Transient separation of 2 chemicals in 1D, due to their different diffusion rates¶

Starting with identical concentrations in one of the bins, they initially separate...¶

until they eventually attain identical concentrations at equilibrium¶

For a complete separation by means of membranes, see experiment membranes_3

TAGS : "diffusion 1D", "basic"¶

In [1]:

LAST_REVISED = "May 30, 2025"
LIFE123_VERSION = "1.0.0rc3"        # Library version this experiment is based on

In [2]:

#import set_path                    # Using MyBinder?  Uncomment this before running the next cell!

In [3]:

#import sys
#sys.path.append("C:/some_path/my_env_or_install")   # CHANGE to the folder containing your venv or libraries installation!
# NOTE: If any of the imports below can't find a module, uncomment the lines above, or try:  import set_path   

from life123 import BioSim1D, ChemData, PlotlyHelper, check_version

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

Prepare the initial system¶

with identical bin concentrations of the chemicals A and B, in the center bin.
B diffuses much faster than A

In [5]:

chem_data = ChemData(names=["A", "B"], diffusion_rates=[0.1, 2.], plot_colors=["turquoise", "green"])

bio = BioSim1D(n_bins=9, chem_data=chem_data)

In [6]:

bio.set_bin_conc(bin_address=4, chem_label="A", conc=10.)
bio.set_bin_conc(bin_address=4, chem_label="B", conc=10.)

bio.describe_state()

SYSTEM STATE at Time t = 0:
9 bins and 2 chemical species

Out[6]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	0.0	0.0	0.0	0.0	10.0	0.0	0.0	0.0	0.0
1	B	2.0	0.0	0.0	0.0	0.0	10.0	0.0	0.0	0.0	0.0

In [7]:

bio.system_snapshot()

Out[7]:

	A	B
0	0.0	0.0
1	0.0	0.0
2	0.0	0.0
3	0.0	0.0
4	10.0	10.0
5	0.0	0.0
6	0.0	0.0
7	0.0	0.0
8	0.0	0.0

In [8]:

bio.visualize_system(title_prefix="Diffusion.  The 2 concentrations initially perfectly overlap")   # Line curve view

In [9]:

bio.system_heatmaps(title_prefix="Diffusion")

In [ ]:

Request history-keeping for some bins¶

In [10]:

# Request to save the concentration history at the bin with the initial concentration injection, 
# and at a couple of other bins
bio.enable_history(bins=[0, 2, 4], frequency=3, take_snapshot=True)    

History enabled for bins [0, 2, 4] and chemicals None (None means 'all')

In [ ]:

Initial Diffusion Step¶

In [11]:

# Advancing to time t=1, with time steps of 0.05
bio.diffuse(total_duration=1.0, time_step=0.05)

Out[11]:

{'steps': 20, 'system time': '1'}

In [12]:

bio.describe_state()

SYSTEM STATE at Time t = 1:
9 bins and 2 chemical species

Out[12]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	0.000026	0.001203	0.039743	0.829776	8.258503	0.829776	0.039743	0.001203	0.000026
1	B	2.0	0.355341	0.656747	1.201979	1.773354	2.025158	1.773354	1.201979	0.656747	0.355341

The 2 chemicals `A` and `B`, previously identical in concentrations, have separated due to their different diffusion rates¶

Notice how B has spread out far more than A

In [13]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [14]:

bio.system_heatmaps(title_prefix="Diffusion")

In [ ]:

Let's advance the diffusion¶

In [15]:

bio.diffuse(total_duration=2.0, time_step=0.05)

Out[15]:

{'steps': 40, 'system time': '3'}

In [16]:

bio.describe_state()

SYSTEM STATE at Time t = 3:
9 bins and 2 chemical species

Out[16]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	0.001863	0.024620	0.254045	1.731402	5.976139	1.731402	0.254045	0.024620	0.001863
1	B	2.0	0.993368	1.048422	1.132834	1.207131	1.236490	1.207131	1.132834	1.048422	0.993368

`B` is close to equilibrium, while `A` is nowhere near it!¶

In [17]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [18]:

bio.system_heatmaps(title_prefix="Diffusion")

In [ ]:

Let's advance the diffusion some more¶

In [19]:

bio.diffuse(total_duration=2.0, time_step=0.05)

Out[19]:

{'steps': 40, 'system time': '5'}

In [20]:

bio.describe_state()

SYSTEM STATE at Time t = 5:
9 bins and 2 chemical species

Out[20]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	0.010722	0.080956	0.500624	2.086625	4.642148	2.086625	0.500624	0.080956	0.010722
1	B	2.0	1.093790	1.101895	1.114312	1.125232	1.129544	1.125232	1.114312	1.101895	1.093790

Notice that the concentrations of `A` and `B` in the central bin 4 are again getting closer in value¶

In [21]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [22]:

bio.system_heatmaps(title_prefix="Diffusion")

In [ ]:

Finally, let's advance the diffusion to equilibrium¶

In [23]:

bio.diffuse(total_duration=95.0, time_step=0.05)

Out[23]:

{'steps': 1900, 'system time': '100'}

In [24]:

bio.describe_state()

SYSTEM STATE at Time t = 100:
9 bins and 2 chemical species

Out[24]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	1.091824	1.100848	1.114675	1.126835	1.131637	1.126835	1.114675	1.100848	1.091824
1	B	2.0	1.111111	1.111111	1.111111	1.111111	1.111111	1.111111	1.111111	1.111111	1.111111

All bins now have essentially uniform concentration¶

Notice that the concentrations of `A` and `B` in the central bin 4 are again essentially identical; the separation of `A` and `B` is coming to an end¶

In [25]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [26]:

bio.system_heatmaps(title_prefix="Diffusion")

Verify that the total amount of each chemicals hasn't changed¶

In [27]:

# Let's verify mass conservation
bio.check_mass_conservation(chem_label="A", expected=10.)

Out[27]:

True

In [28]:

bio.check_mass_conservation(chem_label="B", expected=10.)

Out[28]:

True

In [ ]:

Visualization of time changes at particular bins¶

Instead of visualizing the entire system at a moment in time, like in the previous diagrams, let's now look at the time evolution of the concentrations of `A` and `B` at selected bins (bins whose history-keeping we requested prior to running the simulation)¶

In [29]:

bin4_hist = bio.conc_history.bin_history(bin_address=4)   # The bin where the initial concentration of `A` and `B` was applied
bin4_hist

Out[29]:

	SYSTEM TIME	A	B
0	0.00	10.000000	10.000000
1	0.15	9.704475	5.600000
2	0.30	9.422007	3.908040
3	0.45	9.151954	3.108505
4	0.60	8.893710	2.652694
...	...	...	...
661	99.35	1.132271	1.111111
662	99.50	1.132123	1.111111
663	99.65	1.131976	1.111111
664	99.80	1.131830	1.111111
665	99.95	1.131685	1.111111

666 rows × 3 columns

In [30]:

bio.plot_history_single_bin(bin_address=4, title_prefix="`A` and `B` separate, until they finally rejoin")

The transient separation of `A` and `B` is best seen by plotting the difference of their concentrations, as a function of time¶

In [31]:

bin4_hist["A"] - bin4_hist["B"]

Out[31]:

0      0.000000
1      4.104475
2      5.513967
3      6.043449
4      6.241017
         ...   
661    0.021160
662    0.021012
663    0.020865
664    0.020719
665    0.020574
Length: 666, dtype: float64

In [32]:

PlotlyHelper.plot_curves(x=bin4_hist['SYSTEM TIME'], y = bin4_hist['A'] - bin4_hist['B'], 
                         x_label="SYSTEM TIME", y_label="[A] - [B]", 
                         colors="purple", title="Transient separation of `A` and `B`, as seen in central bin 4")

In [ ]:

The transient separation of `A` and `B` is also seen in other bins...¶

In [33]:

bio.plot_history_single_bin(bin_address=0, title_prefix="The transient separation of `A` and `B` happens at other bins, too...")

In [34]:

bio.plot_history_single_bin(bin_address=2, title_prefix="The transient separation of `A` and `B` happens at other bins, too...")

In [ ]:

Transient separation of 2 chemicals in 1D, due to their different diffusion rates¶

Starting with identical concentrations in one of the bins, they initially separate...¶

until they eventually attain identical concentrations at equilibrium¶

TAGS : "diffusion 1D", "basic"¶

Prepare the initial system¶

Request history-keeping for some bins¶

Initial Diffusion Step¶

The 2 chemicals A and B, previously identical in concentrations, have separated due to their different diffusion rates¶

Let's advance the diffusion¶

B is close to equilibrium, while A is nowhere near it!¶

Let's advance the diffusion some more¶

Notice that the concentrations of A and B in the central bin 4 are again getting closer in value¶

Finally, let's advance the diffusion to equilibrium¶

All bins now have essentially uniform concentration¶

Notice that the concentrations of A and B in the central bin 4 are again essentially identical; the separation of A and B is coming to an end¶

Verify that the total amount of each chemicals hasn't changed¶

Visualization of time changes at particular bins¶

Instead of visualizing the entire system at a moment in time, like in the previous diagrams, let's now look at the time evolution of the concentrations of A and B at selected bins (bins whose history-keeping we requested prior to running the simulation)¶

The transient separation of A and B is best seen by plotting the difference of their concentrations, as a function of time¶

The transient separation of A and B is also seen in other bins...¶

The 2 chemicals `A` and `B`, previously identical in concentrations, have separated due to their different diffusion rates¶

`B` is close to equilibrium, while `A` is nowhere near it!¶

Notice that the concentrations of `A` and `B` in the central bin 4 are again getting closer in value¶

Notice that the concentrations of `A` and `B` in the central bin 4 are again essentially identical; the separation of `A` and `B` is coming to an end¶

Instead of visualizing the entire system at a moment in time, like in the previous diagrams, let's now look at the time evolution of the concentrations of `A` and `B` at selected bins (bins whose history-keeping we requested prior to running the simulation)¶

The transient separation of `A` and `B` is best seen by plotting the difference of their concentrations, as a function of time¶

The transient separation of `A` and `B` is also seen in other bins...¶