# Child of Osmotic Pressure

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globals [ left-side ;; left side of the membrane right-side ;; right side of the membrane split ;; location of membrane pre-split ;; a value for calculating the change in membrane location equilibrium ;; the difference in number of particles moving each direction across membrane tick-delta ;; how much we advance the tick counter this time through collisions ;; list used to keep track of future collisions particle1 ;; first particle currently colliding particle2 ;; second particle currently colliding ] breed [particles particle] breed [membranes membrane] turtles-own [ speed mass energy old-pos new-pos part-type ] to setup ;; (for this model to work with NetLogo's new plotting features, ;; __clear-all-and-reset-ticks should be replaced with clear-all at ;; the beginning of your setup procedure and reset-ticks at the end ;; of the procedure.) clear-all reset-ticks set-default-shape particles "circle" create-container spawn-particles set split 0 calculate-wall set particle1 nobody set particle2 nobody set collisions [] ask particles [ check-for-wall-collision ] ask particles [ check-for-particle-collision ] my-update-plots end to create-container ask patches with [pycor = max-pycor or pycor = min-pycor] [ ;; sets the walls of the container set pcolor blue ] ask patches with [pxcor = max-pxcor or pxcor = min-pxcor] [ set pcolor blue ] set left-side patches with [pxcor < 0] ;; defines the left and right sides of the container equally set right-side patches with [pxcor > 0] set equilibrium 0 ;; sets equilibrium to starting value (the middle of the container) set split 0 ;; sets the starting location of the membrane ask patches with [pxcor = 0 and pycor != max-pycor and pycor != min-pycor and abs pycor mod 2 != 0] [ sprout-membranes 1 [ ;; create the membrane set shape "square" set color red set size 1.3 set heading 90 ] ] end to setup-particles ;; set variable values for each new particle set speed 0.5 if part-type = "solvent" [ set size 1 ] if part-type = "solute" [ set size 2 ] set mass (size * size) set energy (0.5 * mass * speed * speed) end to-report overlapping? ;; particle procedure ;; here, we use IN-RADIUS just for improved speed; the real testing ;; is done by DISTANCE report any? other turtles in-radius ((size + 2) / 2) with [distance myself < (size + [size] of myself) / 2] end to-report wall-lapping? ;; particle procedure report any? patches in-radius ((size + 2) / 2) with [pcolor = blue] or any? membranes in-radius ((size + 2) / 2) end to start-loc move-to one-of patches with [pcolor = black] ;; randomly distribute them throughout the world while [overlapping?] [ move-to one-of patches with [pcolor = black] ] while [wall-lapping?] [ move-to one-of patches with [pcolor = black] ] end to start-right-loc move-to one-of right-side with [pcolor = black] ;; randomly distribute them throughout the world while [overlapping?] [ move-to one-of right-side with [pcolor = black] ] while [wall-lapping?] [ move-to one-of right-side with [pcolor = black] ] end to start-left-loc move-to one-of left-side with [pcolor = black] ;; randomly distribute them throughout the world while [overlapping?] [ move-to one-of left-side with [pcolor = black] ] while [wall-lapping?] [ move-to one-of left-side with [pcolor = black] ] end to spawn-particles if solute = "Sugar" [ ;; checks to see the identity of the solute based on the chooser output-type "C12H22O11" ;; write the chemical formula in the output area create-particles solute-left [ ;; create solute particles based on slider value set color white set shape "circle" set part-type "solute" ;; define these as solute particles setup-particles start-left-loc ] create-particles solute-right [ ;; create solute particles based on slider value set color white set shape "circle" set part-type "solute" setup-particles start-right-loc ] ] if solute = "Sodium Chloride" [ output-type "NaCl" create-particles solute-left [ set color white set shape "circle" set part-type "solute" setup-particles start-left-loc ask patch-here [ sprout-particles 1 [ ;; solutes that dissociate in water split into their respective ions set color white ;; since NaCl splits into two ions, one new turtle is sprouted for each solute molecule set shape "circle" set part-type "solute" setup-particles fd 0.2 ] ] ] create-particles solute-right [ set color white set shape "circle" set part-type "solute" setup-particles start-right-loc ask patch-here [ sprout-particles 1 [ set color white set shape "circle" set part-type "solute" setup-particles fd 0.2 ] ] ] ] if solute = "Magnesium Chloride" [ output-type "MgCl2" create-particles solute-left [ set color white set shape "circle" set part-type "solute" setup-particles start-left-loc ask patch-here [ sprout-particles 2 [ ;; splits into 3 ions..so 2 new turtles are sprouted set color white set shape "circle" set part-type "solute" setup-particles fd 0.2 ] ] ] create-particles solute-right [ set color white set shape "circle" set part-type "solute" setup-particles start-right-loc ask patch-here [ sprout-particles 2 [ set color white set shape "circle" set part-type "solute" setup-particles fd 0.2 ] ] ] ] if solute = "Aluminum Chloride" [ output-type "AlCl3" create-particles solute-left [ set color white set shape "circle" set part-type "solute" setup-particles start-left-loc ask patch-here [ sprout-particles 3 [ ;; splits into 4 ions...so 3 new turtles are sprouted set color white set shape "circle" set part-type "solute" setup-particles fd 0.2 ] ] ] create-particles solute-right [ set color white set shape "circle" set part-type "solute" setup-particles start-right-loc ask patch-here [ sprout-particles 3 [ set color white set shape "circle" set part-type "solute" setup-particles fd 0.2 ] ] ] ] create-particles (100 - count particles with [part-type = "solute" and xcor < split]) [ ;; creates 1000 water molecules set color blue + 2 set part-type "solvent" ;; define these as solvents setup-particles start-left-loc ] create-particles (100 - count particles with [part-type = "solute" and xcor > split]) [ ;; creates 1000 water molecules set color blue + 2 set part-type "solvent" ;; define these as solvents setup-particles start-right-loc ] end to particle-jump ask particles with [xcor >= pre-split and xcor <= split and part-type = "solutes"] [ ;; check for solutes in the way of a membrane jump set xcor (split - pre-split) + 1 ; recalculate-particles-that-just-collided ] ask particles with [xcor <= pre-split and xcor >= split and part-type = "solutes"] [ set xcor (pre-split - split) + 1 ; recalculate-particles-that-just-collided ] end to calculate-wall set pre-split split ;; save location of the membrane before it moves let nudge ((equilibrium * -1) / 10) ;; calculates the amount to move the membrane set split split + nudge ;; set split to be the new location of the membrane particle-jump ;; move solutes in the way of the membrane jump ask membranes [ set xcor xcor + nudge ;; move membrane turtles according to nudge value ] set left-side patches with [pxcor < split] ;; redefine right and left sides set right-side patches with [pxcor > split] set equilibrium 0 ;; reset equilibrium so we can keep track of what happens during the next tick end to go if count turtles-on right-side = 0 or count turtles-on left-side = 0 [ ;; stops model if membrane reaches either edge user-message "The membrane has burst! Make sure you have some solute on both sides!" stop ] ask particles [ set old-pos xcor ] choose-next-collision ask particles with [ part-type = "solute" ] [ ;; bouncing for solutes hitting membrane if any? membranes in-cone 2 180 [ set heading (- heading) ] ] ask particles [ jump speed * tick-delta ] perform-next-collision ask particles with [ part-type = "solvent" ] [ if old-pos < split and xcor >= split [ ;; if a solvent moves from the left of the membrane to the right of the membrane set equilibrium equilibrium + 1 ;; add one to equilibrium ] if old-pos > split and xcor <= split [ ;; if a solvent moves from the right of the membrane to the left of the membrane set equilibrium equilibrium - 1 ;; subract one from equilibrium ] ] calculate-wall ;; recalculate the new location of the wall tick-advance tick-delta recalculate-particles-that-just-collided my-update-plots end to recalculate-particles-that-just-collided ;; Since only collisions involving the particles that collided most recently could be affected, ;; we filter those out of collisions. Then we recalculate all possible collisions for ;; the particles that collided last. The ifelse statement is necessary because ;; particle2 can be either a particle or a string representing a wall. If it is a ;; wall, we don't want to invalidate all collisions involving that wall (because the wall's ;; position wasn't affected, those collisions are still valid. ifelse is-turtle? particle2 [ set collisions filter [item 1 ? != particle1 and item 2 ? != particle1 and item 1 ? != particle2 and item 2 ? != particle2] collisions ask particle2 [ check-for-wall-collision ] ask particle2 [ check-for-particle-collision ] ] [ set collisions filter [item 1 ? != particle1 and item 2 ? != particle1] collisions ] if particle1 != nobody [ ask particle1 [ check-for-wall-collision ] ] if particle1 != nobody [ ask particle1 [ check-for-particle-collision ] ] ;; Slight errors in floating point math can cause a collision that just ;; happened to be calculated as happening again a very tiny amount of ;; time into the future, so we remove any collisions that involves ;; the same two particles (or particle and wall) as last time. set collisions filter [item 1 ? != particle1 or item 2 ? != particle2] collisions ;; All done. set particle1 nobody set particle2 nobody end ;; check-for-particle-collision is a particle procedure that determines the time it takes ;; to the collision between two particles (if one exists). It solves for the time by representing ;; the equations of motion for distance, velocity, and time in a quadratic equation of the vector ;; components of the relative velocities and changes in position between the two particles and ;; solves for the time until the next collision to check-for-particle-collision let my-x xcor let my-y ycor let my-particle-size size let my-x-speed speed * sin heading let my-y-speed speed * cos heading ask other particles [ let dpx (xcor - my-x) ;; relative distance between particles in the x direction let dpy (ycor - my-y) ;; relative distance between particles in the y direction let x-speed (speed * sin heading) ;; speed of other particle in the x direction let y-speed (speed * cos heading) ;; speed of other particle in the x direction let dvx (x-speed - my-x-speed) ;; relative speed difference between particles in x direction let dvy (y-speed - my-y-speed) ;; relative speed difference between particles in y direction let sum-r (((my-particle-size) / 2 ) + (([size] of self) / 2 )) ;; sum of both particle radii ;; To figure out what the difference in position (P1) between two particles at a future ;; time (t) will be, one would need to know the current difference in position (P0) between the ;; two particles and the current difference in the velocity (V0) between the two particles. ;; ;; The equation that represents the relationship is: ;; P1 = P0 + t * V0 ;; we want find when in time (t), P1 would be equal to the sum of both the particle's radii ;; (sum-r). When P1 is equal to is equal to sum-r, the particles will just be touching each ;; other at their edges (a single point of contact). ;; ;; Therefore we are looking for when: sum-r = P0 + t * V0 ;; ;; This equation is not a simple linear equation, since P0 and V0 should both have x and y ;; components in their two dimensional vector representation (calculated as dpx, dpy, and ;; dvx, dvy). ;; ;; By squaring both sides of the equation, we get: ;; (sum-r) * (sum-r) = (P0 + t * V0) * (P0 + t * V0) ;; When expanded gives: ;; (sum-r ^ 2) = (P0 ^ 2) + (t * PO * V0) + (t * PO * V0) + (t ^ 2 * VO ^ 2) ;; Which can be simplified to: ;; 0 = (P0 ^ 2) - (sum-r ^ 2) + (2 * PO * V0) * t + (VO ^ 2) * t ^ 2 ;; Below, we will let p-squared represent: (P0 ^ 2) - (sum-r ^ 2) ;; and pv represent: (2 * PO * V0) ;; and v-squared represent: (VO ^ 2) ;; ;; then the equation will simplify to: 0 = p-squared + pv * t + v-squared * t^2 let p-squared ((dpx * dpx) + (dpy * dpy)) - (sum-r ^ 2) ;; p-squared represents difference ;; of the square of the radii and the square of the initial positions let pv (2 * ((dpx * dvx) + (dpy * dvy))) ;; vector product of the position times the velocity let v-squared ((dvx * dvx) + (dvy * dvy)) ;; the square of the difference in speeds ;; represented as the sum of the squares of the x-component ;; and y-component of relative speeds between the two particles ;; p-squared, pv, and v-squared are coefficients in the quadratic equation shown above that ;; represents how distance between the particles and relative velocity are related to the time, ;; t, at which they will next collide (or when their edges will just be touching) ;; Any quadratic equation that is a function of time (t) can be represented as: ;; a*t*t + b*t + c = 0, ;; where a, b, and c are the coefficients of the three different terms, and has solutions for t ;; that can be found by using the quadratic formula. The quadratic formula states that if a is ;; not 0, then there are two solutions for t, either real or complex. ;; t is equal to (b +/- sqrt (b^2 - 4*a*c)) / 2*a ;; the portion of this equation that is under a square root is referred to here ;; as the determinant, D1. D1 is equal to (b^2 - 4*a*c) ;; and: a = v-squared, b = pv, and c = p-squared. let D1 pv ^ 2 - (4 * v-squared * p-squared) ;; the next test tells us that a collision will happen in the future if ;; the determinant, D1 is > 0, since a positive determinant tells us that there is a ;; real solution for the quadratic equation. Quadratic equations can have solutions ;; that are not real (they are square roots of negative numbers). These are referred ;; to as imaginary numbers and for many real world systems that the equations represent ;; are not real world states the system can actually end up in. ;; Once we determine that a real solution exists, we want to take only one of the two ;; possible solutions to the quadratic equation, namely the smaller of the two the solutions: ;; (b - sqrt (b^2 - 4*a*c)) / 2*a ;; which is a solution that represents when the particles first touching on their edges. ;; instead of (b + sqrt (b^2 - 4*a*c)) / 2*a ;; which is a solution that represents a time after the particles have penetrated ;; and are coming back out of each other and when they are just touching on their edges. let time-to-collision -1 if D1 > 0 [ set time-to-collision (- pv - sqrt D1) / (2 * v-squared) ] ;; solution for time step ;; if time-to-collision is still -1 there is no collision in the future - no valid solution ;; note: negative values for time-to-collision represent where particles would collide ;; if allowed to move backward in time. ;; if time-to-collision is greater than 1, then we continue to advance the motion ;; of the particles along their current trajectories. They do not collide yet. if time-to-collision > 0 [ ;; time-to-collision is relative (ie, a collision will occur one second from now) ;; We need to store the absolute time (ie, a collision will occur at time 48.5 seconds. ;; So, we add clock to time-to-collision when we store it. ;; The entry we add is a three element list of the time to collision and the colliding pair. set collisions fput (list (time-to-collision + ticks) self myself) collisions ] ] end ;; determines when a particle will hit any of the four walls or membrane to check-for-wall-collision ;; particle procedure ;; right & left walls let x-speed (speed * sin heading) if x-speed != 0 [ ;; solve for how long it will take particle to reach right wall let right-interval (max-pxcor - 0.5 - xcor - size / 2) / x-speed if right-interval > 0 [ assign-colliding-wall right-interval "right wall" ] ;; solve for time it will take particle to reach left wall let left-interval ((- max-pxcor) + 0.5 - xcor + size / 2) / x-speed if left-interval > 0 [ assign-colliding-wall left-interval "left wall" ] ] ; ;; membrane ; if x-speed != 0 and part-type = "solute" and xcor < split ; [ ;; solve for how long it will take particle to reach the membrane from the left ; let left-interval2 (split - xcor - size / 2) / x-speed ; if left-interval2 > 0 ; [ assign-colliding-wall left-interval2 "membrane-left" ] ] ; if x-speed != 0 and part-type = "solute" and xcor < split ; [ ;; solve for time it will take particle to reach the membrane from the right ; let right-interval2 (split - xcor + size / 2) / x-speed ; if right-interval2 > 0 ; [ assign-colliding-wall right-interval2 "membrane-right" ] ] ;; top & bottom walls let y-speed (speed * cos heading) if y-speed != 0 [ ;; solve for time it will take particle to reach top wall let top-interval (max-pycor - 0.5 - ycor - size / 2) / y-speed if top-interval > 0 [ assign-colliding-wall top-interval "top wall" ] ;; solve for time it will take particle to reach bottom wall let bottom-interval ((- max-pycor) + 0.5 - ycor + size / 2) / y-speed if bottom-interval > 0 [ assign-colliding-wall bottom-interval "bottom wall" ] ] end to assign-colliding-wall [time-to-collision wall] ;; particle procedure ;; this procedure is used by the check-for-wall-collision procedure ;; to assemble the correct particle-wall pair ;; time-to-collision is relative (ie, a collision will occur one second from now) ;; We need to store the absolute time (ie, a collision will occur at time 48.5 seconds. ;; So, we add clock to time-to-collision when we store it. let colliding-pair (list (time-to-collision + ticks) self wall) set collisions fput colliding-pair collisions end to choose-next-collision if collisions = [] [ stop ] ;; Sort the list of projected collisions between all the particles into an ordered list. ;; Take the smallest time-step from the list (which represents the next collision that will ;; happen in time). Use this time step as the tick-delta for all the particles to move through let winner first collisions foreach collisions [ if first ? < first winner [ set winner ? ] ] ;; winner is now the collision that will occur next let dt item 0 winner ;; If the next collision is more than 1 in the future, ;; only advance the simulation one tick, for smoother animation. set tick-delta dt - ticks if tick-delta > 1 [ set tick-delta 1 set particle1 nobody set particle2 nobody stop ] set particle1 item 1 winner set particle2 item 2 winner end to perform-next-collision ;; deal with 3 possible cases: ;; 1) no collision at all if particle1 = nobody [ stop ] ;; 2) particle meets wall if is-string? particle2 [ if particle2 = "left wall" or particle2 = "right wall" ;or particle2 = "membrane-left" or particle2 = "membrane-right" [ ask particle1 [ set heading (- heading) ] stop ] if particle2 = "top wall" or particle2 = "bottom wall" [ ask particle1 [ set heading 180 - heading ] stop ] ] ;; 3) particle meets particle ask particle1 [ collide-with particle2 ] end to collide-with [other-particle] ;; particle procedure ;;; PHASE 1: initial setup ;; for convenience, grab some quantities from other-particle let mass2 [mass] of other-particle let speed2 [speed] of other-particle let heading2 [heading] of other-particle ;; modified so that theta is heading toward other particle let theta towards other-particle ;;; PHASE 2: convert velocities to theta-based vector representation ;; now convert my velocity from speed/heading representation to components ;; along theta and perpendicular to theta let v1t (speed * cos (theta - heading)) let v1l (speed * sin (theta - heading)) ;; do the same for other-particle let v2t (speed2 * cos (theta - heading2)) let v2l (speed2 * sin (theta - heading2)) ;;; PHASE 3: manipulate vectors to implement collision ;; compute the velocity of the system's center of mass along theta let vcm (((mass * v1t) + (mass2 * v2t)) / (mass + mass2) ) ;; now compute the new velocity for each particle along direction theta. ;; velocity perpendicular to theta is unaffected by a collision along theta, ;; so the next two lines actually implement the collision itself, in the ;; sense that the effects of the collision are exactly the following changes ;; in particle velocity. set v1t (2 * vcm - v1t) set v2t (2 * vcm - v2t) ;;; PHASE 4: convert back to normal speed/heading ;; now convert my velocity vector into my new speed and heading set speed sqrt ((v1t * v1t) + (v1l * v1l)) ;; if the magnitude of the velocity vector is 0, atan is undefined. but ;; speed will be 0, so heading is irrelevant anyway. therefore, in that ;; case we'll just leave it unmodified. set energy (0.5 * mass * speed ^ 2) if v1l != 0 or v1t != 0 [ set heading (theta - (atan v1l v1t)) ] ;; and do the same for other-particle ask other-particle [ set speed sqrt ((v2t ^ 2) + (v2l ^ 2)) set energy (0.5 * mass * speed ^ 2) if v2l != 0 or v2t != 0 [ set heading (theta - (atan v2l v2t)) ] ] end to my-update-plots set-current-plot "Water#" set-current-plot-pen "left" plot count particles with [pxcor < split and part-type = "solvent"] set-current-plot-pen "right" plot count particles with [pxcor > split and part-type = "solvent"] end

There is only one version of this model, created almost 10 years ago by Nathan Holbert.

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**Parent:** Osmotic Pressure

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