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Here the osmotic pressure is a two-dimensional surface pressure, with units of energy per area. In the dilute regime, independent whole chains provide the translational entropy, soIn the semidilute polymer solution, use a correlation blob containing $g$ monomers. Within it the chain still follows the planar good-solvent law $\xi \sim \ell g^{3/4}$. Its area fraction is $\varphi \sim g{\ell }^2/{\xi }^2\sim g^{-1/2}$, so $g\sim {\varphi }^{-2}$ and $\xi \sim \ell {\varphi }^{-3/2}$. There is about one independent thermal contribution per blob area. The semidilute osmotic pressure of a two-dimensional polymer solution is thusIt is independent of total chain length once each chain contains many blobs. The two estimates agree at ${\varphi }^{\ast }\sim N^{-1/2}$, where each gives $(k_{B}T/{\ell }^2)N^{-3/2}$.

Write $s$ for the number of grafted chains per unit area and $\Sigma =s{a}^2$ for its dimensionless form; this separates the physical grafting density from a dimensionless coverage. A free dilute chain has size $R_0\sim a{N}^{\nu }$, with $\nu =1/2$ in a theta solvent and approximately $3/5$ in a good solvent. An isolated end-grafted chain loses translations and configurations that cross the substrate, but retains the same leading size exponent if the surface is nonadsorbing. When the lateral graft spacing $D\sim s^{-1/2}$ exceeds $R_0$, the layer consists of polymer mushrooms. When coils overlap, crowding stretches the chains away from the surface into a polymer brush.

The polymer mushroom-to-brush crossover isFor the stipulated statistical-segment interpretation of $N$ and $a$, the scaling estimates areThus ${\Sigma }_{\ast }\simeq 2.50\times 10^{-3}$ and $7.54\times 10^{-4}$ respectively. The numerical constants are scaling conventions: using a radius of gyration rather than an end-to-end size changes order-one amplitudes, not the powers of $N$.

In the Alexander-de Gennes brush model, take blobs of lateral size $D$. Within a blob, $D\sim a{g}^{\nu }$; beyond it, the $N/g$ blobs are aligned away from the wall. ThusWith coarse-grained segment volume $a^3$, conservation gives the polymer-brush volume fraction through $\varphi H\sim N{a}^{3}s$, henceThe theta-solvent brush height includes three-segment repulsion even though the effective two-segment interaction vanishes. These are scaling estimates for a roughly uniform brush, not an assertion that every chain end lies at the same height in an actual brush.

The many-segment-blob requirement $g\gg 1$ gives $D\gg a$, or $\Sigma \ll 1$. Together with overlap, the regime is $N^{-2\nu }\ll \Sigma \ll 1$. At the dense-brush packing limit, $\varphi \sim 1$, $g\sim 1$ and $H\sim Na$. Here $s\sim a^{-2}=2.04\mathrm{n}{\mathrm{m}}^{-2}$ and $H\sim 280\mathrm{nm}$. If one formally continues a dense incompressible layer above this boundary, conservation would instead giveThis is the requested formal dense-layer expression, but for $\Sigma >1$ it gives $H>Na$, which violates finite extensibility of a polymer chain. There is no physically admissible continuation above this limit in the stated segment-volume model. A Kuhn length segment is not necessarily a cube of material: with an independently specified material volume $v_0$ per counted segment, the dense-layer law is $H=N{v}_{0}s$, and the packing bound is $s\le a/v_0$. The data do not determine $v_0$ separately. The simple model therefore establishes its own breakdown, rather than a universal microscopic maximum grafting density.

grafting-to uses preformed chains whose lengths and dispersities can be characterized before attachment. As coverage rises, each incoming coil must penetrate a crowded brush and lose conformational entropy to bring its end to the surface; the growing insertion barrier limits achievable coverage. grafting-from supplies small monomers to surface-bound growing chains and avoids insertion of a whole coil. Choose grafting-from for high grafting density. Initiator coverage and reaction control still matter, and inaccessible active sites, termination and unequal growth can broaden the resulting chain-length distribution. Neither route removes the elastic and packing costs of the final brush.

A generic application is steric stabilization of synthetic colloids in a good solvent. Bringing coated particles together compresses and interpenetrates their brushes, increasing osmotic pressure and conformational free-energy cost. This repulsion can oppose aggregation. The chains must remain attached and sufficiently cover the surface; poor-solvent collapse, sparse coverage or interparticle bridging can defeat stabilization.

Interpret the stated free energy as per lattice site of volume $v_0$. Flory–Huggins solution theory counts polymer chains, giving the entropy term $(\phi /N)\ln \phi$, and solvent molecules, giving $(1-\phi )\ln (1-\phi )$. The interaction term penalizes unlike contacts when $\chi >0$; a simple contact model has $\chi =z[{ϵ}_{ps}-({ϵ}_{pp}+{ϵ}_{ss})/2]/(k_{B}T)$. Connectivity reduces polymer translational entropy relative to independent monomers. This is an incompressible mean-field model with composition-independent $\chi$ for the calculations below.^[ref]

Expanding the dilute osmotic pressure identifies the effective second virial coefficient. A good solvent has $\chi <1/2$ and positive excluded volume, promoting swollen chains. A poor solvent has $\chi >1/2$, favouring contraction and potentially phase separation. The theta-solvent condition is $\chi =1/2$ in this long-chain mean-field description: the second virial term vanishes, but higher terms remain. For finite $N$, this condition is not the exact critical point, which instead has ${\chi }_c=(1+N^{-1/2}{)}^2/2$.

Let $g=f_{\mathrm{mix}}/(k_{B}T)$. The osmotic pressure in Flory-Huggins solution theory isThus $\Pi =(k_{B}T/v_0)[\phi /N+(1/2-\chi ){\phi }^2+{\phi }^3/3+\cdots ]$. For mass concentration $c={\rho }_p\phi$ and chain molar mass $M_n=N{\rho }_p{v}_0{N}_A$,osmometry uses a membrane permeable to solvent but retaining every polymer chain, at known uniform temperature. Extrapolate $\Pi /c$ to zero concentration to remove interactions; the intercept gives the number-average molar mass. Aggregation, leakage of small chains, ions with a Donnan contribution, or a poor-solvent two-phase sample invalidate the simple molecular-weight interpretation.

The measured ratios give an exactly linear reduced-pressure fit:The reduced osmotic-pressure intercept is $185\mathrm{Pa}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}$. The data determine $M_n/T=0.04494\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$; the measurement temperature is needed to convert this into an absolute molar mass. For an explicitly assumed room temperature $T=300\mathrm{K}$, $M_n\simeq 13.5\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}=1.35\times 10^4\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The boxed expression is the general answer for an arbitrary experimental temperature.

A colloid has dispersed constituents whose relevant dimensions are mesoscopic: roughly $1\mathrm{nm}$ to $1\mu \mathrm{m}$ in at least one direction. Interfaces and Brownian motion are consequently important. The size range is a useful convention, not a sharp boundary separating all physical behaviour. A sol of fine silicon dioxide particles in water illustrates solid in liquid; an oil-in-water emulsion illustrates liquid in another immiscible liquid; and a foam illustrates gas dispersed in a liquid, with thin intervening films supplying the small structural dimension. A colloid need not be a thermodynamic equilibrium state.^[ref]

Unprotected particles attract through Van der Waals forces. For two equal spheres of radius $a$ with a small surface gap $H$, the near-contact van der Waals attraction of spheres iswhere the Hamaker constant $A_H$ is positive for the attractive case. The large number of interacting constituent molecules can give a well much deeper than $k_{B}T$, even though individual molecular attractions are weak. Brownian motion brings particles together, and a sufficiently deep attractive well makes separation unlikely. This gives particle aggregation; solid particles can remain distinct within a cluster rather than merging into one continuous solid. Emulsion droplets can additionally coalesce if the separating film ruptures.

Charge stabilization of colloids supplies a repulsive barrier. Like-charged surfaces carry compensating counterions and electric double layers; bringing the layers together costs free energy. DLVO theory combines this repulsion with Van der Waals forces. A barrier large compared with $k_{B}T$ can make aggregation slow, even when sticking would lower the final free energy. Adding salt generally shortens the Debye–Hückel screening length and can lower the barrier; surface charge regulation and specific ion interactions also matter. A negatively charged particle and a positively charged particle need not repel.

Steric stabilization instead coats each particle with adsorbed or grafted polymer chains. In a good solvent, overlap and compression of these layers raise the osmotic pressure and reduce chain entropy, opposing close contact. Electrosteric stabilization combines the two mechanisms. Dense, securely bound layers matter: incomplete adsorption can allow polymer bridging, while nonadsorbing dissolved polymers can create an attractive depletion force. Poor solvent quality can collapse a protective layer. Reducing the density mismatch slows sedimentation but does not remove the interparticle attraction, so uniform suspension and resistance to aggregation are different requirements.

The left panel is an original qualitative potential with arbitrary coefficients: it illustrates a barrier and its reduction by shorter-ranged repulsion, without fitting an experimental DLVO theory curve or extrapolating the continuum attraction to molecular contact. The right panel plots the derived quasi-steady concentration profile in dimensionless variables.

In diffusion-controlled growth, attachment is rapid and delivery through the surrounding medium is the bottleneck. The growing surface depletes the neighboring solute, so a substantial concentration gradient develops. In reaction-controlled growth, arrival is fast compared with incorporation; many arrivals fail to attach, and the surface concentration stays near the reservoir value. The analogous distinction for particle aggregation is between near-unit sticking probability in diffusion-limited cluster aggregation and barrier-suppressed sticking in reaction-limited cluster aggregation. Here the subsequent calculation concerns accretion onto one growing particle, rather than a collision kernel for two mobile clusters.

The printed time-dependent equation is the diffusion equation, not the usual Poisson equation. For constant diffusion coefficient $D$, no bulk sources and negligible advection, Fick's laws give the number flux $\mathbf{j}=-D\nabla c$. Local conservation of mass, expressed here as conservation of the accreting constituent, then givesAn arbitrary time-independent exterior profile does not solve the full moving-boundary problem exactly. The required elementary result uses a quasi-steady approximation: diffusion establishes the exterior profile much faster than the particle radius changes. At each instantaneous radius $a$, set $\partial c/\partial t\simeq 0$. Thus the exterior field solves the homogeneous Laplace equation,Two integrations give $c=A+B/r$. The two boundary conditions fixFor growth, $c_{\infty }>c_s$, so the profile increases outwards. Its radial flux is negative, and the positive inward arrival rate isA material density is needed to turn an arrival rate into a radial velocity. Introduce the constant constituent number density $c_p$ inside the particle, using the same units as $c_s,c_{\infty }$. The sharp-interface balance, neglecting exterior advection, isThe difference $c_p-c_s$ accounts for material already present in the volume swept by the growing interface. In the usual dilute-feed approximation $c_s\ll c_p$, it can be replaced by $c_p$. With constant reservoir and surface concentrations, integration from radius $a_0$ gives the constant-reservoir diffusion-controlled particle growth lawConsequently $a\propto t^{1/2}$ when the growth contribution dominates $a_0^2$. For a finite seed at the literal limit $t\to 0$, the increment $a-a_0$ is instead initially proportional to $t$. Here “early time” means before appreciable reservoir depletion or interactions with other particles, while already beyond the rapid initial establishment of the diffusion field. Indeed, the ratio of diffusion-adjustment time $a^2/D$ to radial-growth time $a/\dot{a}$ is $(c_{\infty }-c_s)/(c_p-c_s)$; small supersaturation relative to the particle density makes the quasi-steady approximation self-consistent. No initial concentration field was specified, so an exact transient solution is not uniquely determined.

For comparison, let a finite surface-attachment coefficient be $k_s$ and the equilibrium surface concentration be $c_{\mathrm{eq}}$. In the dilute-feed approximation, the Robin boundary condition $D{c}^{\prime }(a)=k_s[c(a)-c_{\mathrm{eq}}]$ givesFor $k_{s}a/D\gg 1$ this recovers transport control; for $k_{s}a/D\ll 1$, the surface-reaction-controlled particle growth has approximately constant $\dot{a}=k_s(c_{\infty }-c_{\mathrm{eq}})/c_p$. Thus reaction control gives $a-a_0\propto t$, in contrast to the diffusion-controlled square-radius law. No numerical $c_p$ or $k_s$ is supplied or needed for these scaling results.

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.

Two stabilization mechanisms. Steric stabilization coats each colloid with adsorbed or grafted polymer layers. Bringing two particles close overlaps or compresses those layers, increasing their free energy through loss of chain conformations and, in a suitable solvent, increased local osmotic pressure. The resulting repulsion can keep the cores apart. A well-solvated nonionic coating can remain effective when added salt would screen electrostatic forces. Coatings can fail through desorption, poor solvent conditions or bridging, so simply adding arbitrary free polymer does not guarantee stabilization.

Charge stabilization of colloids instead uses similarly charged surfaces and their diffuse electric double layers. Their approach costs electrostatic and ionic free energy. In DLVO theory, that repulsion competes with attractive Van der Waals forces. It is particularly sensitive to salt concentration, counterion valence and surface ionization.

Bjerrum length. The standard Bjerrum length is the separation at which the Coulomb interaction energy of two elementary charges equals thermal energy:At $T\simeq 298\mathrm{K}$ and ${\epsilon }_r=78$, this isIt compares ionic electrostatic coupling with thermal agitation. For ions of signed valences $z_1,z_2$, $U/(k_{B}T)=z_1{z}_2{\ell }_B/r$. The standard reference length uses $e$, not the whole colloid charge: assigning a colloid charge $Ze$ does not change the solvent's tabulated ${\ell }_B$.

Screening and the printed normalization. In a symmetric $z:z$ electrolyte, $z$ is the ionic valence magnitude, $e$ the elementary electric charge, $n_{\infty }$ the bulk number density of each ionic species, ${\epsilon }_0$ the vacuum permittivity, ${\epsilon }_R$ the solvent's relative permittivity, $k_B$ the Boltzmann constant and $T$ absolute temperature. Mobile ions rearrange around a charged surface; the Debye–Hückel screening length ${\kappa }^{-1}$ sets the range of the resulting weak potential.

There is an SI normalization error in the original PDF. Combining the Boltzmann distribution $n_{\pm }=n_{\infty }{e}^{\mp ze\psi /(k_{B}T)}$ with the SI Poisson equation and linearizing at $|ze\psi |\ll k_{B}T$ givesThe extra $4\pi$ in the PDF denominator is inconsistent with those SI charge, permittivity and density conventions. Keeping the printed algebra would make its screening length $\sqrt{4\pi }$ times the SI value at the same ion density. No numerical screening length is required here, and the qualitative dependence ${\kappa }^{-1}\propto n_{\infty }^{-1/2}$ is the same in either normalization. Added salt shortens the range of repulsion, allowing the attractive potential to capture particles more readily.

Interaction-potential sketches. Let $h$ be the surface separation between two spheres of radius $a$. A useful schematic DLVO theory potential has the formaway from the microscopic contact region. Here $A_H>0$ is a Hamaker constant; excluded-core repulsion and molecular details regularize contact. The exponential is the double-layer repulsion and the inverse-gap term the short-gap Van der Waals force contribution. A fixed $B$ illustrates the changing screening range; actual charge regulation can change the repulsive amplitude as well.

At low salt there is a high positive barrier between a deep primary minimum near contact and the dispersed state at $U(\infty )=0$. A shallow secondary minimum can occur outside the barrier and support weak reversible association. Increasing salt narrows the repulsive range and lowers the barrier. Thermally activated primary aggregation becomes appreciable when the barrier is only a few $k_{B}T$; the idealized barrier-free condition defines the fast-aggregation threshold associated with the critical coagulation concentration. Above that threshold, collisions can result in rapid capture in the primary minimum.

Aggregate structures. Well above the critical coagulation concentration, the sticking probability is close to one. In diffusion-limited cluster aggregation, the first encounter attaches particles or clusters at accessible outer branches before they can explore many alternative configurations. The result is an open, tenuous structure. Typical ideal three-dimensional cluster models have mass fractal dimension $d_f\simeq 1.8$.

Near the threshold on its lower-salt, slow-aggregation side, a remaining barrier makes most encounters unsuccessful. Reaction-limited cluster aggregation then permits repeated approaches and orientations before attachment, commonly giving denser clusters with $d_f\simeq 2.1$. Since particle count scales as $N\propto (R_g/a{)}^{d_f}$, the larger exponent means more particles at a given large cluster size. This explanation does not require already formed bonds to rearrange: unsuccessful encounters themselves alter which configurations eventually bind.

The side of the threshold matters. At the ideal barrier-free threshold itself, or just above it, aggregation can already be diffusion limited; proximity to the critical coagulation concentration alone does not universally imply a denser reaction-limited structure. The quoted dimensions are typical of irreversible approximately isotropic cluster models, not exact values for every colloid. Strong bond rearrangement or anisotropic attractions can change this comparison.

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 dissolved, nonadsorbing polyethylene glycol, the usual new interaction is a depletion force. Polymer centres are excluded from a layer of thickness of order their radius of gyration $R_g$ around each particle. When the particles approach, their exclusion layers overlap. This increases the volume accessible to polymer and its translational entropy, lowering the colloid pair free energy. Equivalently the surrounding polymer osmotic pressure pushes the particles together.

In the dilute-polymer Asakura-Oosawa model, let $h=R-2a$ be the surface gap and $R_e=a+R_g$. Thenfor $2a\le R<2R_e$, and $U_{\mathrm{dep}}=0$ for $R\ge 2R_e$. Here ${\Pi }_p\simeq n_p^{(\mathrm{polymer})}{k}_{B}T$ is the reservoir polymer osmotic pressure. The overlap-volume formula follows by adding two equal spherical cap volumes with height $R_e-R/2$. Thus the added attraction has range about $2R_g$ and grows with polymer concentration. The total interaction is approximately hard exclusion plus screened electrostatic repulsion plus Van der Waals force and depletion attractions.

Increasing polymer concentration therefore creates a family of phase diagrams, rather than a unique shift determined by polymer size alone. Where the attraction overcomes repulsion, colloid-poor fluid can coexist with a colloid-rich phase, and fluid–crystal coexistence generally broadens. For sufficiently long-ranged attraction, stable fluid–fluid separation can occur. For the stipulated small polymer, the attraction is short-ranged: fluid–fluid separation can be metastable to crystallization, and a rapid attraction increase can instead produce an arrested colloidal gel. Residual long-ranged repulsion can oppose bulk separation. Simulations and free-volume calculations confirm that longer-ranged electrostatic repulsion can counteract depletion-driven separation.^[ref]

The nonadsorption assumption matters. If the polyethylene glycol adsorbs to the particles, a dense attached layer can give steric stabilization, whereas chains attaching to two particles can produce polymer bridging. The paper does not specify surface affinity or polymer concentration, so depletion is the standard conditional answer, not a prediction of an inevitable particular final phase.

Use SI units, uniform absolute permittivity $\epsilon$ and a neutral bulk reservoir. Local equilibrium of ideal mobile ions gives $n_i=n_{i,\infty }{e}^{-z_{i}e\psi /(k_{B}T)}$. Combining their charge with the Poisson equation yieldsThe reference is $\psi =0$ in the bulk. Boundary data specify fixed charge, fixed potential or a surface reaction/charge regulation law. These are physically different problems. The mean-field theory assumes point ions, thermal equilibrium and a continuum solvent; ion-size packing, dielectric structure and strong electrostatic correlations can invalidate it.

For a symmetric monovalent electrolyte let $n_{\infty }$ count each sign and define $u=e\psi /(k_{B}T)$, ${\kappa }_D^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$. Outside fixed sources,For $|u|\ll 1$, the Debye-Hückel approximation is ${\nabla }^{2}u={\kappa }_D^{2}u$. A planar wall bounding electrolyte in $z>0$ has $\psi ={\psi }_0{e}^{-{\kappa }_{D}z}$; assuming zero field on the other side, its charge is $\sigma =\epsilon {\kappa }_D{\psi }_0$. A sphere of radius $a$ and total charge $Q$ instead givesThe factors arise from geometry and boundary conditions; planar screening cannot be substituted unchanged for a small sphere. The weak-potential condition must hold where the potential is largest, particularly at the surface.

The nonlinear planar Poisson-Boltzmann profile is also exactly integrable. Multiply $u^{\prime \prime }={\kappa }_D^2\sinh u$ by $u^{\prime }$ and use $u,u^{\prime }\to 0$ at infinity:The sign selects a profile decaying to the bulk for either sign of surface potential. Integration givesFor weak $u_0$ this reduces to the linear solution. Strong surface potentials have a bounded far-field amplitude $4\gamma$, while the near-wall ion density still grows strongly. There is no need to permit the negative ion densities produced by an invalid extrapolation of a linearized density formula.

The same first integral yields the Grahame equation and its diffuse-layer differential capacitance per unit area:These describe the diffuse layer. A compact interfacial layer adds a voltage drop and can place its capacitance in series with $C_D$; a measured electrode capacitance need not equal the ideal diffuse contribution.^[ref]

For comparison, a planar counterion-only double layer next to a negative plane has positive counterions and no salt reservoir. Its profile is algebraic:Integrating $en(z)$ over $z>0$ gives $|\sigma |$, checking charge neutrality. This problem cannot be obtained by putting a finite reservoir density into the symmetric-electrolyte answer without changing its boundary conditions.

Applications include screening around synthetic colloids, diffuse-layer electrode capacitance and forces between charged plates. In planar geometry, the bulk-subtracted normal stress is constant:At the midpoint of identical plates, symmetry sets $E=0$, so the remaining excess ionic osmotic pressure determines their repulsion. A comparison with force or capacitance measurements should vary salt and surface potential independently, test charge regulation and adsorption, and distinguish compact-layer effects from failures of diffuse screening. Nonlinear Poisson-Boltzmann theory retains finite-potential Boltzmann populations, but still omits correlations and finite-ion-volume saturation.

Sedimentation equilibrium occurs when the downward tendency of suspended particles is balanced by their diffusion, leaving a stationary concentration profile with no net particle flux. For dilute identical particles in an isothermal fluid, let $m_b=({\rho }_p-{\rho }_f)V_p$ be their buoyant mass. Balancing the gradient of ideal osmotic pressure $n{k}_{B}T$ against gravity givesHere height increases upward, and heavy particles have $m_b>0$.

The profile is a miniature atmosphere: thermal motion spreads particles upward while gravity favours lower positions. The Perrin sedimentation experiment measured particle counts at different microscope focal heights and used this competition to infer a molecular energy scale.^[ref] A stationary profile requires sufficiently slow background flow and controlled temperature; a growing deposit during ongoing settling is a different situation.

Take identical rigid spheres of radius $a$, particle density ${\rho }_p$ and displaced liquid density ${\rho }_f$, at uniform absolute temperature $T$. Let $z$ increase upwards from a reference layer and write $\Delta \rho ={\rho }_p-{\rho }_f>0$. The buoyancy correction gives the effective gravitational massThe actual grain mass is ${\rho }_{p}V$; the displaced liquid subtracts ${\rho }_{f}V$ from the effective gravitational mass. In a dilute, noninteracting particle suspension, the Boltzmann distribution therefore givesHere $n$ is number density and $k_B$ is the Boltzmann constant. Equivalently, the dilute osmotic pressure $\Pi =n{k}_{B}T$ supports the grains above it: $d\Pi /dz=-n{m}_{b}g$. Dividing by $n$ and integrating yields the same exponential function. No dynamic viscosity appears in this equilibrium argument. If ${\rho }_p<{\rho }_f$, the population instead increases upwards; at equal densities this gravitational model predicts uniform $n$.

A microscope counts a finite optical slab, not an infinitesimal plane. For equal viewed area $A$ and slab thickness $\delta$, an ideal uniform detection window centred at $z$ givesThus equal optical volumes preserve the exponential function count ratio exactly, even for finite $\delta$, provided the windows lie wholly inside the accessible chamber and have the same detection response. Unequal depth response or overlapping a boundary can corrupt this inference. Repeated observations estimate the mean count; closely spaced visits need not be independent because grains remain in view.

For an original chosen model, set $a=0.250\mu \mathrm{m}$, ${\rho }_p=1200\mathrm{kg}{\mathrm{m}}^{-3}$, ${\rho }_f=1000\mathrm{kg}{\mathrm{m}}^{-3}$, $T=298\mathrm{K}$ and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$. These are illustrative inputs, not Jean Baptiste Perrin's measurements. Using $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$ givesIf equal observation effort gives expected count $400$ in the reference layer, the expected counts $20$, $40$ and $60\mu \mathrm{m}$ higher are approximately $214.3$, $114.8$ and $61.49$. Fractions describe expected values over observations, not fractional grains. Using the actual grain mass instead of its buoyancy-corrected value would make the inferred Avogadro constant six times too small for these chosen densities.

Selective permeability means that a cell membrane permits different substances to pass at different rates under comparable conditions. Both the lipid bilayer and the available membrane proteins contribute to this selection.

Artificial boundaries can also have selective permeability. For osmosis, specify which solutes cross and over what observation interval: a boundary that retains one solute need not retain another. A semipermeable membrane is the ideal solvent-only boundary used in the osmotic pressure calculation.

In a planar good-solvent solution, a correlation blob of $g$ monomers has size $\xi \sim \ell g^{3/4}$. Its area fraction is $\varphi \sim g{\ell }^2/{\xi }^2\sim g^{-1/2}$, giving $g\sim {\varphi }^{-2}$ and $\xi \sim \ell {\varphi }^{-3/2}$. Each blob contributes order $k_{B}T$ to the osmotic free energy, so the surface osmotic pressure is $\Pi \sim k_{B}T/{\xi }^2\sim (k_{B}T/{\ell }^2){\varphi }^3$. This regime requires many blobs per chain and area fraction below close packing. In the dilute regime entire chains are the independently translating objects and $\Pi \sim k_{B}T\varphi /(N{\ell }^2)$. The two expressions match at the two-dimensional polymer overlap fraction ${\varphi }^{\ast }\sim N^{-1/2}$.

Solute potential is the contribution to water potential from dissolved material, with pure water assigned zero. In the nearly incompressible description, ${\Psi }_s=(RT/{\bar{V}}_w)\ln a_w$. In a dilute ideal chemical solution, ${\Psi }_s\simeq -RT{c}_{\mathrm{osm}}$, where $c_{\mathrm{osm}}$ is the osmotic concentration of dissolved particles. Thus solute potential is negative while the corresponding osmotic pressure is positive: ${\Psi }_s\simeq -\Pi$. This sign difference distinguishes a contribution to water potential from the compensating mechanical pressure.^[ref]

A solvent under conditions where the leading effective pair excluded volume of a long dilute polymer vanishes. In the simplest Flory–Huggins solution theory this means $\chi =1/2$. The chain is approximately an ideal chain, while three-segment interactions can still affect the osmotic pressure of a concentrated solution. A particular polymer-solvent pair often meets this condition only near its theta temperature.