Phase Behavior and Phase Transitions in AB- and ABA-type Microphase-Separated Block Copolymers

Abstract

Currently held mean-field theories for microphase-separation in AB-type diblock and ABA-type triblock copolymers are reviewed and their limitations are highlighted. Numerical predictions, based on these theories, for the design of such block copolymers are also presented. It is emphasized that the use of a numerical algorithm leading to successful design and synthesis of block copolymers in terms of order–disorder transition temperature (T _ODT) is critically dependent upon the accuracy of the temperature-dependent interaction parameter. Specifically, the available temperature-dependent interaction parameters are often obtained using the molecular weights which are much lower than the molecular weights of the constituent blocks, in spite of the fact that the interaction parameters are molecular weight dependent. Two as yet unresolved issues, finite molecular weight effect and the phase behavior and phase transitions in highly asymmetric block copolymers, are discussed. These issues are fundamental enough to require a fresh look, particularly from a theoretical point of view, because the currently held mean-field theory cannot explain every conceivable phase behavior and phase transitions experimentally observed in block copolymers.

1 Appendix A Derivation of (9)

According to Helfand [7], the free energy difference between the microphase-separated and homogeneous states is given by

$$\begin{array}{rcl} \frac{\Delta {G}_{m}} {V {k}_{\mathrm{B}}T}& =& \frac{1} {V {k}_{\mathrm{B}}T}\int \nolimits \nolimits \mathrm{d\mathbf{r}}\,\left \{\Delta {f}^{{_\ast}}\left ({\varphi }_{ A}\left (\mathbf{r}\right ),{\varphi }_{B}\left (\mathbf{r}\right )\right ) - {\varphi }_{A}\left (\mathbf{r}\right ) \frac{\partial \Delta {f}^{{_\ast}}} {\partial {\varphi }_{A}\left (\mathbf{r}\right )} - {\varphi }_{B}\left (\mathbf{r}\right ) \frac{\partial \Delta {f}^{{_\ast}}} {\partial {\varphi }_{B}\left (\mathbf{r}\right )}\right \} \\ & & -\frac{{n}_{\mathrm{c}}^{t}} {V } \ln {Q}_{\mathrm{c}} - \frac{\alpha \left ({N}_{A}\bar{{v}}_{0A}\right )\left ({N}_{B}\bar{{v}}_{0B}\right )} {{\left [\left ({N}_{A}\bar{{v}}_{0A}\right ) + \left ({N}_{B}\bar{{v}}_{0B}\right )\right ]}^{2}} \\ \end{array}$$

(A1)

in which Δf ^∗ appearing inside the bracket of the first term on the right-hand side is the free energy density of a mixture having the local volume fractions φ _A (r) and φ _B (r) at position r, which are different from the average volume fractions of A and B blocks in the homogeneous state. The second term on the right-hand side of (A1) represents the conformational entropy contribution, $S = -{k}_{\mathrm{B}}\ln {\left ({Q}_{\mathrm{c}}\right )}^{{n}_{\mathrm{c}}^{t} }$, and the last term on the right-hand side represents the Flory–Huggins interaction energy density in the homogeneous state, G _o = αf(1 − f) in which α (having the units of mol ∕ cm³) is the interaction parameter which is related to the Flory–Huggins interaction parameter χ by α = χ ∕ V _ref with V_ref being the molar volume of a reference component and f is the average volume fractions of block A defined by ${N}_{A}\bar{{v}}_{0A}/\left ({N}_{A}\bar{{v}}_{0A} + {N}_{B}\bar{{v}}_{0B}\right )$ with N _k being the degree of polymerization of component k(k = A or B) and $\bar{{v}}_{ok}$ being the monomeric molar volume of component k(k = A or B).

By defining the first term inside the bracket on the right-hand side of (A1) as ${W}_{\xi }(\mathbf{r})$,

$$-{W}_{\xi }\left (\mathbf{r}\right ) = \frac{1} {{k}_{\mathrm{B}}T}\Delta {f}^{{_\ast}}\left ({\varphi }_{ A}\left (\mathbf{r}\right ),{\varphi }_{B}\left (\mathbf{r}\right )\right ) - {\varphi }_{A}\left (\mathbf{r}\right ) \frac{\partial \Delta {f}^{{_\ast}}} {\partial {\varphi }_{A}\left (\mathbf{r}\right )} - {\varphi }_{B}\left (\mathbf{r}\right ) \frac{\partial \Delta {f}^{{_\ast}}} {\partial {\varphi }_{B}\left (\mathbf{r}\right )} $$

(A2)

we have

$$\frac{\Delta {G}_{m}} {V {k}_{\mathrm{B}}T} = -\frac{1} {V }\int \nolimits \nolimits \mathrm{d\mathbf{r}}\,{W}_{\xi }\left (\mathbf{r}\right ) -\frac{{n}_{\mathrm{c}}^{t}} {V } \ln {Q}_{\mathrm{c}} - \frac{\alpha \left ({N}_{A}\bar{{v}}_{0A}\right )\left ({N}_{B}\bar{{v}}_{0B}\right )} {{\left [\left ({N}_{A}\bar{{v}}_{oA}\right ) + \left ({N}_{B}\bar{{v}}_{oB}\right )\right ]}^{2}}$$

or

$$\begin{array}{rcl} \frac{\Delta {G}_{m}} {{k}_{\mathrm{B}}T\,{n}_{\mathrm{c}}^{t}}& =& - \frac{1} {{n}_{\mathrm{c}}^{t}} \int \nolimits \nolimits \mathrm{d\mathbf{r}}\,{W}_{\xi }\left (\mathbf{r}\right ) -\ln {Q}_{\mathrm{c}} - \frac{V } {{n}_{\mathrm{c}}^{t}}\, \frac{\alpha \left ({N}_{A}\bar{{v}}_{0A}\right )\left ({N}_{B}\bar{{v}}_{0B}\right )} {{\left [\left ({N}_{A}\bar{{v}}_{0A}\right ) + \left ({N}_{B}\bar{{v}}_{0B}\right )\right ]}^{2}} \\ & =& - \frac{1} {{n}_{\mathrm{c}}^{t}} \int \nolimits \nolimits \mathrm{d\mathbf{r}}\,{W}_{\xi }\left (\mathbf{r}\right ) -\ln {Q}_{\mathrm{c}} - \frac{\alpha \left ({N}_{A}\bar{{v}}_{0A}\right )\left ({N}_{B}\bar{{v}}_{0B}\right )} {\left [\left ({N}_{A}\bar{{v}}_{0A}\right ) + \left ({N}_{B}\bar{{v}}_{0B}\right )\right ]} \end{array}$$

(9)

in which use was made of ${V }_{\mathrm{c}} = {N}_{A}\bar{{v}}_{0A} + {N}_{B}\bar{{v}}_{0B}$ and $V = {n}_{\mathrm{c}}^{t}{V }_{\mathrm{c}}$.

2 Appendix B Derivation of (19)

NIA assumes that the interfacial thickness (λ_I) of a block copolymer is much smaller than its domain sizes, D _A and D _B (i.e., λ_I ≪ D _A + D _B ) and thus D = D _A + D _B . Then we have

$$\frac{D} {{N}_{A}\bar{{v}}_{0A} + {N}_{B}\bar{{v}}_{0B}} = \frac{{D}_{A}} {{N}_{A}\bar{{v}}_{0A}} = \frac{{D}_{B}} {{N}_{B}\bar{{v}}_{0B}} $$

(B1)

yielding

$${D}_{A} = \frac{{N}_{A}\bar{{v}}_{0A}D} {{N}_{A}\bar{{v}}_{0A} + {N}_{B}\bar{{v}}_{0B}};\quad {D}_{B} = \frac{{N}_{B}\bar{{v}}_{0B}D} {{N}_{A}\bar{{v}}_{0A} + {N}_{B}\bar{{v}}_{0B}} $$

(B2)

where N _A and N _B are the degrees of polymerization of block chain A and block chain B, respectively, and $\bar{{v}}_{0A}$ and $\bar{{v}}_{0B}$ are the monomeric molar volumes of block chain A and block chain B, respectively.

Now, the area of the interface, S, is defined by [34]

$$S = V/\left (D/2\right ) = 2{n}_{\mathrm{c}}^{t}{V }_{\mathrm{ c}}/D = 2{n}_{\mathrm{c}}^{t}\left ({N}_{ A}\bar{{v}}_{0A} + {N}_{A}\bar{{v}}_{0B}\right )/D $$

(B3)

in which V is the volume of the system, D denotes the size of the microdomains, ${n}_{\mathrm{c}}^{t}$ is the total number of block copolymer chains, and V _c is the molecular volume of one block copolymer chain. In (B3) the following relationships are used: (i) $V = {n}_{\mathrm{c}}^{t}{V }_{\mathrm{c}}$ and (ii) ${V }_{\mathrm{c}} = {N}_{A}\bar{{v}}_{0A} + {N}_{B}\bar{{v}}_{0B}$. Thus, from (B3) we have

$$S/{n}_{\mathrm{c}}^{t} = 2\left ({N}_{ A}\bar{{v}}_{0A} + {N}_{B}{v}_{0B}\right )/D $$

(B4)

Substitution of (B4) into the left-hand side of (17) gives

$$\frac{2\gamma } {{k}_{\mathrm{B}}T}\left ({N}_{A}\bar{{v}}_{0A} + {N}_{B}\bar{{v}}_{0B}\right )/D = - \frac{1} {{n}_{\mathrm{c}}^{t}} \int \nolimits \nolimits \mathrm{d}\mathbf{r}{W}_{\xi }^{I}\left (\mathbf{r}\right ) $$

(B5)

From (16) with m = 2. 5 and c = 0. 085, we have

$$-\ln \,{Q}_{k} = 0.085{\left [{D}_{k}/2{R}_{g,k}\right ]}^{2.5} = 0.085{\left ({6}^{1/2}/2\right )}^{2.5}{\left ( \frac{{D}_{k}} {{b}_{k}{N}_{k}^{1/2}}\right )}^{2.5} $$

(B6)

in which use was made of the definition, ${R}_{g,k} = {b}_{k}{N}_{k}^{1/2}/{6}^{1/2}$. Substitution of (B2) into (B6) gives

$$-\ln {Q}_{A} -\ln {Q}_{B} = 0.141\frac{{\left ({N}_{A}^{1/2}\,\bar{{v}}_{0A}/{b}_{A}\right )}^{2.5} +{ \left ({N}_{B}^{1/2}\,\bar{{v}}_{0B}/{b}_{B}\right )}^{2.5}} {{\left ({N}_{A}\bar{{v}}_{0A} + {N}_{B}\bar{{v}}_{0B}\right )}^{2.5}} {D}^{2.5} $$

(B7)

Finally, substitution of (10), with the aid of (11) and (B7), and (B5) into (9) gives (19).

3 Appendix C Derivation of (34) and (37)–(39)

C.1 Derivation of (34)

According to the RPA, we have

$${\psi }_{i}\left (q\right ) = -\beta \tilde{{S}}_{ij}\left (q\right ){U}_{j}\left (q\right ) = -\beta {S}_{ij}\left (q\right ){U}_{j}^{eff}\left (q\right ) $$

(C1)

where i = A and B, j = A and B for a diblock copolymer, β = 1 ∕ k _B T, and ${U}_{j}^{\mathit{eff }}\left (q\right )$ is given by

$${U}_{j}^{eff}\left (q\right ) = {U}_{ j}\left (q\right ) + {\phi }_{jm}{\psi }_{m}\left (q\right ) + \Theta $$

(C2)

in which Θ is the self-consistent potential acting on all monomers of blocks A and B, and

$${\phi }_{jm} = \left (1/\beta \right ){\chi }_{jm} $$

(C3)

From (C2) we have

$$\begin{array}{rcl}{ U}_{A}^{\mathit{eff }}\left (q\right ) - {U}_{ B}^{\mathit{eff }}\left (q\right )& =& {U}_{ A}\left (q\right ) - {U}_{B}\left (q\right ) + \left ({\phi }_{AA} - {\phi }_{AB}\right ){\psi }_{A}\left (q\right ) + \left ({\phi }_{BA} - {\phi }_{BB}\right ){\psi }_{B}\left (q\right ) \\ & =& {U}_{A}\left (q\right ) - {U}_{B}\left (q\right ) -\left (2/\beta \right )\chi {\psi }_{A}\left (q\right ) \\ \end{array}$$

(C4)

in which use was made of χ _AB = χ _BA = χ, χ _AA = χ _BB = 0, and ψ _A (q) = − ψ _B (q). Also, from (C1), we have

$$\begin{array}{rcl}{ \psi }_{A}\left (q\right )& +& {\psi }_{B}\left (q\right ) = 0 = -\beta \left ({S}_{AA}\left (q\right ){U}_{A}^{\mathit{eff }}\left (q\right ) + {S}_{ AB}\left (q\right ){U}_{B}^{\mathit{eff }}\left (q\right )\right. \\ & +& \left.{S}_{BA}\left (q\right ){U}_{A}^{\mathit{eff }}\left (q\right ) + {S}_{ BB}\left (q\right ){U}_{B}^{\mathit{eff }}\left (q\right )\right ) \\ \end{array}$$

(C5)

Rearrangement of (C5) with the aid of the relationship S _AB (q) = S _BA (q) gives

$${U}_{B}^{eff}\left (q\right )/\,{U}_{ A}^{eff}\left (q\right ) = -\left (\,{S}_{ AA}\left (q\right ) + {S}_{AB}\left (q\right )\right )/\left ({S}_{AB}\left (q\right ) + {S}_{BB}\left (q\right )\right ) $$

(C6)

On the other hand, from (C1) and ψ _A (q) = − ψ _B (q), we have

$$\tilde{{S}}_{AA}\left (q\right ) =\tilde{ {S}}_{BB}\left (q\right ) = -\tilde{{S}}_{AB}\left (q\right ) = -\tilde{{S}}_{BA}\left (q\right ) =\tilde{ S}\left (q\right ) $$

(C7)

Using (C7) and (C1), we have

$$\tilde{S}\left (q\right )\left ({U}_{A}\left (q\right ) - {U}_{B}\left (q\right )\right ) = {S}_{AA}\left (q\right ){U}_{A}^{eff}\left (q\right ) + {S}_{ AB}\left (q\right ){U}_{B}^{eff}\left (q\right ) $$

(C8)

Also, from (C1), we have

$${\psi }_{A}\left (q\right ) = -\beta \tilde{S}\left (q\right )\left ({U}_{A}\left (q\right ) - {U}_{B}\left (q\right )\right ) $$

(C9)

From (C4) and (C9) we have

$${U}_{A}^{eff} - {U}_{ B}^{eff}\left (q\right ) = \left (1 + 2\chi \tilde{S}\left (q\right )\right )\left ({U}_{ A}\left (q\right ) - {U}_{B}\left (q\right )\right ) $$

(C10)

and from (C7) and (C10) we have

$${ U}_{A}^{eff}-{U}_{ B}^{eff}\left (q\right ) = \left (1 + 2\chi \tilde{S}\left (q\right )\right )\left ({S}_{ AA}\left (q\right ){U}_{A}^{eff}\left (q\right ) + {S}_{ AB}\left (q\right ){U}_{B}^{eff}\left (q\right )\right )/\tilde{S}\left (q\right ) $$

(C11)

Finally, from (C6) and (C11) we have

$$1/\tilde{S}\left (q\right ) = \left ({S}_{AA}\left (q\right ) +\; 2{S}_{AB}\left (q\right ) + {S}_{BB}\left (q\right )\right )/\left ({S}_{AA}\left (q\right ){S}_{BB}\left (q\right ) - {S{}_{AB}}^{2}\left (q\right )\right ) - 2\chi $$

(34)

C.2 Derivation of (37) and (39)

S _AA (q) can be expressed by

$$\begin{array}{rcl}{ S}_{AA}\left (q\right )& =& \frac{1} {N}{\int \nolimits \nolimits }_{0}^{fN}{\int \nolimits \nolimits }_{0}^{fN}\exp \left (-y\left \vert i - j\right \vert \right )\mathrm{d}j\mathrm{d}i = \frac{2} {N}{\int \nolimits \nolimits }_{0}^{fN}{\int \nolimits \nolimits }_{0}^{i}\exp \left (-y\left (i - j\right )\right )\mathrm{d}j\mathrm{d}i \\ & =& \frac{2} {N}{\int \nolimits \nolimits }_{0}^{fN}\exp \left (-yi\right )\left (1/y\right )\left (\exp \left (yi\right ) - 1\right )\mathrm{d}i = \frac{2} {yN}{\int \nolimits \nolimits }_{0}^{fN}\left (1 -\exp \left (-yi\right )\right )\mathrm{d}i \\ & =& \frac{2} {yN}\left (fN + \frac{\exp \left (-fNy\right ) - 1} {y} \right ) = \frac{2} {{y}^{2}N}\left (fNy +\exp \left (-fNy\right ) - 1\right ) \\ & =& \frac{2N} {{\left (yN\right )}^{2}}\left (fNy +\exp \left (-fNy\right ) - 1\right ) = \frac{2N} {{x}^{2}} \left (fx +\exp \left (-fx\right ) - 1\right ) = Ng\left (f,x\right )\end{array}$$

(37)

in which y = q ² b ² ∕ 6 and x = yN = q ² Nb ² ∕ 6 are used. S _BB (q) can be obtained from (37) by changing the integration range from f N to (1 − f)N. Further, S _AB (q) can be expressed by

$$\begin{array}{rcl}{ S}_{AB}\left (q\right )& =& \frac{1} {N}{\int \nolimits \nolimits }_{fN}^{N}{ \int \nolimits \nolimits }_{0}^{fN}\exp (-y\left (i - j\right )\mathrm{d}j\mathrm{d}i= \frac{1} {N}\left [{\int \nolimits \nolimits }_{fN}^{N}\exp \left (-yi\right )\mathrm{d}i\right ]\left [{\int \nolimits \nolimits }_{0}^{fN}\exp \left (yj\right )\mathrm{d}j\right ] \\ & =& \frac{1} {N}\left [\frac{-\exp \left (-Ny\right ) +\exp \left (-fNy\right )} {y} \right ]\left [\frac{\exp \left (fNy\right ) - 1} {y} \right ] \\ & =& \frac{1} {{y}^{2}N}\left [-\exp \left (-\left (1 - f\right )Ny\right ) +\exp \left (-Ny\right ) + 1 -\exp \left (-fNy\right )\right ] \\ & =& \frac{N} {2{\left (yN\right )}^{2}}\left [\begin{array}{l} 2\left \{Ny +\exp \left (-Ny\right ) - 1\right \} - 2\left \{fNy +\exp \left (-fNy\right ) - 1\right \}\\ - 2\left \{\left (1 - f \right ) Ny +\exp \left (-\left (1 - f \right ) Ny \right ) - 1 \right \} \end{array} \right ] \\ & =& \frac{N} {2} \left [g\left (1,x\right ) - g\left (f,x\right ) - g\left (1 - f,x\right )\right ] \end{array}$$

(39)

4 Appendix D Derivation of (50)

The difference in free energy density (Φ)( = ΔG _m ∕ k _B T) between the microphase-separated and disordered states is given by [33]

$$\Phi \left [\varphi \right ] = \frac{1} {2\lambda }\left ({{r}_{n}}^{2} -{ {r}_{ 0}}^{2}\right ) + \frac{d} {\tilde{{N}}^{1/2}}\left ({{r}_{n}}^{1/2} -{ {r}_{ 0}}^{1/2}\right ) -\frac{2} {3}n{\theta }_{n}{{a}_{n}}^{3} + \frac{1} {2}n{\eta }_{n}{{a}_{n}}^{4} $$

(D1)

in which $\tilde{N}$ is defined by (48), λ and d are defined in (49), n describes the type of microdomain structure (i.e., n = 1 for lamellar microdomains, n = 3 for cylindrical microdomains, and n = 6 for body-centered-cubic (bcc) lattice), and θ _n and η _n depend on the type of microdomain structure; namely (i) θ₁ = 0 and η₁ = − λ ∕ 2 for n = 1, (ii) θ₃ = μ and η₃ = − λ ∕ 2 for n = 3, and (iii) θ₆ = − 2μ and η₆ = 3λ ∕ 2 for n = 6 in which μ = NΓ ₃ ∕ c ³ with N being the degree of polymerization, Γ ₃ being the third-order vertex function (see (33)), and c being defined in (49). In reference to (D1), r _n and r _o are the inverse susceptibilities of the microphase-separated and disordered states, respectively, and they are given by

$${r}_{0} = \tau + \mathrm{d}\lambda /{\left ({r}_{0}\tilde{N}\right )}^{1/2} $$

(D2)

and

$${r}_{n} = \tau + \mathrm{d}\lambda /{\left ({r}_{n}\tilde{N}\right )}^{1/2} + n\lambda {{a}_{ n}}^{2} $$

(D3)

and a _n is the amplitude of the concentration of block A and is related to the order parameter ψ in the Leibler theory [13]. Also, r _n can be expressed in term of a _n by [33]

$${r}_{n} = {\theta }_{n}{a}_{n} - {\eta }_{n}{{a}_{n}}^{2} $$

(D4)

τ appearing in (D2) and (D3) is defined by

$$\tau = \left [\frac{2{\left (\chi N\right )}_{s} - 2\chi N} {{c}^{2}} \right ] = \left [\frac{F\left ({x}^{{_\ast}},f\right ) -\, 2\chi N} {{c}^{2}} \right ] $$

(D5)

which is a measure of the distance from the spinodal, where use was made of (χN) _s = F(x ^∗, f) ∕ 2 with x ^∗ (or q ^∗) being the value of x (or q) at which $\tilde{S}(q)$ defined by (42) becomes a maximum.

Let us now calculate τ for n = 1 (lamellar microdomains). From (D3) and (D4) we obtain

$$\tau = -{r}_{1}\left [1 + \mathrm{d}\lambda /\left ({{r}_{1}}^{3/2}\tilde{{N}}^{1/2}\right )\right ] $$

(D6)

On the other hand, a rearrangement of (D2) gives

$$\tau = {r}_{0}\left [1 -\mathrm{d}\lambda /\left ({{r}_{0}}^{3/2}\tilde{{N}}^{1/2}\right )\right ] $$

(D7)

From the point of view of dimensional analysis, the second term inside the bracket of both (D6) and (D7) should be dimensionless and thus we have

$${r}_{1} = {k}_{1}{\left (\mathrm{d}\lambda \right )}^{2/3}\tilde{{N}}^{-1/3};{r}_{ 0} = {k}_{0}{\left (\mathrm{d}\lambda \right )}^{2/3}\tilde{{N}}^{-1/3} $$

(D8)

in which k ₁ and k ₀ are constants. From (D6)–(D8) we have

$$-{k}_{1}\left (1 +{ {k}_{1}}^{-3/2}\right ) = {k}_{ 0}\left (1 -{ {k}_{0}}^{-3/2}\right )$$

or

$$\left ({{k}_{1}}^{1/2} -{ {k}_{ 0}}^{1/2}\right ) = \left ({k}_{ 1} + {k}_{0}\right ){\left ({k}_{1}{k}_{0}\right )}^{1/2} $$

(D9)

Note that at a transition temperature Φ = 0 (see (D1) for the definition of Φ). Substitution of (D8) into (D1), with the aid of (D4), gives

$$\begin{array}{rcl} \Phi \left [\phi \right ] ={ \mathrm{d}}^{4/3}{\lambda }^{1/3}\tilde{{N}}^{-2/3}\,\left [\left (1/2\right )\left ({{k}_{ 1}}^{2} -{ {k}_{ 0}}^{2}\right ) + \left ({{k}_{ 1}}^{1/2} -{ {k}_{ 0}}^{1/2}\right ) -{ {k}_{ 1}}^{2}\right ] = 0& & \\ \end{array}$$

or

$$\begin{array}{rcl} \left ({{k}_{1}}^{1/2} -{ {k}_{ 0}}^{1/2}\right ) = \left ({{k}_{ 1}}^{2} +{ {k}_{ 0}}^{2}\right )/2& & \\ \end{array}$$

(D10)

From (D9) and (D10) we have

$$\left ({k}_{1} + {k}_{0}\right ){\left ({k}_{1}{k}_{0}\right )}^{1/2} = \left ({{k}_{ 1}}^{2} +{ {k}_{ 0}}^{2}\right )/2$$

or

$$\sqrt{R}(R + 1) = ({R}^{2} + 1)/2 $$

(D11)

with R = k ₁ ∕ k ₀, yielding R = 5. 27451. Also, from (D10) we have

$${ {k}_{0}}^{1/2}\left ({R}^{1/2} - 1\right ) ={ {k}_{ 0}}^{2}\left ({R}^{2} + 1\right )/2 $$

(D12)

yielding k ₀ = 0. 200801. Then, we obtain k ₁ = 1. 059138. Finally, from (D6) we have

$$\tau = -\left ({k}_{1} +{ {k}_{1}}^{-1/2}\right ){\left (\mathrm{d}\lambda \right )}^{2/3}\tilde{{N}}^{-1/3} = -2.0308{\left (\mathrm{d}\lambda \right )}^{2/3}\tilde{{N}}^{-1/3} $$

(D13)

Rearrangement of (D5) gives

$${ \left (\chi N\right )}_{t} ={ \left (\chi N\right )}_{s} -\left (1/2\right )\,{c}^{2}\tau $$

(D14)

Substituting values of c, d and λ for f = 0. 5 into (D14), we obtain

$${ \left (\chi N\right )}_{t} = 10.495 + 41.022/\tilde{{N}}^{1/3}$$

(50)

for which use was made of the numerical values: ${x}^{{_\ast}}\,=\,3.7852,{\partial }^{2}F\left (f,x\right )/\partial {x}^{2}\left \vert {}_{{x}^{{_\ast}}}\right. = 0.9624$, and λ = 106. 18 given in Table 1 of Fredrickson and Helfand [33].