Rheology control by branching modeling

Transcription

1 Volha Shchetnikava J.J.M. Slot Department of Mathematics and Computer Science TU EINDHOVEN April 11, 2012

2 Outline Introduction Introduction Mechanism of Relaxation Introduction of Entangled Polymers relaxation of linear polymers Open Problems in in dynamic tube dilation Molecular Rheology activated primitive path Complex fluctuations Architectures branch point diffusion Nonlinear Flows hierarchical relaxation DYNACOP Progress on on Generalized Models for Rheology Theory Theory and of and Entangled Simulation Branched Polymers Conclusions Topological structure of LDPE Time-marching model Presentation Outline

3 Introduction LDPE molecules have a highly branched structure characterized by: Broad molecular weight distribution Both long and short side chains are present Irregularly spaced branches Transition from short to long chain branching at M e Exhibit strain hardening in uniaxial extensional flow Exhibit strain softening in shear flow

4 Introduction There is an intimate relations polymer structure, rheology an mate relationship There is an intimate between relationship between polymer structure, rheology re, rheology and processing and processing. nal viscosity and elts (Laun, 1984) Uniaxial extensional viscosity of various PE melts Typical film film blowing operation in in polymer companies polymer companies Uniaxial extensional viscosity and of various PE melts (Laun, 1984)

5 Introduction Branching structure LCB characterization Affect Rheological behavior Rheological behavior (Experiments): sensitive to LCB cannot determine the LCB structure quantitatively Molecular rheological theory (): number of branches length of branches position of branch point

6 Step Strain Experiment Introduction Mechanism STRESS of Relaxation RELAXATION of Entangled Polymers Sample is initially at rest At time t = 0, apply instantaneous shear strain γ0 The shear relaxation modulus Relaxation modulus G(t, γ0) σ(t)/γ0 of linear (2-1) polymers For small strains, the modulus does not depend on strain. Linear viscoelasticity corresponds to this small strain regime. Linear response means that stress is proportional to the strain, and thus the modulus is independent of strain. Shear strain Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Oscillatory shear σ(t) = G(t)γ 0 σ(t) G(t)γ0 (2-3) σ(t) = G(t)γ 0 [G (ω)sin(ωt)+g (ω)cos(ωt)] Figure 1: Stress Relaxation modulus of linear polymers. A is monodisperse with Mw < MC, B is monodisperse with M MC, and C is polydisperse with Mw MC. Linear polymers are viscoelastic liquids. Parameters: The Plateau modulus, G0 N The molecular weight between two entanglements, M e,0 1 The Rouse time of a segment between two entanglements, τ e

7 The Tube framework Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation The The tube Tube model Model offers - de Gennes, a simple Edwards, framework Doi (1970 s) for understanding entangled polymer behavior. Entanglements of one polymer with its neighbors creates a tube-like The existence regionof that other confines chains the polymer to a quasi-one-dimensional constraints motionof on a test a short chain time scale to a tube-like region. Sketch of a tube, where b is a Kunh length, a is the tube diameter, L is the contour length of the polymer itself and L tube is the length The test of chain the can tubeonly escape by diffusing along the tube axis. This process is called reptation.

8 Basic processes of relaxation Relaxation Mechanisms (by motion of the chain) Reptation Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Pierre de Gennes The test chain can only escape by diffusion along the tube axis (reptation) d L 3 Primitive path fluctuations, in which the ends of the chain randomly pull away from the ends of the tube Constraint release where portion of a chain can be relaxed locally

9 Dynamic tube dilation Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation The polymer fraction already relaxed = solvent Marrucci, 1985 Global effect Local effect New equilibrium state: Increase of the tube diameter a and of the M e Decrease of L eq Speeds up the polymer relaxation On the reptation and the fluctuation processes Inter-relationship between all the relaxation mechanisms

10 Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Introduction Tube model for branched polymers Branched Polymer Dy A star polymer cannot A star polymer cannot reptate Star polymers breathing modes reptate instead it must instead it must relax by deep arm A star polymer cannot relax by deep arm retractionstar polymers breathing modes reptate instead it must retraction Becomes exponentially more difficult to relax segments closer to branch point. relax by deep arm retraction Becomes exponentially more difficult to relax Becomes exponentially segments closer to branch more difficult to relax point. segments closer to branch point. shoulder in loss modulus =in primitive shoulder loss modulus = primtive path fluctuation Linear polymers Linear Polymers broad range of relaxation times path fluctuation modesmodes ->broad range of shoulder in loss modulus = primtive path fluctuation Linear Polymers modestimes -> broad range of relaxation times relaxation a( ) R exp [U( )] Rouse modes insider the a( ) exp [U( )] tube Rouse modes inside the tube Slide from Daniel Read, Univ. of Leeds Slide from Daniel Read, Univ. of Leeds

11 Tube dilution of stars Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation At long times the outer parts of the arms act as solvent. This means that the number of entanglement constraints effective during relaxation of star arms diminishes with time. Φ = unrelaxed volume fraction Φ = 1 ξ M e(φ) = M e/φ α a(φ) = a/φ α/2 α is a dilution exponent α = 1, 4/3 Dynamic Tube Dilution Appl to Monodisperse Melt of Sta (Ball and McLeish 1989) F = unrelaxed volum M e (F)= dilution = 1, 4/3 F 1

12 Relaxation of asymmetric star When t < τ a, all arms retract while the branch point remains anchored l Relaxation of Asymmetric Star When t = τ a, the short arm has archical Relaxation relaxed Processes and the branch point makes a random hop within the confining point remains tubeanchored. 1. When t< a,, all arms retract while the branch 2. When t= a, the short arm has relaxed and branch point takes a random hop along the When t > τ a, the whole polymer reptates confining with tube. the branch point acting as a fat friction bead 3. When t> a, the whole polymer reptates with the branch point as a ``fat friction bead. entanglement points Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Hierarchical Relaxation Asymmetric Star: Hierarchical Relaxation P a 1. Wh poin bra con the 2. Wh 3. Wh entanglement p e: a) Branch Point Motion: Branch Point Motion: k B T D br p2 a 2 k B T br = D br = 2 p 2 a 2 a ζ br 2τ a McLeish et al., Macromolecules, 32, Arm retraction Time: Arm Retraction Time: τ a = τ 0 Z 1.5 a exp(νz a) a 0 Z a 1.5 exp( Z a ) McLeish et al., Macromolecules, 32, with p 2 = 1/12; smaller for short arms McLeish

13 Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Relaxation of H polymer Branched Polymer Dynamics First, the arms relax by star-like breathing mode Then, the backbone relaxes by reptation - but with friction concentrated at the ends of the chain ot So Successful Prediction: In general not always leading to successful predictions of the experimental data olyisoprene H polymer H110B20A We take D br = p 2 a 2 /2q a, with p 2 = 1/12

14 Linear rheology of arbitrarily branched polymers Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Relaxation of a branched polymer Relaxation of arbitrary branched polymer Occurs from the outside of the polymer towards the inside Occurs from the outside of the polymer towards the inside Slide from Daniel Read, Univ. of Leeds Slide from Daniel Read, Univ. of Leeds

15 Linear rheology of arbitrarily branched polymers Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Relaxation Relaxation of of arbitrary a branched polymer branched polymer Occurs from the outside of the polymer towards the inside Occurs from the outside of the polymer towards the inside Slide from Daniel Read, Univ. of Leeds

16 Linear rheology of arbitrarily branched polymers Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Relaxation Relaxation of of arbitrary a branched branched polymer polymer Sometimes side Sometimes arms relax side arms - relaxation relaxation cannot cannot proceed further until the main arm catches proceed further up. Side until the arms main give arm extra catches friction. up. Side arms give extra friction Slide from Daniel Read, Univ. of Leeds

17 Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Linear rheology of arbitrarily branched polymers Relaxation Relaxation of arbitrary of a branched branched polymer polymer Sometimes side Sometimes arms relax side arms - relaxation relaxation cannot proceed cannot further until the main arm catches proceed up. further Side until arms the main give arm extra catches friction. up. Side arms give extra friction Slide from Daniel Read, Univ. of Leeds

18 Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Linear rheology of arbitrarily branched polymers Relaxation of a branched polymer Relaxation of arbitrary branched polymer Eventually there is an effectively linear section Eventually there which isrelaxes an effectively via reptation, linear with section side-arms which relaxes via reptation, with providing side-arms the friction providing. the friction c.f. H-polymer terminal relaxation Slide from Daniel Read, Univ. of Leeds

19 Relaxation of linear polymers Dynamic tube dilation Activated primitive path fluctuations Branch point diffusion Hierarchical relaxation Linear rheology of arbitrarily branched polymers Relaxation Relaxation of arbitrary of a branched branched polymer polymer and.. relax! And finally relaxed! Slide from Daniel Read, Univ. of Leeds

20 Models Introduction Hierarchical Model (Larson, Park, Wang; 2001, 2005, 2010) Linear, Star, H, Comb Star-linear blends BOB Model (Das et al., 2006, 2008) Linear, Star, H, Comb Star-linear blends Commercial polyolefins van Ruymbeke (2005, 2006, 2007, 2008) Linear, Star, H, Comb, Pom-Pom, Caylee-tree Star-linear blends

21 226 WANG, CHEN, AND LARSON Differences between models TABLE I. Main differences between the hierarchical model Larson 2001 ; Park et al ; current work and the bob model Das et al Element of algorithm Hierarchical Bob Arm retraction potential U eff and relaxation time late Analytical formulas from Milner and McLeish 1997;1998 Numerical evaluation of Taylor expansion at each time step Compound arm fluctuation Entire compound arm fluctuates with lumped branch point frictions Growing portion of compound arm fluctuates Arm retraction in CR-Rouse regime No arm retraction Park et al ; Arm retraction in thin tube arm retraction in fat Larson 2001 and thin tubes this work Branch point friction Time independent Time dependent Reptation In a partly dilated tube In an undilated tube Disentanglement Yes No Dilution exponent =1 Larson 2001 ; =1 =4/3 Park et al Branch point friction p 2 p 2 =1/12 Park et al p 2 =1/40 eral refinements of the relaxation mechanisms and by employing a logarithmic algorithm for calculating the time evolution of the arm retraction coordinate to replace the linear

22 Problems The hierarchical and bob models quantitatively predict the linear rheology of a wide range of branched polymer melts but also indicate that there is still no unique solution to cover all types of branched polymers without case-by-case adjustment of parameters such as the dilution exponent α and the factor p 2 which defines the hopping distance of a branch point relative to the tube diameter. Z. Wang and R. Larson, 2010

23 Topological structure of LDPE Time-marching algorithm Our approach We want to: Understand the role of each generation of segments within molecules in the relaxation of the total ensemble Consider the effect of taking a limited number of generations into account Assume that the rest of the ensemble will relax automatically due to dynamic tube dilation (disentanglement relaxation) We need to: Choose a representative ensemble of molecules Analyse the distribution of generations of segments in the ensemble Find all topologically different architectures belonging to a given generation Extend the time-marching model to treat the relaxation of high enough generations (up to 6?)

24 Reactor Dimension (m) Branching in CSTR Feed condition (kg/s) Topological structure of LDPE Time-marching algorithm T 1 T 2 T 3 Diameter Length Additional feeding positions Monomer CTA Initiator, S Initiator, C Simulation conditions Pre-exponential were factor chosen of scissionto rate (m achieve 3 /(kmol s)) monomer conversion and physical properties of alpha IUPAC LDPE (Tackx and Tacx, 1998), where additional feeding location. polydispersity is 26 and M n = 29kg/mol. Notes: Subscript 0 for temperature means reactor inlet position, while subscripts 1, 2, 3 mean the additional injec dw/d{log(n)} Ψ n 1 x10 6 (kmol/m 3 ) C Chain Length Fig. 3. First and second branchin of first and second moments are 1. Simulation results with topological Fig. 2. Comparative scission MWD (dash-dot Plot line) between andexperimental with linearand scission simulation (dashed result line) (solid line, first branching mome in CSTR (scission paradox). Alpha IUPAC ldpe has M n of 29.0 kg/kmol and D of Simulation results with topological scission model and linear Volha scission Shchetnikava model have J.J.M. Mn of Slot 28.0kg/kmol, Rheology 29.0kg/kmol control and by D of branching 27.8 and modeling gives , which

25 Branching in CSTR Topological structure of LDPE Time-marching algorithm Input data: Chain length Concentration Branching density n 1 c 1 b 1... n n c n b n Number of branch points in a molecule follow a binomial distribution P(n) with respect to the chain length n Bimodal chain length\degree of branching distribution Computational synthesis of architectures for a given combination of chain length and number of branch points (n,n) by a conditional Monte Carlo method

26 Representative ensemble Topological structure of LDPE Time-marching algorithm Binominal distribution P(n) determines the range of chain lengths n u,..., n v which can have N number of branch points. Parameters which control the generation of molecules: bins branch - the grid on the branch point number axis bins length - the grid on the chain length axis binfractionsn - fractional contribution of each bin Each bin is represented by a pair n, N. Number of architectures generated in bin depends on the size of a bin. 0 2 log10(binfractionsnn) log10(n) log10(n)

27 Representative ensemble Topological structure of LDPE Time-marching algorithm Macromolecules are described by graphs (trees) and represented by: Vertices - branch points and arm ends Weight of the edge - molecular weight of the strand The adjacency matrix of a weighted graph We specify an ensemble of a large number of branched molecules by introducing the following parameters: α labels the molecular species 1,..., N s c α indicates the concentration of particular species

28 Seniority Topological structure of LDPE Time-marching algorithm Seniority is a property of an interior segment of a branched molecule and is simply the number of segments (chain portions between branch points) that connects it to the retracting chain end responsible for its relaxation. MODELING OF LINEAR VISCOELASTIC BEHAVIOR OF LOW-DENSI of the previous one seniority represents th a segment relaxes rath molecule. Segments of start to relax first. The return to an equ segment of seniority 1 of the free end, as for st of seniority 2 also beh because on the time sca the segments of seniorit ually reconfiguring and entanglement network that the effective drag Figure 3. The seniority distribution in a branched segments is imposed o molecule. To calculate this parameter for a given segment, one needs to count the number of strands to lower seniority attached arising from the dissi

29 Distribution of seniorities Topological structure of LDPE Time-marching algorithm Seniority Mass fraction % Number fraction %

30 Topological structure of LDPE Time-marching algorithm MWD of seniorities x x x x x x 10 4 x x x x

31 Topologies of seniorities Topological structure of LDPE Time-marching algorithm Seniority Number of topologies Seniority 4 Seniority 3

32 Topologies of seniorities Topological structure of LDPE Time-marching algorithm Seniority % % % 6.62 % 5.13 %

33 Relaxation of a branched polymer Topological structure of LDPE Time-marching algorithm Reptation and contourrelaxation length fluctuations of a branched are polymer considered as simultaneous processes Reptation and contour length fluctuations are considered as Survival probability simultaneous of oriented processes segments is calculated by summing up all contributions x Survival probability of oriented 0 over types of arms and positions along the 1 segments: arms calculated by summing up all contributions over types of arms and positions along the arms 1 G( t) F(t) = ϕ i ( p rept (x i,t).p fluct (x i,t).p envir (x i,t))dx i 0 G N i 0 All types of arms (fractions ϕ i ) x i not relaxed by reptation x i not relaxed by fluctuations x x x i not relaxed by the environment if not relaxed otherwise

34 Time-marching algorithm Topological structure of LDPE Time-marching algorithm Time-marching algorithm No analytical function can be found for complex polymers Molecules Time t Reptation Fluctuations Polymer «solvent» Explicit time-marching algorithm: G(t) G (ω), G (ω) Φ(t i ) Φ(t i-1 ) G(t i ) τ reptation (x,t i ) τ fluctuation (x,t i ) t i-1 t i p survival (x, between t i-1 and t i ) t p survival (x, t i ) = p survival (x, t i-1 ). p survival (x, between t i-1 and t i ) E. van Ruymbeke, R. Keunings, C. Bailly, J.N. N. F. M., 128 (June 2005)

35 Relaxation of a branched polymer How does this molecular section relax? Topological structure of LDPE Time-marching algorithm Relaxation of a branched polymer Reptation Additional friction Contour length Fluctuations 2 Fluctuations modes: x branch =1 x branch =0 Coordinate system: x b =0 x b =1 L eq x b =0 U(x) Relaxed branches U(x) x b =x br x b =1 L eq x EVR et al., Macromol. 06

36 Polydispersity Polydispersity Topological structure of LDPE Time-marching algorithm d dw Polydispersity fixed to 1.05 ( M ) M ) log( Log(M) Explicit timemarching algorithm: Φ DTD (t) calculated at each time step G, G (ω) [Pa] G (ω) [Pa] Φ DTD () t = ϕi ( pre pt ( xi, t). pfluct ( xi,) t ) dxi i Backbone: H=1 to 1.2 a ω (1/sec) ω (1/sec) c G, G (ω) [Pa] ω (1/sec) G (ω) [Pa] Backbone and arms: H=1 to 1.2 Arms: b H=1 to ω (1/sec) d

37 Topological structure of LDPE Time-marching algorithm Thank You!