# Hydrogel A **hydrogel** is a [[Polymer|polymer]] network, held together by crosslinks, that absorbs a large volume of [[Water|water]] and holds it without dissolving. The chains are hydrophilic, so [[Properties_of_water|water]] would carry them apart if it could; the crosslinks stop it, and the material settles at a swelling equilibrium where the pull of mixing balances the elastic pull of the stretched network. The result is a solid that is ninety per cent or more liquid by weight, keeps a shape, transmits load, and lets small molecules [[Diffusion|diffuse]] through it almost as fast as through free water. Gelatin dessert, a soft contact lens, a wound dressing and the extracellular matrix of living [[Bone|tissue]] are all hydrogels. In the microsim below the reader turns one dial, the crosslink density n, and watches a network in water swell or collapse. The equation that answers is the Flory–Rehner condition for swelling equilibrium, `ln(1 - phi) + phi + chi·phi^2 + V_1·n·(phi^(1/3) - phi/2) = 0`, where φ is the volume fraction of polymer in the swollen gel, χ the polymer–solvent interaction parameter and V₁ the molar volume of the solvent.[^flory-rehner] The readouts are the swelling ratio Q = 1/φ and the shear modulus of the swollen gel, `G ~ n·k·T·phi^(1/3)`.[^spec-m36] A second control moves χ, which is what [[Temperature|temperature]] and salt do to a real gel. Three presets sit on the dial: a tissue scaffold, a wound dressing and a soft contact lens. On the [[Materials_science]] flagship this article serves the *Biomaterials* section of Part VII — Research, where the hydrogel is the soft [[Biomaterial|biomaterial]] against which stiff implant materials such as [[Hydroxyapatite|hydroxyapatite]] and [[Nacre|nacre]] are compared. Its sim is a sibling of the [[Rubber_elasticity|rubber elasticity]] sim: the same entropic network, now with a solvent in it. ## Chemistry A hydrogel needs two things at once: chains that water likes, and junctions water cannot undo. Hydrophilic groups — hydroxyl, amide, carboxylate, ether oxygen — make each monomer hydrogen-bond to water, so the free energy of mixing drives the solvent in.[^likharev-sm1] The junctions are covalent bonds formed during synthesis, or physical associations: [[Hydrogen_bond|hydrogen-bonded]] helices, crystallites, entanglements, ionic bridges. Covalent gels cannot be remelted; physical gels dissolve, warm or ion-exchange back into a [[Solution_(chemistry)|solution]], which is why gelatin sets on cooling and melts in the mouth. ### Classification Hydrogels are sorted along four axes, with one answer on each. By junction type they are chemical (covalently crosslinked) or physical. By origin they are natural — polysaccharides and proteins — synthetic, or hybrid. By charge they are neutral, anionic, cationic or zwitterionic; a charged network swells far more than a neutral one at the same crosslink density, because the mobile counter-ions add an osmotic term the neutral Flory–Rehner equation does not carry. By response they are inert or stimulus-responsive. Every axis but charge acts on the page's two dials, χ and n, so a reader who moves those has covered most of the map.[^spec-m36] ### Preparation Three routes dominate. In free-radical copolymerization a hydrophilic monomer is polymerized in water with a small fraction of a difunctional monomer that becomes the crosslink; the crosslinker mole fraction sets n directly, which is why it is the natural control variable. In [[Step-growth_polymerization|step-growth]] gelation two multi-arm precursors react end to end, and the network is far more uniform, because every junction is placed by the chemistry rather than by chance. In physical gelation a dissolved polymer is cooled, acidified or exposed to a crosslinking ion, and junctions form without any new covalent bond. The [[Degree_of_polymerization|chain length]] between junctions is the inverse of n: raising the crosslinker from 0.1 to 1 mol % shortens the strands roughly tenfold, and in the equation V₁·n rises by the same factor. Working the sim through that decade with χ held at 0.45 takes the swelling ratio from Q = 46 to Q = 14 and the modulus from about 4 kPa to about 59 kPa, a gel that drapes replaced by a gel that stands up.[^derived-hg] ### Peptide based hydrogels Short peptides designed to fold into β-sheets stack into nanofibres held by [[Hydrogen_bond|hydrogen bonds]] between backbone amides; above a low concentration the fibres entangle and the suspension stops flowing. Their junctions are many weak contacts rather than few strong ones, so they yield and re-form under shear — the property that lets them be injected through a needle and set again in place. In the sim's variables a peptide gel is a network whose effective n is set by entanglement and rises steeply with concentration, and whose χ sits near the θ value of one-half, where the swelling ratio follows the power law derived below.[^derived-hg] ### Other Polysaccharide gels form by ionic bridging or by helix formation on cooling; protein gels by denaturation and re-association. Synthetic networks of poly(ethylene glycol), poly(vinyl alcohol) and poly(acrylic acid) give control over n that natural polymers do not. Two architectures sit outside the single-network picture. In an interpenetrating network two networks form in the same volume without bonding; in a double network a brittle, densely crosslinked first network is swollen with a loose second one, and the first sacrifices itself to dissipate energy while the second holds the piece together. Both are [[Composite_material|composites]] in which one phase is water, and both fall outside the sim's one-network model. ## Mechanical properties A hydrogel is a solid on the timescale of a push and a liquid on the timescale of an hour, and its stiffness is set not by bond strength but by how many strands per unit volume can be stretched. Moduli span four decades, from under a kilopascal to several megapascals, and every decade is bought with crosslink density.[^derived-hg] ### Rubber elasticity The elastic half of the Flory–Rehner condition is [[Rubber_elasticity|rubber elasticity]] applied to a network already stretched by its own solvent. A dry [[Elastomer|elastomer]] resists deformation because stretching a chain reduces the conformations available to it and so lowers its [[Entropy|entropy]]; the restoring force is `f = -T·(dS/dL)`, proportional to absolute temperature and to the number of strands, not to any bond stiffness.[^likharev-sm1] Swelling stretches every strand isotropically, and the affine model gives the elastic term `V_1·n·(phi^(1/3) - phi/2)`, the φ^(1/3) being the linear expansion factor of the swollen network.[^flory-rehner] Setting the sum to zero states that the [[Chemical_potential|chemical potential]] of water inside the gel equals that outside.[^flory-rehner] The exchange the sim makes visible is between swelling and stiffness. Doubling n at χ = 0.45 from 56 to 111 mol per cubic metre takes Q from 14.1 down to 10.2 and G from 59 to 132 kPa.[^derived-hg] At the θ condition χ = 1/2 the equation solves in closed form. Expanding ln(1 − φ) + φ = −φ²/2 − φ³/3 − … cancels the φ² term against χφ², leaving −φ³/3 + V₁·n·φ^(1/3) ≈ 0, so `phi = (3·V_1·n)^(3/8)` and Q scales as n^(−3/8). The closed form sits within 4 % of the numerical root at V₁n = 10⁻⁴ and within 8 % at 10⁻³, and it is why swelling answers chemistry so sluggishly: eight-fold more crosslinker only halves the water content.[^derived-hg] The modulus readout is the affine estimate and is ILLUSTRATIVE, because the affine assumption overstates G for densely crosslinked networks, where the contact-lens preset lands.[^spec-m36] *Try:* set χ to 0.45 and sweep n, then return n to the wound-dressing preset and sweep χ. The first sweep changes stiffness and water content together; the second changes water content alone until the network collapses. ### Viscoelasticity Under a step of strain the stress in a hydrogel does not stay put: it decays as chains slide past one another and physical junctions detach and re-form. The material is [[Viscoelasticity|viscoelastic]], its response written as a storage modulus G′ and a loss modulus G″ that both depend on frequency. A covalent gel keeps a finite G′ at zero frequency; a physical gel relaxes to zero given long enough, and its terminal relaxation time is the lifetime of a junction. [[Creep_(deformation)|Creep]] and stress relaxation are two views of the same spectrum, and the frequency and [[Temperature|temperature]] axes trade through [[Time–temperature_superposition|time–temperature superposition]]. None of this appears in the sim's equilibrium condition, which describes the end state and says nothing about the road to it. ### Poroelasticity The second, slower relaxation is not molecular at all: it is water moving. Compressing a gel raises the pressure of the fluid in its pores, and the fluid must flow out through the network before the deformation is complete. The flow obeys [[Darcy's_law|Darcy's law]], and the coupled problem gives a [[Fick's_laws_of_diffusion|diffusion]] equation for the solvent with a cooperative diffusivity D of order G·κ/η, where κ is the network permeability and η the [[Viscosity|viscosity]] of water. The equilibration time then goes as the square of the specimen size, τ ≈ L²/D. Taking D = 3×10⁻¹¹ m²/s as an order-of-magnitude value for a soft synthetic network — a figure chosen for illustration, not measured on this page's shelf — a 1 mm slab needs about 9 hours to reach the swelling ratio the sim reports, while a 10 μm feature inside a printed scaffold needs about 3 seconds.[^illustrative-poro] A hundredfold change in size is a ten-thousandfold change in time, which is why thin gels are used wherever response matters. ### Toughness and hysteresis A single-network hydrogel is weak in the way a wet tissue is weak: a crack finds the shortest strand, breaks it, and the load moves to the next strand along the same path. Toughening requires a mechanism that dissipates energy away from the crack tip, and the successful ones all add [[Hysteresis|hysteresis]] — a loading curve that does not retrace on unloading, the missing area being the energy spent. Sacrificial ionic bonds, a brittle first network, and crystalline domains that unfold under load all work this way, and all trade recovery for [[Fracture_toughness|toughness]]. [[Self-healing_material|Self-healing]] gels are the limiting case, built so the sacrificial bonds re-form on their own. ### Environmental response Everything a stimulus does to a hydrogel it does through χ. The mixing term in the equation favours swelling only while χ is below one-half; above it, polymer–polymer contacts are preferred to polymer–water contacts and the network collapses. Holding V₁n at 10⁻³ and walking χ from 0.3 to 0.8 takes Q from 25.9 to 1.9 and the water content from 96 to 47 volume per cent, and most of that change happens across a narrow band near χ = 0.5.[^derived-hg] The sharpness is a genuine [[Phase_transition|phase transition]] of the same mean-field type as the liquid–gas transition, with the same free energy developing two minima.[^likharev-sm4] Thermoresponsive gels are designed so χ crosses one-half within a few degrees of body temperature; the collapse temperature usually quoted for poly(N-isopropylacrylamide) in water is not documented on this page's shelf and is left as a claim to source.[citation needed][^lcst-cn] pH-responsive gels move χ by ionizing their own side groups, which also switches on the counter-ion term the neutral equation omits. ### Additives Fillers change the mechanical answer without changing the network's chemistry. Rigid nanoparticles raise the modulus and act as multifunctional crosslinks, since chains adsorbed on one particle are effectively joined; clays and silica work this way, and the effective n rises although no covalent bond was made. Fibres raise toughness by bridging cracks. Salts and cosolvents move χ instead, and humectants reduce the [[Solubility|activity]] of water so the gel resists drying. Porogens — particles later dissolved away — leave voids that raise permeability and shorten the poroelastic time. ### Processing techniques Casting in a mould, photopolymerizing through a mask and extruding through a needle are the three common routes, and all three contend with a part made at one water content and used at another. A gel cast at φ = 0.25 and then equilibrated to φ = 0.07 grows by a linear factor of (0.25/0.07)^(1/3) = 1.53, so every dimension in the mould must be divided by that number.[^derived-hg] Extrusion [[3D_printing|3D printing]] of hydrogel inks adds the constraints any additive process carries: the Portal Book's [[Design_for_additive_manufacturing|design-for-additive]] chapter tabulates minimum wall thickness and feature size process by process, and although its tables cover metal and thermoplastic processes rather than hydrogel extrusion, the rule they encode — that the smallest printable feature is set by the deposition physics, not by the model — applies unchanged.[^barnes-dfam] For a gel ink the extra rule is that the filament must yield under nozzle pressure and recover its modulus before the next layer lands on it. ### Yield Stress Measurement in Hydrogels A physical gel behaves as a solid below a yield stress τ_y and flows above it, and printing, injection and spreading all depend on where that threshold sits. Two rheometer protocols are standard. A steady flow curve sweeps shear rate and fits the Herschel–Bulkley form `tau = tau_y + K·gamma_dot^n`, reading τ_y as the intercept at zero rate; that fit is a display fit, not a measured law, and is ILLUSTRATIVE wherever it appears. An amplitude sweep instead raises the strain at fixed frequency and takes the yield point where G″ overtakes G′. The two rarely agree, because one measures where flow becomes steady and the other where the elastic response fails, so a yield stress is meaningful only with its protocol attached. ## Applications The uses of hydrogels follow from three properties that no other class of material combines: high water content, a modulus that can be matched to soft tissue, and free diffusion of small solutes. ### Biomaterials Soft tissue has a shear modulus in the kilopascal range, and a hydrogel is the only synthetic material that can meet it. Matching matters mechanically, because a stiff implant carries load the surrounding tissue then stops carrying, and biologically, because cells read the stiffness of what they sit on. The sim's scaffold preset — n = 5.6 mol per cubic metre, χ = 0.45, Q = 46, about 98 volume per cent water and G near 4 kPa — is in that range, and its water content is what lets nutrients reach cells hundreds of micrometres from the nearest vessel.[^derived-hg] [[Biocompatibility|Biocompatibility]] is a separate question from mechanics: the network may be inert while unreacted crosslinker, initiator fragments or degradation products are not, so purification is part of the specification. Degradable gels break down at a rate matched to [[Tissue_engineering|tissue]] ingrowth. ### Soft contact lenses The soft lens was the first mass-market hydrogel. Otto Wichterle and Drahoslav Lím described hydrophilic crosslinked gels for biological use in *Nature* in 1960, and the lens that followed traded the rigidity of earlier materials for a gel that drapes on the cornea.[^wichterle] A lens sits at the stiff end of the range: the sim's lens preset, n = 2,780 mol per cubic metre with χ = 0.7, gives φ = 0.53 and about 44 per cent water by weight.[^derived-hg] The design tension is that the property the eye needs most, oxygen permeability, rises with water content, while handling and optical stability want a stiffer, drier gel; silicone-containing hydrogels break the tension by carrying oxygen through a phase that is not water at all. ### Research Current work runs along the three axes the equation exposes. Toughening attacks the elastic term, adding sacrificial networks so the energy to break a strand is no longer the energy to break the material. Responsiveness attacks χ, building gels that turn a chemical signal into a volume change large enough to act as a valve, a [[Sensor|sensor]] or a soft actuator. Transport attacks the permeability κ, using porogens and printed channels to shorten the poroelastic time. Gels dried without collapse become [[Aerogel|aerogels]], and gels loaded with conductive phases are becoming the interface between electronics and tissue in [[Biomedical_engineering|biomedical engineering]]. *See also:* [[Biomaterial]] · [[Tissue_engineering]] · [[Biocompatibility]] · [[Nacre]] · [[Hydroxyapatite]] · [[Bone]] · [[Rubber_elasticity]] · [[Viscoelasticity]] · [[Polymer_science]] ## References [^flory-rehner]: Flory, Paul J.; Rehner, John (1943). "Statistical Mechanics of Cross-Linked Polymer Networks II. Swelling." *Journal of Chemical Physics* 11 (11): 521–526. https://doi.org/10.1063/1.1723792 — the swelling-equilibrium condition used on this page, in which the mixing terms of the Flory–Huggins free energy are balanced against the affine elastic term. Part I, "Rubberlike Elasticity", appeared in the same volume; its pages and DOI are to pin. [^wichterle]: Wichterle, Otto; Lím, Drahoslav (1960). "Hydrophilic Gels for Biological Use." *Nature* 185 (4706): 117–118. DOI to pin. [^likharev-sm1]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 1, "Review of thermodynamics" (pp. 5–28): free-energy minimization at fixed temperature and the entropic origin of the elastic force (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^likharev-sm4]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 4, "Phase transitions" (pp. 107–142): the mean-field free energy with two minima, and the van der Waals transition whose mathematics the volume collapse of a responsive gel repeats (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^barnes-dfam]: Barnes, John; Simpson, Timothy (2025). *Additive Manufacturing Essentials*. Chapter 4, "Design for Additive Manufacturing" (pp. 75–100): minimum feature size and wall thickness by process, Tables 4.4 and 4.6 (pp. 82–86, 93), and the STL chord-error trade-off (pp. 78–79). The tables cover powder-bed, directed-energy, fused-deposition and sintering processes, not hydrogel extrusion (page to pin). [^illustrative-poro]: ILLUSTRATIVE. The cooperative diffusivity D = 3×10⁻¹¹ m²/s used for the poroelastic estimate is an order-of-magnitude value chosen for this article, not a measurement from any Portal Book on this page's shelf; only the scaling τ ≈ L²/D and the ratio between the two sizes are claimed. Computed for this article: (10⁻³ m)²/D = 3.3×10⁴ s ≈ 9 h and (10⁻⁵ m)²/D = 3.3 s. [^lcst-cn]: Citation needed. The collapse temperature commonly quoted for poly(N-isopropylacrylamide) in water is not given in any Portal Book or open text on this page's shelf. The record that would settle it is a primary polymer-physics paper reporting the lower critical solution temperature with its concentration and heating rate; none is cited here rather than reproduce a number without a source. [^spec-m36]: Matter & Energy Cluster contract, `_registry/plans/MATERIALS_SCIENCE_SECTIONS.md` row M36: sim concept (new sibling of `Rubber_elasticity`), the Flory–Rehner condition `ln(1 - phi) + phi + chi·phi^2 + V_1·n·(phi^(1/3) - phi/2) = 0` solved by `hydro.util.bisect`, the crosslink density n (or χ) as the control, and the readouts Q = 1/φ and `G ~ n·k·T·phi^(1/3)` with contact-lens, wound-dressing and scaffold presets. [^derived-hg]: Computed for this article by bisection on the Flory–Rehner condition of [^flory-rehner], with V₁ = 18×10⁻⁶ m³/mol for water, T = 310 K and G = n·R·T·φ^(1/3). At χ = 0.45: V₁n = 10⁻⁴ (n = 5.6 mol/m³) gives φ = 0.0216, Q = 46.3, 97.8 vol % water and G = 4.0 kPa; V₁n = 10⁻³ (n = 55.6) gives φ = 0.0710, Q = 14.1, 92.9 vol % and G = 59 kPa; V₁n = 2×10⁻³ (n = 111) gives Q = 10.2 and G = 132 kPa. At χ = 0.7 and V₁n = 5×10⁻² (n = 2,780) φ = 0.527, Q = 1.90, 43.8 wt % water at a polymer density of 1.15 g/cm³. The χ walk at V₁n = 10⁻³ gives Q = 25.9, 18.4, 9.5, 3.9, 2.4 and 1.9 at χ = 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8. The θ-solvent closed form φ = (3·V₁n)^(3/8) gives 0.0477 against the exact 0.0460 at V₁n = 10⁻⁴ and 0.1132 against 0.1050 at 10⁻³. The mould-shrinkage factor is (0.25/0.07)^(1/3) = 1.53. ## External links - The Wikipedia pair's *External links* section lists hydrogel reviews and supplier data; the primary paper and Portal Book chapters above are the sources of this page. ## Further reading - Flory, Paul J.; Rehner, John (1943). *Statistical Mechanics of Cross-Linked Polymer Networks*, Parts I and II, *Journal of Chemical Physics* 11 — the elastic and swelling halves of the equation on this page. https://doi.org/10.1063/1.1723792 - Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM*, Chapters 1 and 4, for the free-energy minimization and the mean-field collapse. https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics - Barnes, John; Simpson, Timothy (2025). *Additive Manufacturing Essentials*, Chapter 4, for the design rules that govern printed scaffolds. <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Hydrogel.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Hydrogel* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Hydrogel.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Hydrogel.html" data-title="Hydrogel"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Hydrogel) : [Wikitube](https://en.wikitube.io/wiki/Hydrogel) · pinned revision [1373945145](https://en.wikipedia.org/w/index.php?oldid=1373945145) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M36 · sim pending (matter/Hydrogel).*