A prepolymer resin is made by mixing a difunctional monomer (two groups per molecule

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch Rubber elasticity, swelling and gelation

Problem

A prepolymer resin is made by mixing a difunctional monomer

A_2

(two

A

groups per molecule, reacting only with

B

groups) with an

f

-functional monomer

B_f

. Let

N_A

and

N_B

be the total numbers of

A

and

B

groups present, and define the stoichiometric ratio

r \equiv N_A/N_B \le 1

(a deliberate excess of

B

groups). Let

p_A

and

p_B

be the fractions of

A

and

B

groups, respectively, that have reacted at a given stage of the polymerisation. ((a)) Show that

p_B = rp_A

. ((b)) Using the same branching-tree extinction argument that gives

p_c=1/(f-1)

for a one-component

A_f

system, show that following a bond outward from a

B_f

unit reaches a new branch point with probability

p_Ap_B

, that each new branch point offers

(f-1)

further outgoing arms, and hence that the gel point occurs when

(f-1)p_Ap_B=1

. Derive

p_{A,c}=1/\sqrt{r(f-1)}

. ((c)) Because

p_A

cannot exceed

1

, show that gelation is possible at all only if

r

exceeds a threshold

r_{\min}

, and find

r_{\min}

for

f=4

. ((d)) A diacid (

A_2

) is mixed with a tetrafunctional polyol (

B_4

f=4

). Evaluate

p_{A,c}

for (i)

r=0.30

and (ii)

r=0.50

, and state in each case whether the mixture can gel.