Theoretical Study of Network Formation and Mechanical Properties of

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Theoretical Study of Network Formation and Mechanical Property of Physical Gel with Well-Defined Junction Structure Hiroto Ozaki, and Tsuyoshi Koga J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b05183 • Publication Date (Web): 18 Jul 2016 Downloaded from http://pubs.acs.org on July 23, 2016

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The Journal of Physical Chemistry

Theoretical Study of Network Formation and Mechanical Property of Physical Gel with Well-Defined Junction Structure Hiroto Ozaki and Tsuyoshi Koga∗ Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Katsura, Kyoto 615-8510, Japan E-mail: [email protected] Phone: +81 75 383 2705. Fax: +81 75 383 2705

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Abstract A statistical-mechanical theory of thermo-reversible gelation considering loops for the system consisting of bifunctional polymer units carrying A functional groups and trifunctional units carrying B functional groups at their ends is constructed. We obtain the sol-gel transition line and properties of the post-gel region as functions of the polymer concentration, temperature, association constant and loop parameter by using the present theory. In this paper, we calculate the number concentration of elastically effective chains in the gel region and obtain the shear modulus by an application of the phantom network theory. The shear modulus obtained by this theory is lower than that of conventional theory due to the loop formation. We find that these theoretical results are in good agreement with the experimental data.

1. Introduction Polymer gels, three-dimensional cross-linked polymer network, are classified into mainly two categories, chemical gels and physical gels. Chemical gels are formed by permanent cross-links, while physical gels are formed by weak reversible cross-links such as hydrogen bonds and hydrophobic interaction. These polymer gels have attracted much attention for many applications such as rheology modifiers, soft materials, and drug delivery systems. Mechanical properties of polymer gels have been studied experimentally and theoretically. 1–3 Recently, rheological properties of gels with well-defined structure were experimentally studied. 4–12 These gels receive significant attention, because they make possible rigorous comparison between theory and experiment. For chemical gels, tetra-PEG gel made by end-linking reaction of tetra-arm star polymers is well-known for the highly homogeneous structure. 4,5 The local cyclic defects were quantified experimentally by using 1 H NMR spectroscopy by Lange et al. 6 Zhou et al. have developed simple framework of loop quantification called network disassembly spectrometry and studied the loop formation of the gel consisting of bifunctional and trifunctional units. 7,8 For physical gels, Skrzeszewska et al. have studied 2 ACS Paragon Plus Environment

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the properties of the gel formed by telechelic associating polymers with end blocks which associate into triple helices experimentally and theoretically. 9,10 The research shows that the effect of the smallest cycles (i.e., loops) plays a significant role in the elasticity. Deng et al. have studied the gel formed by bifunctional polymer units carrying A functional groups (A◦2 ) and trifunctional polymer units carrying B functional groups at their ends (B◦3 ). 11,12 The gel has the well-defined structure: B◦3 units are regarded as junctions of the network. These structures of physical gels are controlled by the introduction of non-conventional well-defined physical cross-linking. For the A◦2 /B◦3 system mentioned above, it is expected that the loop formation often occurs because the molecular weight of the A◦2 unit is low. In this paper, we focus on the effect of loop formation on mechanical properties of the gel consisting of A◦2 and B◦3 units, and construct a statistical-mechanical theory of thermo-reversible gelation considering loops consisting of a single chain. Some theoretical studies focused on cyclization have been already published. 13–23 In some previous studies, cycles of all possible sizes are considered. However, in our study, we only take into account of the smallest cycle formation (i.e., loop formation) because the loop is formed with high probability owing to the contribution of entropy term and obviously lowers the modulus of the gel network. Several theories of reversible gelation are available, 24–26 however, in this study, we develop the theory by referring to the theory of molecular association and thermo-reversible gelation by Tanaka and Stockmayer which is based on Flory–Huggins solution theory and classical gelation theory. 26–29 Our theory can be used to derive the sol-gel transition and the properties of the post-gel region as functions of the polymer concentration, temperature, association constant and loop parameter. The mechanical properties are related to the single molecule properties, e.g. the stiffness and the length, through the loop parameter. In the next section, we show the detail of the present theory for the system consisting of A◦2 and B◦3 units. We obtain the free energy change between a reference state and a final solution and get equilibrium conditions for cluster formation. We obtain the gelation



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criterion and investigate the post-gel region by using the present theory. In this paper, we study the post-gel region by the recursive method of Macosko and Miller. 30,31 Furthermore, in section 3, we calculate the shear modulus by using the theory and compare with the experimental results. 12

2. Theory of associating polymer solutions with loops 2.1 Model A2

B3

B1

Hidden functional groups

Figure 1: Schematic figure of clusters formed by A2 , B3 , and B1 units. In this figure, (5, 3, 2)-cluster is shown. In B1 units, there are two hidden functional groups each for A and B.

In this section, we present the detail of our theory. The list of symbols used in the theory is shown in the supporting information. We consider bifunctional polymer units carrying A functional groups at their ends denoted as A◦2 and trifunctional polymer units carrying B functional groups denoted as B◦3 . In the present theory, we consider small loops, which are formed when both end groups of the A◦2 unit associate with B functional groups belonging to the same B◦3 unit. We regard the loop as a pseudomolecule carrying a single effective B functional group denoted as B1 (see figure 1). Thus, we regard the system as a mixture of A2 /B3 /B1 /solvent instead of the original A◦2 /B◦3 /solvent system, where the B3 units have the same structure as B◦3 units, but the total number of B3 units is reduced by the loop 4 ACS Paragon Plus Environment

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formation. Here, we use a lattice theory to derive the free energy of the system. First, we divide the total volume V of the system into small cells of size s of the monomeric unit on a chain. The total number of the microscopic cells is Ω ≡ V /s3 . Let us consider the number of statistical units on an A◦2 unit, a B◦3 unit and solvent are nA , nB and 1, respectively. The number of statistical units on a B1 unit is nA + nB . We assumed that the size of a statistical unit of polymers is the same as that of solvent. Let NA2 , NB3 and NB1 be the number of A2 , B3 and B1 units, respectively, which are determined by the equilibrium condition of the clusters. Let N0 be the number of solvent. Now we define a cluster of type (a, b, l) to consist of a A2 units, b B3 units and l B1 units. We have assumed that the structure of the cluster is a tree-like structure. Therefore, in the present theory, we consider only the smallest loops defined above. The number of (a, b, l)-clusters is Na,b,l and the volume fraction ϕa,b,l is

ϕa,b,l =

na,b,l Na,b,l Ω

(1)

where na,b,l ≡ anA + bnB + l(nA + nB ). The total number of the cells Ω is written as

Ω = N0 +

∑

na,b,l Na,b,l + nA NAG2 + nB NBG3 + (nA + nB )NBG1

(2)

a,b,l

where NAG2 , NBG3 and NBG1 are the number of A2 , B3 and B1 units in the gel network, respectively. The volume fraction of all polymers is written as ∑ ϕ=

a,b,l

na,b,l Na,b,l + nA NAG2 + nB NBG3 + (nA + nB )NBG1 Ω


(3)


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2.2 Free energy We obtain the free energy change at temperature T in three steps shown in figure 2.

β∆F =

∑

Na,b,l ln ϕa,b,l + N0 ln(1 − ϕ) + χϕ(1 − ϕ)Ω

a,b,l

+

∑

( ∆a,b,l Na,b,l + ∆ℓ

a,b,l

∑

) lNa,b,l

+ ∆F G (ϕ)

(4)

a,b,l

where β is defined by β ≡ 1/kB T . Here, kB is the Boltzmann constant. The parameter χ is the Flory’s interaction parameter. In our theory, we assume that the parameter between A2 units and the solvent χ(A2 ;S) and that between B3 units and the solvent χ(B3 ;S) are equal to χ: χ(A2 ;S)=χ(B3 ;S)= χ. On the other hand, the parameter between A2 units and B3 units χ(A2 ;B3 ) is assumed to be zero: χ(A2 ;B3 )= 0. The quantity ∆a,b,l is the free energy change accompanying the formation of an (a, b, l)-cluster from separate a A2 units, b B3 units and l B1 units. Here, ∆1,0,0 = ∆0,1,0 = ∆0,0,1 = 0. The quantity ∆ℓ is the free energy change accompanying the formation of a single B1 unit from an A2 unit and a B3 unit. The quantity ∆F G (ϕ) is the free energy of gel network. This term appears only in the gel region. By differentiation of the free energy eq 4 with respect to the number of (a, b, l)-clusters, the chemical potential is given by

β∆µa,b,l = 1 + ∆a,b,l + l∆ℓ + ln ϕa,b,l − na,b,l ν S + χna,b,l (1 − ϕ)2 + where ν S = (N0 +

∑ a,b,l

na,b,l ′ (1 − ϕ)∆F G (ϕ) (5) Ω

Na,b,l )/Ω is the number concentration of clusters and molecules that ′

possess the degree of freedom for translational motion. The quantities ∆F G (ϕ) is defined by differentiation of ∆F G (ϕ) with respect to ϕ. Now the multiple equilibrium conditions for the (a, b, l)-cluster formation

∆µa,b,l = a∆µ1,0,0 + b∆µ0,1,0 + l∆µ0,0,1


(6)

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(a) Solvent

(b)

Loop Formation Solvent

(c)

Cluster Formation

Mixing (d)

Figure 2: Schematic representation of the free energy change between the reference states and the final solution. (a) The reference state that polymers and solvent molecules being separated. (b) The state that A2 units, B3 units, B1 units and solvent exist. (c) The state that various clusters are formed. (d) The final solution made by mixing various clusters.



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lead to the population distribution,

ϕa,b,l = exp(a + b + l − 1 − ∆a,b,l )ϕa1,0,0 ϕb0,1,0 ϕl0,0,1

(7)

We next consider the energy change ∆a,b,l . We can separate ∆a,b,l into three terms conf bond ∆a,b,l = ∆comb a,b,l + ∆a,b,l + ∆a,b,l

(8)

where ∆comb a,b,l is the contribution of the free energy change arising from the selection of the primary molecules that are bound together into the (a, b, l)-cluster, ∆conf a,b,l is that from the conformational change of the (a, b, l)-clusters, and ∆bond a,b,l is that from the bond formation. The free energy change of the combination ∆comb a,b,l can be derived from the similar procedure of the classical gelation theory. 27 This term is given by [

∆comb a,b,l

ωa,b,l = − ln a b l a!ω1,0,0 b!ω0,1,0 l!ω0,0,1

] (9)

Here, ωa,b,l is given by

ωa,b,l =

2a 3b a!(2b)! (2b + 1 − a)!(a − b − l + 1)!

(10)

The details are shown in the supporting information. The free energy change of the conformation ∆conf a,b,l is given by ∆conf a,b,l = −

) 1 ( dis dis dis dis − lS0,0,1 − bS0,1,0 Sa,b,l − aS1,0,0 kB

(11)

dis where Sa,b,l is the lattice-theoretical entropy of disorientation given by

[

dis Sa,b,l

] na,b,l z(z − 1)na,b,l −2 = kB ln + l∆Sℓ σa,b,l ena,b,l −1


(12)

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where z is the lattice coordination number. The quantity σa,b,l is the symmetry number of the (a, b, l)-cluster, being most probably σa,b,l = 1. Here, we consider the term which represents the conformational entropy change between the reference conformation (an open chain) and excite conformation (a loop) of the A2 unit denoted as ∆Sℓ . The free energy change of bonding ∆bond a,b,l is given by ∆bond a,b,l = β(a + b + l − 1)∆f0

(13)

because there are (a + b + l − 1) bonds in the (a, b, l)-cluster, where ∆f0 < 0 is the free energy change of the bond formation. From eqs 8, 9, 11, and 13, we obtain ∆a,b,l . Combining the results and eq 7, the number concentration of (a, b, l)-clusters is given by

νa,b,l =

2a 3b (2b)! a b l λa+b+l−1 ν1,0,0 ν0,1,0 ν0,0,1 b!l!(2b + 1 − a)!(a − b − l + 1)!

(14)

where λ is the association constant

λ(T ) ≡ (z − 1) exp(−β∆f0 )

(15)

Here, the model lattice coordination number z is replaced by (z − 1). Meanwhile, from the classical gelation theory, νa,b,l is expressed in another way

νa,b,l =

(2b)! 3νB3 w3b−1 (1 − w3 )l b!l!(2b + 1 − a)!(a − b − l + 1)! (1 − pA )a−b−l+1 paB (1 − pB )2b+1−a × pb+l−1 A

(16)

The details are shown in the supporting information. The quantity νB3 is the number concentration of the B3 units and w3 is the probability for a randomly chosen B functional group to belong to B3 units. Similarly, let νA2 be the number concentration of A2 units. The quantities pA and pB are the pseudo extent of association of functional groups A and B,



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respectively, that exclude the hidden functional groups in B1 units(see figure 1). From eqs 14 and 16, we find

ψA pA = ψB pB = λ(1 − pA )ψA (1 − pB )ψB

(17)

where ψA = 2νA2 and ψB = 3νB3 /w3 is the pseudo number concentration of functional groups A and B, respectively.

2.3 Loop formation We consider the chemical equilibrium condition for the formation of a single B1 unit

∆µ0,0,1 = ∆µ1,0,0 + ∆µ0,1,0

(18)

The free energy change ∆ℓ can be separated into three terms ∆ℓ = ∆comb + ∆conf + ∆bond ℓ ℓ ℓ

(19)

The free energy change arising from the number of different ways of constituting the B1 unit by primary molecules ∆comb is given by ℓ ∆comb = − ln 6 ℓ

(20)

For the conformational free energy, ∆conf is given by ℓ ) 1 ( dis dis dis S0,0,1 − S1,0,0 − S0,1,0 kB ( ) nA + nB z − 1 ∆Sℓ = − ln − nA nB e kB

∆conf =− ℓ


(21)

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The free energy change of bonding ∆bond is given by ℓ ∆bond = 2β∆f0 ℓ

(22)

ν0,0,1 = 6ζλ2 ν1,0,0 ν0,1,0

(23)

From eqs 5 and 18–22, we then find

where ζ is loop parameter and given by ( ζ ≃ (z − 1)

−1

exp

∆Sℓ kB

)

( =

3 2πls2

)3/2 vs n−3/2 s

(24)

where ns is the total number of segments composing a single A◦2 unit, ls is the length of the segments, and vs is a small volume in which contains both ends of the A◦2 unit when the loop is formed. For derivation of the formula, we obtain the probability P that the A2 unit be in a configuration corresponding to a loop as P ∼ (3/2πns ls2 )3/2 vs◦ , and regard vs◦ to be proportional to (z − 1) considering the number of way to get into the space: vs◦ = (z − 1)vs . 32 The loop parameter is obtained by using the relation ∆Sℓ = kB ln P . Then from eqs 16, 17 and 23, we find

w3 =

1 1 + hpB

(25)

where h is given by

h ≡ ζλ(1 − pA )(1 − pB )


(26)


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From eq 25, we finally find the relation between the pseudo number density of functional groups A and B (ψA and ψB ) and the real number density (ψA◦ and ψB◦ ) as follows:

ψA =

1 + hpB ψ◦ 1 + h(2pA + pB ) A

(27)

1 + hpB ◦ ψ 1 + 3hpB B

(28)

ψB =

Additionally, the relation between the pseudo extent of association of functional groups A and B (pA and pB ) and the real extent of association (p◦A and p◦B ) are given by p◦A =

1 + h(2 + pB ) pA 1 + h(2pA + pB )

(29)

1 + h(2 + pB ) pB 1 + 3hpB

(30)

p◦B =

respectively. Then from eqs 17, 27 and 28, we obtain pA and pB as functions of λ(T ), ζ, ψA◦ and ψB◦ .

2.4 Sol-gel transition Now we study sol-gel transition theoretically by using the present theory. The gel point is defined as the point at which the weight-average molecular weight of the clusters becomes infinite. 33 To find the gel point, we use the branching coefficient α. The gel point condition is given by

(f − 1)α = 1


(31)

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where f is the junction multiplicity. In this paper, we regard B3 units as network junctions and set to be f = 3. We find the branching coefficient α is given by

α = w3 pA pB

(32)

(see supporting information for details). Therefore, from eqs 25, 31, and 32, the gel condition gives

1 + pB (h − 2pA ) = 0

(33)

Figure 3 shows the gel regions obtained by this theory. The quantities ϕA , ϕB and ϕ0 are the volume fractions of A◦2 , B◦3 and solvent, respectively. In this calculation, we assume that ns = nA = nB = 50. We set the association constant and the loop parameter are λ = 2000 and ζ = 0.001, respectively. In the figure, the solid line indicates the sol-gel transition line for the parameters given above. The dashed line indicates the loop-free gel region (λ → ∞, ζ = 0). In the extreme condition, the gel appears under the condition 1/3 < ψA◦ /(ψA◦ + ψB◦ ) < 2/3. According to eq 24, the loop parameter ζ increases with decreasing ns . Figure 4 shows the dependence of gel regions on the loop parameter. In this calculation, for the confirmation of the effect of loop formation clearly, we assume that ns = nA and the factor of proportionality of ζ is set so as to be ζ = 0.001 when nA = 50. We fix λ = 2000 and nB = 50. The gel region is reduced with decreasing nA because the loop formation is promoted with decreasing nA since the loop parameter ζ increases. On the other hand, the gelation concentration of A◦2 units decreases with decreasing nA . This is because the number concentration of the functional group A increases with decreasing nA at the fixed volume fraction ϕA .



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0.0 1.0 0.2

Sol

0.8

0.4

0.6

Gel 0.6

0.4

0.8

0.2 loop-free system

1.0

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Figure 3: The gel regions on the ternary phase plane of the A◦2 , B◦3 and solvent system. We fix the association constant λ = 2000, the loop parameter ζ = 0.001, and the number of statistical units on each unit ns = nA = nB = 50. The dashed line indicates the loop-free gel region (λ → ∞ and ζ = 0).

0.0 1.0

0.2

Sol 0.4 Decrease in

0.6

0.8

0.6

Gel 0.4

0.8

0.2

1.0 0.0

0.0 0.2

0.4

0.6

0.8

1.0

Figure 4: The gel regions on the ternary phase plane of the A◦2 , B◦3 and solvent system for the number of statistical units on an A◦2 unit nA = 10, 30 and 50. We fix the association constant λ = 2000 and the number of setatistical units on a B◦3 unit nB = 50.


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2.5 Post-gel region The post-gel region is investigated in this section. In this study, we consider that the relation between the total polymer concentrations and the average extent of association in the postgel region is given by eqs 17, 27 and 28. We calculate the gel fraction and the number concentration of elastically effective chains by using the theory developed by Macosko and Miller. 30,31 Using the theory, the gel fraction wG is expressed by [ ] wG = 1 − wA2 P (FA out )2 + wB3 P (FB out )3 + wB1 P (FB out )

(34)

where wA2 , wB3 and wB1 are the weight fraction of A2 , B3 and B1 units, respectively. The quantity P (FA out ) is the probability of the path emerging from an A2 unit is the start of a finite cluster, and given by

P (FA out ) =

1 − pA pB w3 (2 − pB ) pA p2B w3

(35)

Similarly, we define P (FB out ) for the path emerging from an B3 unit. The quantity P (FB out ) is expressed by

P (FB out ) =

1 − pA pB w3 pA pB w3

(36)

Here, we assume that the all microscopic cells have the same weight. Figure 5 shows the gel fraction wG for λ = 2000, nA = nB = 50 and ζ = 0.001. In figure 5, the dashed line indicates the sol-gel transition line. The gel fraction is increased as the point separates from the sol-gel transition line in the gel region. In the gel network, chains that contribute to the elasticity are called elastically effective chains. Scanlan and Case introduced a criterion for the chains theoretically. 34,35 Here, we consider path number of a junction: the number of independent paths connected to the 15 ACS Paragon Plus Environment


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1

0.0 1.0

0.2

0.8

0.4

0.6 0

Gel

0.6

0.4

0.8

0.2

Sol 1.0 0.0

0.0 0.2

0.4

0.6

0.8

1.0

Figure 5: The gel fraction on the ternary phase plane of the A◦2 , B◦3 and solvent system. We fix the association constant λ = 2000, the number of statistical units on each unit nA = nB = 50 and the loop parameter ζ = 0.001. The dashed line indicates the sol-gel transition line. skeletal gel network from the junction. Let µi,k be the number concentration of junctions of multiplicity of k with path number i. The number concentration of elastically effective junctions µeff is generally given by

µeff =

k ∞ ∑ ∑

µi,k

(37)

k=3 i=3

According to Scanlan and Case, the chain is elastically effective if both its ends are connected to the elastically effective junctions. Therefore, the number concentration of the effective chains is given by 1 ∑∑ iµi,k 2 k=3 i=3 ∞

νeff =

k

(38)

In our system, we regard B3 units as junctions. Therefore, we consider that the junction multiplicity is three. Let Pj be the probability that exactly j of three paths are infinite. It


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is expressed by

Pj = 3 Cj P (FB out )3−j [1 − P (FB out )]j

(39)

µeff = P3 νB3

(40)

[(2pA − h)pB − 1]3 p3A p3B

(41)

Then µeff is given by

From eqs 36 and 39,

P3 =

Here, we combine these results with our theory. From eqs 17 and 25,

νB3 =

pA w3 ψB = 3 3λ(1 − pA )(1 − pB )(1 + hpB )

(42)

Considering that each elastically effective junction has three elastically effective chains, νeff is given by

νeff =

[(2pA − h)pB − 1]3 2λp2A p3B (1 − pA )(1 − pB )(1 + hpB )

(43)

The condition at νeff = 0 is in agreement with gel condition given by eq 33. Figure 6 shows the number concentration of elastically effective chains as a function of the polymer concentration for λ = 2000, nA = nB = 50 and ζ = 0.001. The dashed line indicates the sol-gel transition line. In the gel region, the elastically effective chains exist. The number concentration of elastically effective chains increases with the progress of gelation.



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2.6 10-3

0.0 1.0

0.2

0.8

0.4

0.6 0

Gel

0.6

0.4

0.8

0.2

Sol 1.0 0.0

0.0 0.2

0.4

0.6

0.8

1.0

Figure 6: The number concentration of elastically effective chains on the ternary phase plane of the A◦2 , B◦3 and solvent system. We fix the association constant λ = 2000, the number of statistical units on each unit nA = nB = 50 and the loop parameter ζ = 0.001. The dashed line indicates the sol-gel transition line.

3. Comparison of theoretical and experimental results We calculate the shear modulus G0 from the number concentration of the effective chains νeff obtained in the previous section by using following expression: G0 = Fνeff RT

(44)

where R is the gas constant and prefactor F is F = 1 for the affine network theory, while F = 1/3 for the phantom network theory. 1,36–38 We compare the modulus with experimental results. 12 In the experimental system, dynamic covalent bonds are formed between tris[(4formylphenoxy)methyl]ethane and bis(acylhydrazine)-functionalized poly(ethylene oxide) (PEO) in the presence of N ,N -Dimethylformamide and acetic acid (figure 7). 11,12 The physical gel network is formed reversibly under the appropriate condition. The viscoelastic properties obtained by the dynamic mechanical analysis are described by the Maxwell model with a single relaxation time. In particular, for the well-gelated samples (the concentration ≥ 4.2 wt% at 298 K), the plateau of the storage modulus is observed in a wide frequency range 18 ACS Paragon Plus Environment

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(10−1 ≤ ω ≤ 102 rad/s), since the time scale of bond dissociation is much slower than those of observation and the conformational change of polymer chains in the network. In the present paper, we compare our theoretical results with such plateau modulus (at 102 rad/s) obtained by experiment. The number average molecular weight of PEO is Mn ≃ 2000. 12 Therefore, it seems that many loops are formed in the network. bis(acylhydrazine)-functionalized poly(ethylene oxide)

tris[(4-formylphyenoxy) methyl] ethane

Figure 7: The unit constituting the cluster in the experiment. For the comparison between theoretical and experimental results, we now determine the parameters: the association constant λ and the loop parameter ζ. The association constant is determined by the equilibrium constant Keq obtained in the experiment given above. 12 In the experiment, the quantity Keq is expressed by [AB] = Keq [A][B]

(45)

where [A] and [B] are the concentrations of unassociated functional groups A and B, respectively. The quantity [AB] is the concentration of associated A-B pairs. The association is considered to be a second-order association. In the experiment, Keq = 8000 M−1 at 298K. On the other hand, in our theory, for the case of the equimolar functional groups, using eqs



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˜ eq ≡ Keq /s3 is described by 17, 27 and 29, the dimensionless equilibrium constant K ˜ eq = K

1 + h(2 + p) p◦ = λ (1 − p◦ )2 ψ ◦ 1 + hp

(46)

where p◦ ≡ p◦A = p◦B , ψ ◦ ≡ ψA◦ = ψB◦ and p ≡ pA = pB . Here, we set the size of a microscopic cell as s3 = 1 M−1 ≃ 1.7 nm3 to relate experimental results to the lattice theory. Figure 8 30000

20000 0.015 0.010 0.005

8000

10000 5500 0 0

5

10

15

0.014

20

Polymer Concentration [wt%]

˜ eq on the concentration. The Figure 8: Dependence of dimensionless equilibrium constant K association constant λ and the loop parameter ζ are varied as indicated. ˜ eq as a function of polymer concentration. When the loop parameter ζ is zero, the shows K ˜ eq is independent of the polymer concentration. The association is considered to be value K a second-order association if ζ = 0. On the other hand, when the loop parameter ζ is a finite ˜ eq increases with decreasing in the polymer concentration. Therefore, value greater than 0, K the association is promoted in the low concentration regime by the loop formation. ˜ eq by considering the process of the association as shown We explain the behavior of K in figure 9. First, we consider a B3 unit having a single dangling A2 unit. The unassociated B group on the B3 unit associates in either way: the B group associates with an A group belonging to other clusters shown in figure 9 (a) or with the A group belonging to the dangling A2 unit (b) in the figure. The smallest loops are formed by the process (b). Taking into account the process (b), the unassociated B group is easier to encounter an unassocated 20 ACS Paragon Plus Environment

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(a)

(b)

Figure 9: The schematic figure of the process of the association. A group within the low concentration region. Therefore, the loop formation causes the ˜ eq increases promotion of association in the low concentration region. This is the reason why K in the region. In the experiment, the equilibrium constant Keq is estimated to be 8000 M−1 at 298K. In contrast, if the association constant λ is set to be 8000, and the loop parameter ζ is a finite ˜ eq becomes much larger than 8000 in the low concentration region. For value in our theory, K this reason, we set ζ = 0.014, as we discuss later, and λ = 5500 as adjustable parameters. Figure 10 shows that the shear modulus as a function of polymer concentration where the solution contains equimolar functional groups A and B. The symbol indicates experimental data. The solid line indicates theoretical results considering loops by using phantom network theory. The dotted line indicates theoretical results in which loops are completely absent (ζ = 0). We find that the theoretical results considering loops are in good agreement with the experimental results. For this comparison, the loop parameter is treated as an adjustable parameter. We explain the estimation of the loop parameter ζ from a molecular theory. The loop parameter



ζ is given by eq 24, where ls , ns and vs◦ are the segment length, the number of segments, and the small volume in which contains both ends of single A2 unit, respectively. The segment length ls is assumed to be ls = 7.6 Å based on the Kuhn length for PEO. 39 For estimation of the number of segments ns , we calculate the contour length of an A◦2 unit as Lmax = 200 Å where the length of an EO unit is assumed to be 4.4 Å. Then, the number of segments is ns = Lmax /ls ≃ 26. The volume vs◦ is estimated as the volume of the B3 unit: 3

vs◦ = 4π(LB3 )3 /3 Å where LB3 is the length of the arm of the B◦3 unit. Here, we calculate the length LB3 = 7.9 Å based on the molecular structure. From these results, we obtain ζ ≃ 0.012, which is consisted with ζ = 0.014 we used for the comparison. 10 10 10

G0 [Pa]

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10 10 10 10 10

6

5

4

3

2

1

0

298 K

-1

1

10


Figure 10: The shear modulus as a function of polymer concentration. The symbols indicate experimental data. The lines indicate theoretical results. We fix T = 298 K. In the experimental condition, 12 the extent of association is about 0.9 in the regime. The shear modulus obtained by the theory without loops is a very high value. On the other hand, the shear modulus experimentally obtained increases rapidly near the point where the polymer concentration is 4 wt%. It is assumed that this is due to the change in a state of association. In our theory, we classify B3 units into six categories by the state of association. Figure 11 shows the relative populations of their categories as functions of the polymer concentration. In dilute regime, The B3 units composing loops are dominant in clusters. With increasing polymer concentration, the population of those B3 units decreases 22 ACS Paragon Plus Environment

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and the population of B3 units connected to other B3 units increases. As a result, the network formation is promoted in high concentration region. The network formation causes rapid increasing in the shear modulus. Associtation categories of B3 units

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1.0

298 K 0.8

0.6

0.4

0.2

0.0

1

10


Figure 11: The relative populations of six categories of B3 units classified by the state of association as functions of polymer concentration.

4. Conclusions In the present paper, we have constructed a statistical-mechanical theory of thermo-reversible gelation considering loops for the system consisting of bifunctional polymer units and trifunctional units carrying associative groups capable of forming reversible pairwise bonds. We find that the loops considered in our theory are often formed in dilute regime. At the molecular level, when the loops are formed in the network, they behave as ends of the network and lower the connectivity. Therefore, the number concentration of elastically effective chains is decreased. From this point of view, the loop behaves like a dangling chain, however, the difference between loops and dangling chains is that the loops have hidden functional groups. Hence, the loop formation causes decreasing in the elasticity in spite of a progress of the association. In the present study, the amount of loops is overestimated by neglecting the larger-scale cycle formation. In our theory, the larger cycles only allow within the gel 23 ACS Paragon Plus Environment


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implicitly, similar to Flory’s treatment. It is difficult to consider all of these cyclic structures in our theory. However, considering the large cycles as other types of monomer, we can partially treat cycles in our theory. In this article, we construct the theory for the system consisting of trifunctional polymer units and bifunctional polymer units. Our theory is able to apply for other systems, for example, the system with fixed multiple junction, 9 and the system consisting of tetrafunctional polymer units. 40 More detailed studies of other physical gels with well-defined structure will be reported in later publications.

Acknowledgement We wish to acknowledge valuable discussions with Prof. F. Tanaka and Dr. H. Kojima.

Supporting Information Available The list of symbols used in the present theory and the detailed description of classical gelation theory of A2 /Bf /B1 system.

This material is available free of charge via the Internet at

http://pubs.acs.org/.

References (1) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (2) Erman, B.; Mark, J. E. Structures and Properties of Rubberlike Networks; Oxford University Press: Oxford, U.K., 1997. (3) Rubinstein, M.; Colby, R. Polymers Physics; Oxford University Press: Oxford, U.K., 2003.


Page 24 of 29

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(4) Sakai, T.; Matsunaga, T.; Yamamoto, Y.; Ito, C.; Yoshida, R.; Suzuki, S.; Sasaki, N.; Shibayama, M.; Chung, U. Design and Fabrication of a High-Strength Hydrogel with Ideally Homogeneous Network Structure from Tetrahedron-like Macromonomers. Macromolecules 2008, 41, 5379–5384. (5) Akagi, Y.; Matsunaga, T.; Shibayama, M.; Chung, U.; Sakai, T. Evaluation of Topological Defects in Tetra-PEG Gels. Macromolecules 2010, 43, 488–493. (6) Lange, F.; Schwenke, K.; Kurakazu, M.; Akagi, Y.; Chung, U.; Lang, M.; Sommer, J.U.; Sakai, T.; Saalwächter, K. Connectivity and Structural Defects in Model Hydrogels: A Combined Proton NMR and Monte Carlo Simulation Study. Macromolecules 2011, 44, 9666–9674. (7) Zhou, H.; Woo, J.; Cok, A. M.; Wang, M.; Olsen, B. D.; Johnson, J. A. Counting Primary Loops in Polymer Gels. Proc. Natl. Acad. Sci. U. S. A. 2012, 109, 19119– 19124. (8) Zhou, H.; Schön, E.-M.; Wang, M.; Glassman, M. J.; Liu, J.; Zhong, M.; Díaz Díaz, D.; Olsen, B. D.; Johnson, J. A. Crossover Experiments Applied to Network Formation Reactions: Improved Strategies for Counting Elastically Inactive Molecular Defects in PEG Gels and Hyperbranched Polymers. J. Am. Chem. Soc. 2014, 136, 9464–9470. (9) Werten, M. W. T.; Teles, H.; Moers, A. P. H. A.; Wolbert, E. J. H.; Sprakel, J.; Eggink, G.; de Wolf, F. A. Precision Gels from Collagen-Inspired Triblock Copolymers. Biomacromolecules 2009, 10, 1106–1113. (10) Skrzeszewska, P. J.; de Wolf, F. A.; Werten, M. W. T.; Moers, A. P. H. A.; Stuart, M. A. C.; van der Gucht, J. Physical Gels of Telechelic Triblock Copolymers with Precisely Defined Junction Multiplicity. Soft Matter 2009, 5, 2057–2062. (11) Deng, G.; Tang, C.; Li, F.; Jiang, H.; Chen, Y. Covalent Cross-Linked Polymer Gels



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

with Reversible Sol-Gel Transition and Self-Healing Properties. Macromolecules 2010, 43, 1191–1194. (12) Liu, F.; Li, F.; Deng, G.; Chen, Y.; Zhang, B.; Zhang, J.; Liu, C.-Y. Rheological Images of Dynamic Covalent Polymer Networks and Mechanisms behind Mechanical and Self-Healing Properties. Macromolecules 2012, 45, 1636–1645. (13) Harris, F. E. Ring Formation and Molecular Weight Distributions in Branched-Chain Polymers. I. J. Chem. Phys. 1955, 23, 1518–1525. (14) Hoeve, C. A. J. Molecular Weight Distribution of Thermally Polymerized Triglyceride Oils. II. Effect of Intramolecular Reaction. J. Polym. Sci. 1956, 21, 11–18. (15) Kilb, R. W. Dilute Gelling Systems. I. The Effect of Ring Formation on Gelation. J. Phys. Chem. 1958, 62, 969–971. (16) Gordon, M.; Scantlebury, G. R. Theory of Ring-Chain Equilibria in Branched NonRandom Polycondensation Systems, with Applications to POCl3 /P2 O5 . Proc. R. Soc. London, Ser. A 1966, 292, 380–402. (17) Dušek, K.; Gordon, M.; Ross-Murphy, S. B. Graphlike State of Matter. 10. Cyclization and Concentration of Elastically Active Network Chains in Polymer Networks. Macromolecules 1978, 11, 236–245. (18) Spouge, J. L. Equilibrium Ring Formation in Polymer Solutions. J. Stat. Phys. 1986, 43, 143–196. (19) Sarmoria, C.; Miller, D. R. Spanning-Tree Models for Af Homopolymerizations with Intramolecular Reactions. Comput. Theor. Polym. Sci. 2001, 11, 113–127. (20) Erukhimovich, I.; Thamm, M. V.; Ermoshkin, A. V. Theory of the Sol-Gel Transition in Thermoreversible Gels with Due Regard for the Fundamental Role of Mesoscopic


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Cyclization Effects. 1. Thermodynamic and Structural Characteristics of the Gel Phase. Macromolecules 2001, 34, 5653–5674. (21) Dobrynin, A. V. Phase Diagram of Solutions of Associative Polymers. Macromolecules 2004, 37, 3881–3893. (22) Dušek, K.; Dušková-Smrčková, M.; Yang, J.; Kopeček, J. Coiled-Coil Hydrogels: Effect of Grafted Copolymer Composition and Cyclization on Gelation. Macromolecules 2009, 42, 2265–2274. (23) Lang, M.; Schwenke, K.; Sommer, J.-U. Short Cyclic Structures in Polymer Model Networks: A Test of Mean Field Approximation by Monte Carlo Simulations. Macromolecules 2012, 45, 4886–4895. (24) Cohen, R. J.; Benedek, G. B. Equilibrium and Kinetic Theory of Polymerization and the Sol-Gel Transition. J. Phys. Chem. 1982, 86, 3696–3714. (25) Semenov, A. N.; Rubinstein, M. Thermoreversible Gelation in Solutions of Associative Polymers. 1. Statics. Macromolecules 1998, 31, 1373–1385. (26) Tanaka, F.; Stockmayer, W. H. Thermoreversible Gelation with Junctions of Variable Multiplicity. Macromolecules 1994, 27, 3943–3954. (27) Stockmayer, W. H. Theory of Molecular Size Distribution and Gel Formation in Branched-Chain Polymers. J. Chem. Phys. 1943, 11, 45–55. (28) Flory, P. J. Thermodynamics of High Polymer Solutions. J. Chem. Phys. 1942, 10, 51–61. (29) Huggins, M. L. Some Properties of Solutions of Long-Chain Compounds. J. Phys. Chem. 1942, 46, 151–158. (30) Macosko, C. W.; Miller, D. R. A New Derivation of Average Molecular Weights of Nonlinear Polymers. Macromolecules 1976, 9, 199–206. 27 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(31) Miller, D. R.; Macosko, C. W. A New Derivation of Post Gel Properties of Network Polymers. Macromolecules 1976, 9, 206–211. (32) Jacobson, H.; Stockmayer, W. H. Intramolecular Reaction in Polycondensations. I. The Theory of Linear Systems. J. Chem. Phys. 1950, 18, 1600–1606. (33) Flory, P. J. Molecular Size Distribution in Three Dimensional Polymers. I. Gelation. J. Am. Chem. Soc. 1941, 63, 3083–3090. (34) Scanlan, J. The Effect of Network Flaws on the Elastic Properties of Vulcanizates. J. Polym. Sci. 1960, 43, 501–508. (35) Case, L. C. Branching in Polymers. I. Network Defects. J. Polym. Sci. 1960, 45, 397– 404. (36) Flory, P. J.; Gordon, M.; McCrum, N. G. Statistical Thermodynamics of Random Networks [and discussion]. Proc. R. Soc. London, Ser. A 1976, 351, 351–380. (37) James, H. M.; Guth, E. Theory of the Increase in Rigidity of Rubber During Cure. J. Chem. Phys. 1947, 15, 669–683. (38) James, H. M. Statistical Properties of Networks of Flexible Chains. J. Chem. Phys. 1947, 15, 651–668. (39) Kienberger, F.; Pastushenko, V. P.; Kada, G.; Gruber, H. J.; Riener, C.; Schindler, H.; Hinterdorfer, P. Static and Dynamical Properties of Single Poly(Ethylene Glycol) Molecules Investigated by Force Spectroscopy. Single Mol. 2000, 1, 123–128. (40) Fullenkamp, D. E.; He, L.; Barrett, D. G.; Burghardt, W. R.; Messersmith, P. B. MusselInspired Histidine-Based Transient Network Metal Coordination Hydrogels. Macromolecules 2013, 46, 1167–1174.


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Graphical TOC Entry A2

B3

B1

Loop


Theoretical Study of Network Formation and Mechanical Properties of

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