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An incompressible, equal-site-volume mean-field approximation for two polymers of degrees of polymerization $N_A,N_B$. Here $f$ is mixing free energy per site and $\varphi$ is the volume fraction of species $A$. For a composition-independent Flory interaction parameter, local stability requires $1/(N_A\varphi )+1/[N_B(1-\varphi )]-2\chi >0$. The thermodynamic critical point is ${\varphi }_c=\sqrt{N_B}/(\sqrt{N_A}+\sqrt{N_B})$ and ${\chi }_c=(N_A^{-1/2}+N_B^{-1/2}{)}^2/2$.

The product $\chi N$ of the Flory interaction parameter and degree of polymerization, using compatible reference-segment conventions. It compares unlike-contact avoidance with chain entropy; solvent dilution need not change the chemical chain length.

In Flory–Huggins solution theory, $\varphi$ is the volume fraction, $N$ is the number of equal-sized lattice segments per chain, $T$ is temperature, $k_{\mathrm{B}}$ is the Boltzmann constant, and $\chi$ is the dimensionless Flory interaction parameter. The stated $f$ is the mixing Helmholtz free energy per lattice site. The first two terms describe mixing entropy; the last describes the contact-energy contribution.

At fixed $T$ and composition-independent $\chi$,Hence the spinodal isThe homogeneous state is locally unstable when $\chi >{\chi }_s(\varphi )$. The minimum gives the critical point of a Flory–Huggins polymer solution:If $\chi$ depends on temperature, the equation $\chi (T)={\chi }_s(\varphi )$ converts this result into a temperature–composition boundary.

The Flory interaction parameter ${\chi }_{AB}$ measures the effective free energy penalty for unlike A–B contacts relative to like contacts, and $N$ is the total number of statistical segments per chain. Their product is the segregation strength: contact-energy preference competes with the conformational entropy cost of separating covalently joined blocks. $N$ does not grow during solvent evaporation. For an approximately nonselective solvent, increasing polymer concentration instead increases the effective importance of A–B contacts, so a roughly fixed-composition path moves from weak toward stronger effective segregation.

The diagram's labels are DIS: disordered; S: block-copolymer sphere phase; H: hexagonal cylinder block-copolymer phase; G: double gyroid; L: lamellar microphase. At $f_A=0.15$, A is the minority component and B forms the continuous matrix. Reading the supplied diagram vertically at this composition gives disorder initially, a sphere region when segregation first becomes sufficient, and a cylinder region if sufficiently strong segregation is reached within the displayed range. The thin gyroid and central lamellar regions are not the thermodynamic equilibrium path at this composition. The dry endpoint cannot be fixed numerically without the final ${\chi }_{AB}N$ value. The cited theoretical calculation determines thermodynamic equilibrium phase stability, not a universal evaporation trajectory.^[ref]

Crucially, at least one block is glassy at room temperature. Solvent can initially plasticize it, but solvent loss can raise its effective glass-transition temperature above the processing temperature. Chain relaxation then becomes too slow to equilibrate. Even slow evaporation can therefore freeze disorder, spheres, cylinders or another transient morphology before the thermodynamic equilibrium dry state is reached. Solvent concentration, annealing time and segmental mobility must be followed alongside the phase diagram.

The Flory interaction parameter $\chi$ measures the free-energy penalty of unlike block contacts relative to like contacts, in units of $k_{B}T$. The degree of polymerization $N$ counts the reference segments per chain, using the segment-volume convention associated with $\chi$. Their product, the block copolymer segregation strength $\chi N$, compares the tendency to avoid unlike contacts with the entropy cost of organizing and stretching connected chains.

At small $\chi N$, the diblock copolymer is disordered. Increasing $\chi N$ can cause microphase separation: the blocks form distinct domains, but their covalent connection prevents complete macroscopic separation. Domain interfaces cost energy while chain stretching and lost configurations cost entropy; their competition selects a finite domain scale.

A common temperature law is $\chi =A/T+B$, so increasing temperature often decreases segregation when $A>0$. This is not a universal sign law for every polymer pair. For an intact, fixed chain, $N$ does not change just because temperature or solvent concentration changes.

Adding a nonselective good solvent usually dilutes unlike-block contacts and reduces an effective segregation strength; it does not chemically shorten the chain. A selective solvent additionally changes domain swelling and solvent-block interactions. One needs a ternary-mixture description, or a clearly defined effective $\chi$, rather than treating the melt's $\chi$ as a universal solvent-concentration function.^[ref]

For equal-size molecules, let $f$ be the mixing Helmholtz free energy per lattice site. Flory–Huggins solution theory reduces to the regular solution model:The first two terms are minus the combinatorial entropy of mixing; the last is the excess cost of unlike contacts. A positive Flory interaction parameter penalizes such contacts. Additive terms linear in composition change the chosen pure-component reference but do not change stability.

A small composition fluctuation at fixed mean composition costs energy proportional to $f^{\prime \prime }$. Thus local stability requires $f^{\prime \prime }>0$, and the spinodal isInside this curve, $f^{\prime \prime }<0$ and arbitrarily small composition fluctuations grow. The minimum of ${\chi }_s$ is at $\varphi =1/2$, giving the critical value ${\chi }_c=2$. At the thermodynamic critical point, $f^{(2)}=f^{(3)}=0$ while $f^{(4)}>0$.

For $\chi =0$, $f$ is strictly convex with its minimum at $1/2$. At $\chi =2$, the central quadratic term vanishes but a positive quartic term remains. At $\chi =2.5$, there are two minima and a central maximum; the spinodal compositions are $(1\pm \sqrt{1-2/\chi })/2\simeq 0.2764,0.7236$. The common-tangent construction gives coexisting compositions ${\varphi }_b\simeq 0.1448$ and $1-{\varphi }_b$, satisfying $\ln [{\varphi }_b/(1-{\varphi }_b)]+\chi (1-2{\varphi }_b)=0$. Between a binodal endpoint and the spinodal, the homogeneous state is locally stable but metastable; global stability requires the convex envelope of $f$.

Write $z_c$ for the lattice coordination number and ${ϵ}_{AA},{ϵ}_{BB},{ϵ}_{AB}$ for contact energies. Counting the excess of unlike contacts relative to the unmixed reference gives the Flory interaction parameterUnlike contacts that are less favorable than the mean like contact give $\chi >0$. For temperature-independent contact energies, $\chi$ is proportional to $1/T$, so heating weakens this energetic preference relative to the entropy of mixing. Effective parameters in real solutions can also include entropic terms, often represented schematically by $A/T+B$; the pure contact-energy model does not explain every possible temperature dependence.

The Flory interaction parameter measures the energetic preference for unlike rather than like contacts in Flory–Huggins solution theory. For polymer volume fraction $\varphi$ and $N$ lattice sites per chain, the dimensionless mixing Helmholtz free energy per site isThe logarithmic terms represent combinatorial entropy; polymer connectivity reduces the chain's translational mixing entropy by the factor $N$. In a simple contact model $\chi =z[{ϵ}_{ps}-({ϵ}_{pp}+{ϵ}_{ss})/2]/(k_{B}T)$, where $z$ is the coordination number. Positive $\chi$ penalizes polymer-solvent contacts. Increasing it can overcome mixing entropy and cause phase separation; a negative second derivative of the mixing free energy marks the spinodal region. For a long dilute chain, $\chi <1/2$ corresponds to a good solvent, $\chi =1/2$ to the idealized theta condition, and $\chi >1/2$ to a poor solvent. Real effective parameters can depend on temperature and concentration, rather than being universal chemical constants.

Use polymer volume fraction $\varphi$ horizontally and the Flory interaction parameter $\chi$ vertically. Lower $\chi$ gives better solvent quality. The simplest Flory–Huggins solution theory assigns the dimensionless mixing free energy per lattice siteThus $\chi <1/2$ gives positive excluded volume and a good-solvent coil, $\chi =1/2$ is the long-chain theta condition, and $\chi >1/2$ is the poor-solvent side. It is the net two-segment interaction that vanishes at theta, not every molecular force.

At fixed good-solvent quality, the polymer overlap volume fraction ${\varphi }^{\ast }\sim N{a}^3/R_g^3$ separates isolated coils from a semidilute polymer solution. Above overlap, many chains interpenetrate, but there is still solvent on the segment scale. As $\varphi$ approaches order unity the solution is concentrated; these concentration-regime boundaries are crossovers, not first-order phase transitions. The overlap fraction increases from order $N^{-4/5}$ in a strong good solvent toward order $N^{-1/2}$ at theta. The crossover can be expressed as ${\varphi }^{\ast }\sim N^{-4/5}(1-2\chi {)}^{-3/5}$ while $1-2\chi \gg N^{-1/2}$; it saturates at the theta scaling when that condition fails.

On the poor-solvent side there can instead be coexistence of polymer-poor and polymer-rich phases. The binodal follows a common tangent of $f$; the spinodal followsThe critical point of a Flory–Huggins polymer solution isFor finite $N$, ${\chi }_c$ lies above $1/2$. Therefore poor solvent does not automatically mean demixing at every concentration, and an overlap line must not be mistaken for a coexistence boundary. The original figure uses the stated ideal model for an illustrative chain length; it does not assert a measured phase boundary. A vertical temperature axis would additionally require a material-specific relation $\chi (T)$.

The Flory interaction parameter $\chi$ measures the excess cost of unlike segment contacts. In a simple lattice with coordination number $z$, contact energies ${ϵ}_{AA},{ϵ}_{BB},{ϵ}_{AB}$ and temperature $T$,A positive Flory interaction parameter penalizes unlike contacts; a negative value favors them. Mixing still depends on the balance with entropy of mixing, rather than the sign of $\chi$ alone.

For two polymers of degrees of polymerization $N_A,N_B$, let $\varphi$ be the volume fraction of species $A$. The binary Flory-Huggins polymer blend has mixing free energy per lattice siteThe logarithmic terms favor mixing. Their factors $1/N_A$ and $1/N_B$ arise because an entire chain translates as one object; connecting many segments greatly reduces the mixing entropy per segment. Hence long polymers can demix for a much smaller positive Flory interaction parameter than small molecules.

For composition-independent $\chi$, the local stability criterion isPositive curvature means stability to small composition fluctuations, zero curvature locates the spinodal, and negative curvature means spontaneous growth of fluctuations. Actual two-phase coexistence is determined by the common-tangent construction, or equality of both species' chemical potentials; its boundary is the binodal. A locally stable but globally unfavourable homogeneous mixture can lie between the binodal and spinodal.

Setting the second and third derivatives of $f$ to zero givesFor the symmetric Flory-Huggins polymer blend, ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. For a polymer solution with $N_B=1$ and long chains of species $A$, ${\chi }_c\to 1/2$, consistent with the theta solvent limit. An effective Flory interaction parameter can depend on temperature and composition; in the simple energetic case $\chi \propto 1/T$, heating encourages mixing, but that temperature trend is not universal.

The original illustration uses stipulated $N_A=N_B=100$. At $\chi =0.01$ the mixing free energy is convex, whereas at $\chi =0.03$ a central unstable region appears. The horizontal common-tangent construction in the latter case joins the symmetric coexisting compositions; its contacts differ from the spinodal boundaries. These are model curves, not supplied experimental data.

Meaning of the mixing free energy. The supplied expression is the Flory–Huggins solution theory mixing free energy per lattice site, or per solvent-sized reference cell. Although the question calls it “per particle,” it cannot mean per polymer molecule: one chain occupies $N$ sites. Let the site volume be $v_0$, the total number of sites be $M=V/v_0$, and the chain and solvent counts be $n_p,n_s$, sowhereThe first term is the polymer's translational entropy of mixing. Its factor $1/N$ expresses that $N$ connected segments translate as one molecule. The second is the solvent's translational mixing entropy. Both are negative in the mixing free energy for $0<\varphi <1$, favoring mixing.

The final term is a mean-field excess contact contribution. The Flory interaction parameter $\chi$ measures the relative preference for unlike polymer-solvent contacts compared with like contacts. In the simplest lattice model with coordination number $z$ and pair energies ${ϵ}_{ps},{ϵ}_{pp},{ϵ}_{ss}$,Positive $\chi$ penalizes unlike contacts and promotes separation; negative $\chi$ favors them. Effective real-solution values can also include non-enthalpic effects, so their temperature dependence need not be exactly a constant divided by $T$. The subsequent derivatives assume composition-independent $\chi$ as in the supplied model.

Osmotic pressure and the special value one half. At fixed chain count, $\partial \varphi /\partial V=-\varphi /V$. Differentiating $F_{\mathrm{mix}}=(V/v_0)k_{B}Tg(\varphi )$ with respect to volume, relative to the pure-solvent reference, givesThe resulting osmotic pressure in Flory-Huggins solution theory isThe leading term is the ideal osmotic pressure of chains, since their number density is $\varphi /(N{v}_0)$. At $\chi =1/2$ the second-virial term vanishes: effective two-segment attraction and excluded-volume repulsion balance. This is the theta solvent condition for a sufficiently long dilute chain; the positive third-order contribution remains, so a finite-concentration solution is not a completely noninteracting gas. The site-volume factor is necessary to give physical pressure units.

For $\chi <1/2$, the effective pair excluded volume is positive. In a good solvent, the isolated chain expands relative to a Gaussian coil, with $R\sim b{N}^{\nu }$, $\nu \simeq 0.588$ or $3/5$ in the Flory approximation. For $\chi >1/2$, attraction wins in a poor solvent and a sufficiently long chain collapses toward a dense polymer globule, with asymptotic size $R\propto N^{1/3}$. Close to the theta condition, finite-chain crossover effects matter. At $\chi =1/2$, $R\sim b{N}^{1/2}$ to leading scaling order. This theta criterion is not the exact finite-$N$ phase-separation critical point; the lattice model has ${\varphi }_c=1/(1+\sqrt{N})$ and ${\chi }_c=(1+1/\sqrt{N}{)}^2/2$.

Overlap concentration. The polymer overlap concentration is reached when there is roughly one isolated coil in its own pervaded volume. If the coil size is $R$ and a repeat unit occupies volume $v_0$,With $R\sim b{N}^{\nu }$, the general scaling law isThe usual single-segment-volume estimate $v_0\sim b^3$ gives ${\varphi }^{\ast }\sim N^{1-3\nu }$: $N^{-1/2}$ in a theta solvent, approximately $N^{-4/5}$ in a good solvent, and order one for compact globules. A coil has no sharp surface, so choosing a radius of gyration or a geometric coil volume changes numerical prefactors.

For the stated polystyrene at $N=1500$ and $\chi =1/2$, the conventional scaling estimate isThis is a volume-fraction estimate, not a molar concentration. It uses the implicit site-volume approximation; the question does not provide an independent repeat-unit volume to determine a more precise prefactor.

Semidilute length scale. Beyond overlap, a semidilute polymer solution is characterized locally by a concentration blob size $\xi$. Suppose one blob contains $g$ statistical units. At the theta condition the subchain is Gaussian, so $\xi \sim b\sqrt{g}$, while matching its internal concentration to the bulk gives $\varphi \sim g{v}_0/{\xi }^3$. Eliminating $g$ gives the concentration-blob size in a theta solvent:Using the supplied statistical step length $b=0.67\mathrm{nm}$ and $\varphi =0.05$,The same convention gives $g\sim {\varphi }^{-2}=400<N=1500$, and the whole ideal chain has $R\sim b\sqrt{N}\simeq 26\mathrm{nm}$. Thus the blob is smaller than the chain. Because $\varphi /{\varphi }^{\ast }\simeq 1.94$ is only modestly above overlap, the value is a scaling estimate in a crossover region, not a precision asymptotic measurement. A statistical step length is not automatically a physical monomer diameter: if $v_0$ differs from $b^3$, the more general estimates are ${\varphi }^{\ast }\sim (v_0/b^3)N^{-1/2}$ and $\xi \sim v_0/(b^2\varphi )$, with the missing prefactor left explicit.

Solvent quality. The Flory interaction parameter $\chi$ measures the free-energy cost of polymer–solvent contacts relative to like contacts. In the simplest lattice model,where $z_c$ is the coordination number and the $ϵ$ values are contact energies. For lattice-site volume $v_0$, the effective two-segment excluded volume is $v_{\mathrm{eff}}=v_0(1-2\chi )$. Its sign determines the leading tendency of a long dilute flexible chain.

For $\chi <1/2$, $v_{\mathrm{eff}}>0$: a good solvent favors a swollen coil because segment encounters are effectively repulsive. For $\chi =1/2$, the leading attraction and repulsion cancel, giving approximately ideal chain statistics. For $\chi >1/2$, effective attraction favors contraction in a poor solvent, eventually producing a compact polymer globule under sufficiently poor conditions. This does not mean that every finite-chain solution with $\chi >1/2$ is automatically phase-separated; that also depends on composition and chain length.

The condition $\chi =1/2$ is the theta temperature condition for that polymer–solvent pair. For an energetic model $\chi =A/T$, it occurs at $T=\Theta =2A$. More generally, if $\chi =A/T+B$, it occurs when $A/\Theta +B=1/2$. It is not the absence of all intermolecular interactions and does not require the polymer and solvent to be chemically identical.

Chain dimensions. Interpret $R$ as the root-mean-square end-to-end distance of a polymer chain. Let $a$ denote the supplied statistical step length, calibrated to the same count $N$ as the degree of polymerization. At the theta condition, random-walk statistics giveFor a good solvent, the Flory approximation balances entropic stretching and two-segment interaction:Minimization gives $R^5\sim v_{\mathrm{eff}}{a}^2{N}^3$, henceThus the familiar good-solvent scaling is $R\sim a{N}^{3/5}$ when its nonuniversal solvent-quality amplitude is suppressed. With the additional lattice estimate $v_0\simeq a^3$, the more explicit form is $R\sim a(1-2\chi {)}^{1/5}{N}^{3/5}$. This swollen-coil estimate is used away from the narrow theta crossover; it must not be extrapolated to predict $R=0$ as $\chi \uparrow 1/2$. The crossover occurs when $(1-2\chi )N^{1/2}$ is of order one. Nearer theta, the ideal-coil size supplies the appropriate limit.

Numerical sizes. The theta value isFor $\chi =0.3$, the plain good-solvent scaling givesIf the solvent-quality amplitude is retained with $v_0=a^3$, it instead givesThe distinction is an explicitly stated modeling convention: the question does not give a microscopic site volume or an absolute swollen-coil prefactor. Both estimates describe a coil larger than the theta coil. Here $(1-2\chi )\sqrt{N}\simeq 15.5$, so a swollen-coil estimate is reasonable. Neither result is an exact microscopic prediction from the supplied data alone.

Overlap concentration. The polymer overlap concentration corresponds to one chain per coil volume. Using the scaling convention $V_{\mathrm{coil}}\sim R_{\theta }^3$ givesThe first is chain number density; the second counts repeat units. Thus the repeat-unit molar concentration is approximately $0.143\mathrm{mol}{\mathrm{L}}^{-1}$. If the lattice-site volume is taken as $v_0=a^3$, this is equivalentlyFor a mass-concentration estimate, a polystyrene repeat unit ${\mathrm{C}}_8{\mathrm{H}}_8$ has molar mass about $104.15\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, so this scaling convention givesThere is no sharp coil surface fixing an exact numerical prefactor. For example, using the often adopted geometric definition $V_{\mathrm{coil}}=4\pi R_g^3/3$, with the ideal chain radius of gyration $R_g=R_{\theta }/\sqrt{6}=10.6\mathrm{nm}$, gives $c^{\ast }\simeq 52\mathrm{g}{\mathrm{L}}^{-1}$. The invariant content is one chain per coil volume and the theta scaling ${\varphi }^{\ast }\propto N^{-1/2}$; the radius and coil-volume convention must accompany a numerical value.

Osmotic pressure. In the given lattice expression, $\varphi$ is a dimensionless polymer volume fraction, not a dimensional number density, and $f$ is the mixing free energy per lattice site. Introduce $v_0$ to obtain a physical osmotic pressure. If $M=N_s+N{N}_p$ is the number of sites and $\varphi =N{N}_p/M$, differentiation of $F_{\mathrm{mix}}=Mf(\varphi )$ with respect to the solvent count at fixed chain count givesBalancing this change in solvent chemical potential against the pressure contribution $v_0\Pi$ gives $\Pi =(\varphi f^{\prime }-f)/v_0$. For composition-independent $\chi$,and therefore the osmotic pressure in Flory-Huggins solution theory isExpanding the logarithm as $\varphi \to 0$ givesThe first term is $n_p{k}_{B}T$ with $n_p=\varphi /(N{v}_0)$ now denoting chain number density: a whole chain supplies one translational contribution. The sign of the quadratic correction reproduces the solvent-quality classification. It is positive for a good solvent, zero at the theta temperature, and negative for a poor solvent. This is the same two-segment interaction that swells or contracts a single coil. At theta, a positive third-order contribution remains; the chain is approximately ideal in its large-scale conformation, not a gas of segments with no interactions at all.

For two equal-site-volume polymers of chain length $N$, the mixing free energy per lattice site is $f/(k_{B}T)=[\varphi \ln \varphi +(1-\varphi )\ln (1-\varphi )]/N+\chi \varphi (1-\varphi )$. Each chain is one translating object, reducing the entropy of mixing per monomer by $N$ relative to the regular solution model. Its curvature is $f^{\prime \prime }/(k_{B}T)=1/[N\varphi (1-\varphi )]-2\chi$. The symmetric thermodynamic critical point has ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. Long chains can therefore demix under a very small unfavorable Flory interaction parameter.