Lecture 8: Network Formation, Statistical Approach, Pw Based, a Word on MWD for Nonlinear Polymerization

10.569 Synthesis of Polymers Prof. Paula Hammond Lecture 8: Network Formation, Statistical Approach, Pw Based, A Word on MWD for Nonlinear Polymerization Network formation Consider first case: a—b + a—a a [a3] monomers so far have been difunctional O H2N R O O HOC HO COH R' COH R" difunctional = 2 groups that can react (participate in rxn) OH trifunctional: HO R" OH OH a—b a—a b—a b—a b—a a b a b a b a Always end in ‘a’ group Forms branches, not crosslinks What is the molecular weight or MW distribution? (Peebles, Schaefgen, Flory) for a—b + af systems f = func of af π = conversion of b-groups (b=minority) (N B )o r= ≤ 1.0 (N A )o frπ + 1 − rπ 1 − rπ ( f − 1)2 (rπ )2 + (3 f − 2 )rπ + 1 pw = ( frπ + 1 − rπ )(1 − rπ ) pn = Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date. pw pn =z= 1 + frπ ( frπ + 1 − rπ )2 polydispersity Consider limit as r → 1.0 lim z = r→1.0 pw pn = 1+ 1 f ⎛ r → 1.0 ⎞ ⎜⎜ ⎟⎟ ⎝ π = 1.0 ⎠ Specific cases: Let f = 1 ⇒ a—b + a—x end capper same as a—b + end capper ⇒ z = 2.0 ⇒ same result discussed for general difunctional systems let f = 2 → a—b + a—a linear polymer z = 1 + ½ = 1.5 → narrower MWD does this make sense? Recall: a—b + a—a : b—a b—a b—a b—a b a—a b—a b—a b—a Join 2 chains together with a—a Get rid of extremeties of MWD. Longer chains are joined together with short chain. Long chains can join w/long chains but much less likely. Consider an example with af: 1 mol a—b + 10-2 mol a3: (0.01) f=3 1 b group to 1.03 a groups Note: for r = 0.97 and typical a—b polymerization: p n = 66 = 10.569, Synthesis of Polymers, Fall 2006 Prof. Paula Hammond 1+ r 1− r Lecture 8 Page 2 of 6 Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date. r= 1 ≅ 0.97 1.03 pn = 3(0.97 ) − 0.97 + 1 ≅ 99 1 − 0.97 Systems forming networks Soluble fraction → “sol” Polymer chains, oligomers, monomers not connected to network Gel fraction → insoluble, intractable ⇒ constitutes network Gel point: π at which an infinite network is formed ⇒ πc Above gel point πc: Determine πc as: Carothers gel fraction ↑ sol fraction ↓ pn → ∞ pw → ∞ pn → ∞ Consider simplest case: Equal # of a,b functional groups Define f AVG = ∑N f ∑N i i i fi = functionality of given monomer Ni = # of molecules with fi O O 3 HOC COH f AVG = fi ≥ 1.0 3(2 ) + 1(3) + 1.5(2 ) 12.0 = = 2.18 3 + 1 + 1.5 5.5 OH 1 HO 1.5 CH2CH CH2OH HOCH2CH2OH Can define conversion No = initial # of monomer molecules N = # of remaining molecules Initial # of functional groups = fAVGNo 10.569, Synthesis of Polymers, Fall 2006 Prof. Paula Hammond Lecture 8 Page 3 of 6 Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date. O 6 HCOH 6 OH r=1.0 ⇒ # of functional groups reacted = 2(No-N) For each a—b reaction, lose 1a, 1b ⇒ 2 functional groups For each a—b reaction, decrease # of molecules by 1 # of functional groups reacted 2(N o − N ) = a π = total # of functional gruops N o f AVG b pn = No (see original definition) N # avg degree of polymerization Rearrange N o f AVG π = 2 N o − 2 N a : N o (πf AVG − 2 ) = −2 N No −2 = N πf AVG − 2 N 2 pn = = N o 2 − πf AVG Universal expression Carothers Equation Works for all the cases, but fAVG must be adjusted when r ≠ 1.0 Rearrange Carothers: π= 2 f AVG − 2 p n f AVG Consider gel point: pn → ∞ ⇒ πc = e.g. 2 f AVG π at gel point general expression for Carothers more functionality → gel at lower conversion f AVG = 2.18 2 πc = = 0.92 2.18 Consider less perfect case: Ni fi (Na )o f AVG = only time for r = = 1.0 (Nb)o Ni ∑ ∑ If r ≠ 1.0 ⇒ only gain in increased MW + crosslinking when using –a + –b (i.e. deficient group quantity determines how many of these reactions take place) 10.569, Synthesis of Polymers, Fall 2006 Prof. Paula Hammond Lecture 8 Page 4 of 6 Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date. Excess of b only decreases p n by end capping Count only a groups + 2 f AVG = 2 (∑ N f ) ∑N ai ai i f A,i = function in - a of monomer i N A,i = # of molecules of monomer i Example: HOCH2CH2OH A O B fi 2 fA = fa,i = 2 4 fB = fb,i = 2 1 fC = fa,i = 3 O HOC COH OH C N HOCH2CH2CHCH2OH Enter body copy in Verdana 10. 8b, 7a deficient where a = OH O b = COH f AVG = πc = 2(N a f a + N c f c ) 2((2 )(2 ) + (1)(3)) = 2.0 = 2 + 4 +1 Na + Nb + Nc 2 = 1.0 2.0 Only at full conversion do you form network. π c > 1.0 ⇒ physically impossible to create network Case of exact stoichiometry: doesn’t matter which is deficient. Let NB = 3.5 πc = 2 = 0.93 2.15 Other case: Flory-Stockmayer pw → ∞ Generalized cases: 10.569, Synthesis of Polymers, Fall 2006 Prof. Paula Hammond Lecture 8 Page 5 of 6 Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date. πc = 1 {r ( f w,A − 1)( f w,B − 1)}1/ 2 f w,A ∑f = ∑f f w,B ∑f = ∑f A,i N A,i i A,i N A,i B,i r= 2 ∑f ∑f 2 where fw,i can be ≥ 1.0 N B,i i B,i N B,i A,i N A,i B,i N B,i For our earlier example, Carothers: ≤ 1.0 π c = 0.90 (NB = 3.5) π c = 0.93 Lower πc → longest chains form more of infinite network 10.569, Synthesis of Polymers, Fall 2006 Prof. Paula Hammond Lecture 8 Page 6 of 6 Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date.