a random access memory (RAM)  108 , and a storage device  110 . The ROM  106 , the RAM  108 , and the storage device  110  are communicatively coupled to the processing module  102 . 
     The rheometer  100  further includes a communication port  112  configured to transmit instructions to control the piston  120 , a display  114 , a keyboard  116 , and an input device  118 . The input device  118  may include a card reader, an optical disk drive or any other device that allows the rheometer  100  to receive input from the on-site technicians. In some embodiments, the input device  118  is configured to input computer instructions (software algorithms) stored in a non-transitory computer-readable medium  120 , and the processing module  102  is configured to execute operations for performing the method  20  according to the computer instructions. In some embodiments, the processing module  102  reads software algorithms from the input device  118  or the storage device  110 , executes the calculation steps, and stores the calculated result in the RAM  108 . 
     In our present work, at least some the operations of the method  20  to obtain the extensional viscosity η E  has been implemented in commercial simulation software, Moldex3D (CoreTech System Co. of Taiwan), and the rheometer  100  carries the commercial simulation software to obtain the extensional viscosity η E . 
     One aspect of the present disclosure provides a method of measuring an extensional viscosity of a polymer melt and a capillary injection system. The method includes operations of: based on a weighted GNF viscosity model, obtaining a viscosity profile of the polymer melt according to a transport equation, a Navier Stokes equation, and a Trouton function; measuring a pressure drop of the polymer melt; obtaining a general viscosity of the polymer melt from the viscosity profile according to the pressure drop, wherein the general viscosity comprises a shear viscosity of the polymer melt and the extensional viscosity of the polymer melt; and extracting the extensional viscosity from the general viscosity. 
     Another aspect of the present disclosure provides a capillary injection system for measuring an extensional viscosity of a polymer melt, including a barrel, a capillary, a piston, a first pressure transducer, a second pressure transducer, and a rheometer. The barrel has a cavity configured to contain a polymer melt. The capillary is coupled to the barrel. The piston is coupled to the barrel, and configured to provide a force to the polymer melt to make the polymer melt flow through the capillary. The first pressure transducer is configured to measure a first pressure of the polymer melt at an inlet of the capillary. The second pressure transducer is configured to measure a second pressure of the polymer melt at an outlet of the capillary. The rheometer is configured to perform operations of: based on a weighted GNF viscosity model, obtain a viscosity profile of the polymer melt according to a transport equation, a Navier Stokes equation, and a Trouton function; obtain a general viscosity of the polymer melt from the viscosity profile according to the first pressure and the second pressure, wherein the general viscosity comprises a shear viscosity of the polymer melt and the extensional viscosity of the polymer melt; and extract the extensional viscosity from the general viscosity. 
     Although the present disclosure and its advantages have been described in detail, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the disclosure as defined by the appended claims. For example, many of the processes discussed above can be implemented in different methodologies and replaced by other processes, or a combination thereof. 
     Moreover, the scope of the present application is not intended to be limited to the particular embodiments of the process, machine, manufacture, composition of matter, means, methods and steps described in the specification. As one of ordinary skill in the art will readily appreciate from the present disclosure, processes, machines, manufacture, compositions of matter, means, methods, or steps, presently existing or later to be developed, that perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein, may be utilized according to the present disclosure. Accordingly, the appended claims are intended to include within their scope such processes, machines, manufacture, compositions of matter, means, methods and steps.